Wind Energ. Sci.

, 7, 1171–1181, 2022
https://doi.org/10.5194/wes-7-1171-2022
© Author(s) 2022. This work is distributed under
the Creative Commons Attribution 4.0 License.

Damage equivalent load synthesis and stochastic

extrapolation for fatigue life validation
Anand Natarajan
Department of Wind Energy, Technical University of Denmark, Frederiksborgvej 399, Roskilde 4000, Denmark
Correspondence: Anand Natarajan (anand_nk@hotmail.com)

Received: 1 November 2021 – Discussion started: 8 November 2021

Revised: 10 April 2022 – Accepted: 3 May 2022 – Published: 2 June 2022

Abstract. Present verification of the fatigue life margins on wind turbine structures utilizes damage equivalent
load (DEL) computations over a limited time duration. In this article, a procedure to determine long-term fa-
tigue damage and remaining life is presented as a combination of stochastic extrapolation of the 10 min DEL
to determine its probability of exceedance and computationally fast synthesis of DELs using level crossings
of a Gaussian process. Both the synthesis of DELs and long-term stochastic extrapolation are validated using
measured loads from a wind farm. The extrapolation for the blade root flap and tower base fore–aft damage
equivalent moment is presented using a three-parameter Weibull distribution, whereby the long-term damage
equivalent load levels are forecast for both simulated and measured values. The damage equivalent load magni-
tude at a selected target probability of exceedance provides an indicator of the integrity of the structure for the
next year. The extrapolated damage equivalent load over a year is validated using measured multi-year damage
equivalent loads from a turbine in the Lillgrund wind farm, which is subject to wakes. The simulation of damage
equivalent loads using the method of level crossings of a Gaussian process is shown to be able to reconstruct the
damage equivalent load for both blade root and tower base moments. The prediction of the tower base fore–aft
DEL is demonstrated to be feasible when using the Vanmarcke correction for very narrow band processes. The
combined method of fast damage equivalent load computations and stochastic extrapolation to the next year
allows a quick and accurate forecasting of structural integrity of operational wind turbines.

1 Introduction ferent amplitudes. Damage equivalent load provides a mea-

surable quantity from an operating turbine that can be com-
The fatigue damage on wind turbine structures is strongly pared with results from simulations made in the design phase,
influenced by the stochastic inflow to the turbine, which is whereas Miner’s damage is an abstract quantity of the ratio
mainly composed of wind turbulence and spatial variations of the number of cycles that is difficult to quantify. Therefore
such as shear and veer (Dimitrov et al., 2017). For offshore verification of fatigue life on wind farms is usually made us-
substructures, irregular ocean waves also result in dynamic ing DELs and not damage.
loads leading to fatigue damage. The Palmgren–Miner rule The conventional wind turbine fatigue design process con-
(Miner, 1945) is the standard approach followed in the de- siders a set of aeroelastic load simulation results under nor-
sign of wind turbines by which it is ensured that the linear mal operation and transients and standstill under normal tur-
damage sum over an intended lifetime is lower than unity bulence conditions, whereby the load cycles determined over
after considering required safety margins. Since the Miner a short period of time at each mean wind speed are assumed
damage variable is not a physically measurable quantity, a to repeat continually over the turbine’s full expected life-
substitute for damage is used, which is the damage equiva- time. However, it is seldom accurate to consider that the
lent load (DEL). This is the load level at a particular number load cycle amplitudes and cycle counts seen in simulations
of cycles which results in the same damage as the original of a few hours (depending on the number of turbulence
summation of a multitude of different load cycles with dif-

Published by Copernicus Publications on behalf of the European Academy of Wind Energy e.V.
1172 A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation

seeds) can be taken to repeat over 25 years, a typical wind The fatigue damage as expressed in terms of the DEL is
farm lifetime. The International Electrotechnical Commis- strongly dependent on the wakes within wind farms (Gallinos
sion (IEC) 61400-1 standard (IEC, 2019) informally recom- et al., 2016) due to strong correlation of several load compo-
mends a stochastic extrapolation process to determine the nents with the wind turbulence in the wake. This implies that
amplitude and cycle count of the largest amplitude loads as varying atmospheric conditions such as the wind direction
part of the fatigue design process. Moriarty et al. (2004) used and stability can influence the wake turbulence and thereby
extrapolation of load amplitudes to determine the long-term also change the DELs on the turbine of interest. Since the
fatigue damage equivalent load, in a manner similar to ex- turbulence in wakes is a function of a multitude of variables
treme load extrapolation. Since the load amplitude is extrap- such as the turbine position, the wind direction, upstream tur-
olated, the probability of the load amplitude and number of bine thrust, and upstream turbine yaw, it is not readily feasi-
cycles is conditional on the joint distribution of turbulence ble to quantify the cumulative distribution function (CDF)
and mean wind speed. However the DEL is insensitive to iso- of the wake turbulence. Herein, a novel approach is put for-
lated changes in load amplitudes for fixed wind turbulence, ward to determine the return period of the DEL magnitudes
and therefore extrapolation using the load amplitudes may conditional on the mean wind speed, wherein the variation
be conservative as such a process also takes into account iso- in the DEL is considered to directly correlate to the variation
lated amplitude extremes within 10 min. Load measurements in the wake turbulence. It is also common that many wind
from wind farms show a wide variation in damage equiva- farms possess a wind turbine that is instrumented with load
lent loads, much more than seen in the conventional design sensors and from which only the 10 min load statistics are
process due to varying inflow turbulence. The question then archived. This usually implies the 10 min mean, standard de-
needs to be addressed as to whether larger damage equiva- viation, maximum, and minimum are available. The DEL is
lent loads seen in measurements as compared to simulated often not stored, as its computation requires rainflow count-
values encountered in the design process using limited load ing or similar procedures to be available on the turbine com-
simulations are indicative of reduced structural reliability and puter, which is often not the case. In such situations, it is
decreased lifetime. essential that the DEL can be computed from the measured
To better quantify fatigue life over a long time, it is needed standard deviations of the loads. This is not a prevalent prac-
to perform stochastic extrapolation of the short-term DELs tice but is straightforward to map the standard deviation of
to determine the probability of exceeding the DEL magni- loads to the DEL if the underlying stochastic process is as-
tude over the long term and thereby determine the life of sumed to be Gaussian or Poisson. In the following sections,
the structure. In the present work, the DEL itself is extrap- the 1-year DEL at the blade root and tower base of wind tur-
olated as a stochastic variable and taken to be fully corre- bines in a farm is predicted both using stochastic extrapola-
lated to the wind turbulence. The DEL being an aggregated tion of the measured DEL and through synthesis of the DELs
quantity is not affected by isolated changes in load ampli- using the measured load standard deviations over limited in-
tudes over 10 min, but the change in DEL that is modeled tervals. The method of synthesis of DELs using 10 min statis-
is due to change in turbulence at a given mean wind speed. tics also allows the computation of DELs without the need
Therefore, the probability distribution of DEL is conditional for detailed turbine information to be present, such as is re-
only on the mean wind speed, as is the case with wind turbu- quired for an aeroelastic model. This allows the wind farm
lence. Extrapolation of extreme loads (Natarajan and Verelst, owner to simulate DELs quickly without access to detailed
2012) is mandated by IEC 61400-1 to determine the 50-year turbine information that is possessed only by the wind tur-
ultimate design load level, but there is no mandatory require- bine manufacturer.
ment presently to extrapolate fatigue DEL. The DEL is an Some authors refer to extrapolation to imply the prediction
aggregated quantity over a period of 10 min or more, the of DELs for a future time interval based on measured DELs
value of which is relatively stable and may not change sig- and wind conditions in the past (Hübler et al., 2018b, a).
nificantly for isolated load excursions resulting from the ran- In the present work, extrapolation of DEL is defined as a
domness of a stationary process. However, the real condi- stochastic methodology to determine the return period of
tions on the wind farm have varying wind turbulence inten- increasing DEL magnitudes outside the domain of present
sities for a given mean wind speed, thus resulting in varying results. Continuous monitoring and assessment of the tur-
damage equivalent loads. The resulting DEL values from dif- bine structural life is crucial since the costs of unplanned
ferent 10 min simulations over different wind turbulence at a downtime and repairs outweigh the cost of monitoring; also,
given mean wind speed can be extrapolated so that the proba- early correction of wind farm operation ensures safety of
bility of exceeding a target damage equivalent load level over structures for their intended lifetime and for life extension.
a long-term period can be determined. This is more accurate Wind farm operational correction can be carried out by der-
and realistic than the conventional process used today of as- ating upstream wind turbines to reduce the wake turbulence
suming that the same load cycles over a limited set of load generated by those turbines and thus lower loads on the
simulations are prevalent for the entire life of the turbine. downstream wind turbines (Dimitrov and Natarajan, 2021;
Munters and Meyers, 2018). In the next sections, the abil-

Wind Energ. Sci., 7, 1171–1181, 2022 https://doi.org/10.5194/wes-7-1171-2022

A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation 1173

ity to use measured or simulated 10 min load statistics to di- Here gi is the number of standard deviations, σL , away from
rectly quantify DELs and the use of the stochastic extrapola- the mean, µL . The number of load cycles with amplitude
tion methodology to forecast the return period of DELs are greater than Li (upcrossings) is given by (Meirovitch, 2001)
explained, which leads to a criterion to determine if the mea-
L2
sured DELs on a wind turbine are within design limits. − i
2
2σL
Nci = νe , (4)
2 Methodology
where ν is the mean crossing frequency, which here will be
assumed to be the first rotational frequency (P ) of the rotor
Given load cycles over 10 min intervals as obtained through
for the blade flap loads and the first natural frequency for the
rainflow counting of the output time series of aeroelastic sim-
tower base in the loading direction of interest. Based on the
ulations, the annual 10 min damage equivalent load at a given
Rayleigh distribution decay rate, a load amplitude bin can be
mean wind speed bin is conventionally computed as
determined which provides one load cycle in that amplitude
Pn
! m1 bin; that is the number of upcrossings of level Li minus up-
i=1 ni Lm
i crossings of Li + 1Li is unity. Equations (3) and (4) assume
Leq|v = 6Nv , (1)
Neq a Gaussian process with a single mean crossing frequency
of interest. For broadband Gaussian processes or Poisson
where Nv is the number of hours in a year at the mean wind processes, the methods of Cramer–Leadbetter (Cramer and
speed; Neq is the equivalent number of cycles, v; ni is the Leadbetter, 1967) can be used, which have also been proven
number of load cycles of amplitude, Li ; and m is an expo- for extreme value analysis (Madsen et al., 1986). However
nent of the SN curve of the material. The load time series for highly damped structures, the energy in the stochastic re-
used to compute ni and Li in Eq. (1) are conventionally re- sponse is representative of a very narrow band process, and
sults from rainflow counting of limited aeroelastic simula- in such cases, Eq. (4) can overly magnify the rate of level
tions performed under conditions important for fatigue dam- crossings. Vanmarcke (1975) prescribed a correction factor
age. The duration of the load time series over all mean wind to accurately determine the level crossings of very narrow
speeds and operational conditions simulated is seldom more band processes, which states that the rate of crossings of such
than a few days, and the same load cycles are assumed equiv- a process is
alently prevalent for the lifetime of the wind turbine. In prac-
tice, a load measurement campaign will provide a wide scat- L2
i
− 2
ter in DELs (Barber et al., 2016), so it may be difficult to Nvi = νv e 2σL
, (5)
ascertain, based on the limited simulations, the design DEL
value that is to be considered for assuring that the magni- where
tude of the measured DELs are within acceptable limits to  √ 
− 2π(1−α 2 )0.6 Li
ensure structural integrity. Instead of assuming the same lim- σL
1 − e
ited number of load cycles repeated for the lifetime of the νv = ν  . (6)

L2
turbine in Eq. (1), it is more realistic to determine the prob- − i2
2σL
ability of exceedance of DELs over a period and thereby de- 1−e
termine the number of cycles for each DEL magnitude. α is the Vanmarcke bandwidth parameter, which for a very
While frequency domain DEL computation methods such narrow band process will approach unity. The Vanmarcke
as Dirlik’s method (Ragan and Manuel, 2007) have been used correction regulates the peak crossings of different load lev-
by some, these methods require knowledge of the load spec- els by conditioning them on the bandwidth parameter α. Thus
trum and may not always be the best choice for design val- this correction can be directly applied for the computation of
idation due to erroneous spectra. On the other hand, level- DELs of processes with different bandwidths.
crossing methods need not always require availability of the For a Gaussian process with a known standard deviation,
process spectrum. If the 10 min statistics of the standard de- Eq. (1) can also be written as
viation, maxima, and minima of loads are available, then as-
suming the load amplitude can be expressed as a Gaussian Pn
! m1
i=1 ni gim
process, the probability of crossing an amplitude level, Li , Leq|v = 6Nv σLm . (7)
follows a Rayleigh distribution (Meirovitch, 2001); that is Neq

L2 We assume gi is a standardized random normal variable be-

Li − i2
pdf(Li ) = 2 e 2σL , (2) tween [min, max] recorded loads that provides a unit cy-
σL cle for each amplitude level. Equation (7) in combination
where with Eqs. (4)–(6) allows very fast computation of the dam-
age equivalent load over a time interval of relevance, with-
Li = gi σL . (3) out the need for aeroelastic load simulations. The only inputs

https://doi.org/10.5194/wes-7-1171-2022 Wind Energ. Sci., 7, 1171–1181, 2022

1174 A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation

needed are the 10 min mean, standard deviation, minimum

and maximum load levels, and the dynamic frequency of in-
terest. This also implies that the DEL values over several in-
put wind conditions (wind turbulence, mean wind speed) can
be determined over a long time, without limiting the DEL to
only a few 10 min computations per mean wind speed as is
the practice today.

2.1 Extrapolation of fatigue

The DEL values show a wide scatter based on the varying

inflow turbulence due to wake conditions and other contribu-
tions at any given mean wind speed. While knowledge of the
10 min load statistics will enable the ready reconstruction of
the DEL as per the previous section, it is also essential to de-
termine the probability of exceedance of a DEL magnitude, Figure 1. Blade root flap damage equivalent moment extrapolation
that is, to determine the magnitude of the DEL that can pos- at 10 m s−1 for low wind turbulence.
sess a 1-year return period or a 10-year return period. The
DEL is a stochastic variable that is dependent on wind tur-
bulence, wind direction, etc., and herein it is assumed that practice (IEC, 2019) of simulating loads for a minimal dura-
the DEL follows a Weibull distribution with three param- tion and assuming the same load cycles are applicable for the
eters. Hoole et al. (2019) showed that the three-parameter full lifetime.
Weibull distribution was more accurate in modeling fatigue The results from stochastic extrapolation of simulated
life as compared to two-parameter distributions. Extrapola- blade root flap damage equivalent moment, obtained from
tion of the tail of the Weibull distribution will provide the limited aeroelastic simulations, are shown in Fig. 1. The ex-
long-term DEL values and the corresponding return period trapolation in Fig. 1 uses Eqs. (8) and (9) at a mean wind
for those values. The cumulative distribution function (CDF) speed of 10 m s−1 using simulated blade root moments under
of the DEL magnitudes over different 10 min intervals may normal turbulence inflow. Figure 1 shows that the extrapola-
be determined using their median rank (Hoole et al., 2019), tion of the probability of exceedance monotonically reduces
from which a three-parameter Weibull distribution can be fit, up to very low values of 10−15 , and this methodology can
whereby the probability of a damage equivalent load magni- therefore be used to extrapolate DELs across different mean
tude, d, is given as wind speeds and wind turbulence levels to their asymptotic
target probability of exceedance.
d−γ

β
F (d) = 1 − e− α . (8)
2.2 Measurement data
The three-parameter Weibull distribution is fit to the median
Loads and supervisory control and data acquisition
rank, which can be computed using DEL values from aeroe-
(SCADA) measurements from a 2.3 MW wind turbine within
lastic simulations, from field measurements, or as obtained
the Lillgrund offshore wind farm (Dimitrov and Natarajan,
with the Gaussian process approach described in the previous
2021) are used in the validation of the methodology de-
subsection. The median rank is fit to measured or simulated
scribed in the above subsections. The blade root and tower
DEL using the empirical relationship for Weibull distribu-
base loads on a fully instrumented turbine (C-08) that is the
tions:
middle turbine in the bottom row shown in Fig. 2 are used
i in this process. Depending on the wind direction, varying
mR(Li ) = , (9) inflow turbulence is experienced by this turbine due to the
N +1
wake effects, equivalent to added turbulence. This results in
where i is the sorted rank of the load Li and N is the number a wide scatter in damage equivalent moments at any given
of DEL values used in the fitting process. mean wind speed, an example of which is shown in Fig. 3.
Extrapolation of the fitted CDF is made to a minimum of Simulated DEL using the conventional tools depicted in
a 1-year return period, whereby the DEL magnitude with a Fig. 4 does not provide such a variation as seen in Fig. 3 due
1-year return period is identified. This provides a readily im- to limited 10 min samples, and therefore the stochastic mod-
plementable technique to compute the DEL for a long dura- els in Eqs. (4)–(6) are needed to replicate the variation in
tion of at least 1 year without requiring simulation of the full DEL seen in measurements. Extrapolation to a 1-year prob-
period. It also provides a robust DEL value over a long time, ability of exceedance is made using both a small subset of
rather than applying the conventional methodology used in the measured DELs and a stochastic simulation set. The 1-

Wind Energ. Sci., 7, 1171–1181, 2022 https://doi.org/10.5194/wes-7-1171-2022

A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation 1175

variety of wind and wake conditions over a year with the re-
spective measured DELs.
Further time-domain aeroelastic simulations using the
HAWC2 software (Larsen and Hansen, 2012) are also made
to compute the blade root and tower base damage equivalent
moments of the 2.3 MW wind turbine. The DELs obtained
from all three methods, that is, Gaussian process analysis,
aeroelastic simulations, and field measurements, are extrap-
olated to determine the 1-year DEL value. Based on this ex-
trapolation, a criterion is established in the following sections
that allows lifetime assessment of blades and towers.

4 Results

4.1 Extrapolation with measured loads and

comparisons with aeroelastic simulation results
Figure 2. Layout of the Lillgrund wind farm.
Figure 3 describes the measured blade root flap damage
equivalent moments and tower base fore–aft damage equiva-
lent moments over a 1-year period normalized by a character-
year extrapolated DEL values for both the tower base and the
istic value and as a function of mean wind speed for turbine
blade root are validated with the measured DELs over a year.
C-08. The large spread in the measured DELs is due to the
varying turbulence from the wake as a function of wind direc-
3 Fatigue from Gaussian process analysis tion. The turbine loads are also simulated using the properties
of the 2.3 MW wind turbine in the HAWC2 aeroelastic tool
The blade root flap moment and the tower base fore–aft mo- with the Technical University of Denmark (DTU) controller.
ment are the main focus points, as these two moments are Figure 4 depicts the normalized blade root flap and tower
strongly driven by wind turbulence and wake effects. The life base fore–aft DELs using the results from a limited number
consumption of the blade and support structure within a wind of aeroelastic load simulations over all IEC 61400-1 turbu-
farm are highly dependent on these load components. For the lence classes and with 12 random seeds of wind turbulence
blade root flap moment, the primary frequency of interest for at each mean wind speed. Figure 4 shows a typical number of
fatigue damage is the rotor rotational speed, or the “P ” fre- simulation-based results as made in the final turbine design
quency. While the blade flap moments also contain multiples and type certification.
of P , such as 2P or 3P , these are of much lower energy con- The scatter observed in Fig. 4 is due to the different IEC
tent and the primary energy content is the P frequency. For class turbulence levels used in the simulation along with the
the support structure, the primary excitation frequency is usu- randomness of the turbulence seed, while the scatter in DELs
ally the first natural frequency of the structure. It is assumed in Fig. 3 is due to the significant difference in turbulence due
that the turbine designs have been made as to not result in to different wake situations at different mean wind speeds
resonant excitation. While the tower fore–aft moment spec- and in different wind directions. Therefore there is a signif-
tra have a peak at their natural frequency, it is assumed that icantly greater variation in DELs in the measured signals in
this is not caused by excitation from rotor harmonics, such as Fig. 3 versus the simulated DELs in Fig. 4 for most of the
3P . wind speeds. In Fig. 3, there are far fewer measurements
Under the above conditions of turbine operation, the blade obtained at the higher mean wind speeds above 14 m s−1
root flap damage equivalent moment is narrow-band Gaus- as compared to lower mean wind speeds. Further at higher
sian and the tower base fore–aft damage equivalent mo- mean wind speeds, the wake effects in the measurements are
ment is equivalent to a very narrow band Gaussian process. reduced due to the lower thrust on the upstream wind tur-
The reason for the very narrow band assumption for the bine, and therefore the variation in turbulence seen by the
tower base fore–aft moment is due to the strong aerodynamic measurement turbine is smaller at higher mean wind speeds.
damping of the support structure during turbine operation, as However the load simulations in Fig. 4 included wind inflow
also later explained through Fig. 9b. This implies that Eq. (4) with the highest IEC turbulence class “A” and therefore is
can be used to compute the DEL cycles for the blade root flap supposed to represent a safe upper load level. The question
moment and Eqs. (5) and (6) may be used to compute DEL arises as to whether the measured DEL magnitudes on the ac-
cycles for the tower fore–aft moment. Finally Eq. (7) is used tual turbine imply that the structures on the turbine are close
to compute the DELs. To validate the above assumptions, the to their intended fatigue lifetime or that their reliability is
computed DELs are compared for each 10 min period for a sufficiently intact to enable continued operation for further

https://doi.org/10.5194/wes-7-1171-2022 Wind Energ. Sci., 7, 1171–1181, 2022

1176 A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation

Figure 3. Measured 10 min (a) blade root flap and (b) tower base fore–aft damage equivalent moments, on the C-08 turbine. The coefficient
of variation (CoV) of blade root flap DELs below rated wind speed is 0.49 and above rated wind speed is 0.17. The CoV of tower base
fore–aft (FA) DELs below rated wind speed is 0.47 and above rated wind speed is 0.29.

Figure 4. Simulated 10 min (a) blade root flap damage equivalent moment and (b) tower base fore–aft damage equivalent moment, over
different mean wind speeds and turbulence. The CoV of blade root flap DELs below rated wind speed is 0.31 and above rated wind speed is
0.22. The CoV of tower base FA DELs below rated wind speed is 0.31 and above rated wind speed is 0.23.

years. This question can be answered only if the probability be seen that the tail of the extrapolated fitted distribution
of exceedance of the DEL magnitudes is assessed. The wind corresponding to the 1-year exceedance probability matches
turbine structure is designed to meet a target annual probabil- the median rank of the measured DELs very well. This pro-
ity of failure in fatigue, and since the DEL is representative cess can therefore be replicated at all mean wind speeds
of the damage suffered, the probabilities of exceedance of the and over all turbulence levels to determine DEL magnitudes
DEL magnitudes over a 1-year return period are compared. with multi-year return periods. The resulting DEL magnitude
The stochastic extrapolation of the DELs using the three- probability can be weighted with the probability of mean
parameter Weibull distribution is validated using the mea- wind speed to determine the DEL over all mean wind speeds.
sured blade root flap DEL. The measured turbulence varia- The method proposed is applicable for any number of
tion at each mean wind speed is divided into 50 bins, and DELs from different load components that are strongly de-
one 10 min DEL is taken from each bin to compute the me- pendent on wind turbulence. While the influence of the tail
dian rank. The three-parameter Weibull distribution is then region is greater for larger material exponents such as for
fit to the resulting median rank subset. This stochastic fit is blades, enabling a longer-term DEL prediction, it is not in-
then extrapolated to a 1-year probability of exceedance and significant for steel towers. Therefore the methodology pre-
compared with the global median rank of the 1-year mea- sented herein seeks to use limited load measurements and
sured DELs to validate the approach. The results are shown load simulation results to enable a comparison of the cor-
in Fig. 5 for two different mean wind speeds, wherein it can responding estimated 1-year DELs between the field instal-

Wind Energ. Sci., 7, 1171–1181, 2022 https://doi.org/10.5194/wes-7-1171-2022

A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation 1177

Figure 5. Validation of the stochastic extrapolation method with the measured 1-year median rank of the DEL.

Figure 6. Comparison of the 10 min (a) extrapolated blade root flap bending damage equivalent moments and (b) the extrapolated tower
base fore–aft damage equivalent moments, using measurements. For the blade root, extrapolated curves begin at the far left (5 m s−1 – lowest
DEL), move to the right with increasing mean wind speed until 11 m s−1 (highest DEL), and then move left again with increasing mean
wind speed. For the tower base, extrapolated curves begin at the middle at 5 m s−1 and move to the right (highest DEL) until 8 m s−1 , before
moving to the left again.

lation and design. The long-term DEL magnitude should be The same extrapolation can also be performed using only
bounded with increase in time, and the tail of the DEL should the simulated DEL values for the same two load sensors, and
be accurately represented, which is what is shown to be the the results are shown in Fig. 7. The simulated DELs cover
case in Fig. 5. As can be seen, the rate of increase in DEL all IEC turbulence classes, and these are representative of the
reduces significantly with reduction in the probability of ex- turbulence levels experienced by the actual turbine. However
ceedance and asymptotically approaches the empirical dis- the simulated results use the 90 % quantile of turbulence,
tribution from the measured DELs. This implies that the de- whereas the measured turbulence covers a range of quan-
sired annual or long-term probability of failure can be ac- tiles. Consequently, it can be compared if the extrapolation
tively measured. using the simulated DELs in Fig. 7 has a higher 1-year DEL
Based on the validations shown in Fig. 5, Fig. 6 displays magnitude at the 1-year probability of exceedance than the
the long-term extrapolated values for the blade root flap dam- measured 1-year DEL magnitudes over different mean wind
age equivalent moment and tower base damage equivalent speeds. The weighted probability of the DEL with the an-
moment to very low probabilities of exceedance, from which nual probability of mean wind speeds can be quantified to
the DEL magnitude corresponding to the desired exceedance enable a definite conclusion on a target annual DEL magni-
probability can be determined. It can be seen that extrapo- tude, above which the turbine structure can be said to possess
lation using the three-parameter Weibull distribution repro- a diminished annual reliability level.
duces the same trend as the empirical distribution using the It should be noted that while the process of verification
measurements while allowing for predicting the DEL mag- of the structural integrity in fatigue is presented, the quan-
nitudes asymptotically to lower probabilities of exceedance tification of the structural reliability or remaining life of the
for the blade root and tower base over various mean wind specific operational turbine is not made herein since the ac-
speeds. tual design loads of the specific turbine used in its design are
not available.

https://doi.org/10.5194/wes-7-1171-2022 Wind Energ. Sci., 7, 1171–1181, 2022

1178 A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation

Figure 7. Comparison of the 10 min (a) extrapolated blade root flap bending damage equivalent moments and the (b) extrapolated tower
base fore–aft damage equivalent moment, using simulations.

Figure 8. Comparison of the 10 min blade root flap bending damage equivalent moments between the Gaussian process method and mea-
surement data on the C-08 turbine.

4.2 Extrapolation with Gaussian process analysis (GPA) Figure 8 compares the resulting blade root flap damage
equivalent moment from the GPA with the measured DELs
Since aeroelastic simulation is time-consuming and therefore
at two different mean wind speeds. The results in Fig. 8 show
provides limited DEL results, the methods for narrow-band
that for all the different wind turbulence variations with var-
and very narrow band processes as explained in previous sec-
ious wake angles, the simple Gaussian process analysis pro-
tions are used to directly simulate 1 year of DELs for the
vides a similar quantification of the true DEL to that seen
C-08 wind turbine blade root and tower base. The 10 min
in measurements. For these blade root flap DELs, Eq. (4)
measured load statistics are used to determine the DEL val-
is used directly without the Vanmarcke correction to obtain
ues. Using load levels within the measured minimum and
the number of crossings of different levels. However if the
maximum load level and using the measured 10 min stan-
same method is used (i.e., without the Vanmarcke correc-
dard deviation as σL , a 1-year DEL simulation is made using
tion) for the tower base fore–aft damage equivalent moment,
Eqs. (4)–(7), which requires only a few seconds on a stan-
then as seen in Fig. 9a, the tower base fore–aft DELs are
dard laptop computer. Many load measurement statistics of-
greatly amplified in comparison to the measured DELs. The
ten do not possess DEL magnitudes, and under such condi-
power spectrum of the tower base fore–aft (FA) moment is
tions, the DEL magnitudes can be re-created using Eqs. (4)–
compared with the blade root flap in Fig. 9b, from which it
(7). If measured 10 min load statistics are unavailable, then
can be seen that the first peak for the tower moment has a
the aeroelastic simulations can be used to determine a range
much smaller energy rise relative to its starting point than
of mean and standard deviations relevant for the 10 min load
the corresponding first peak in the blade moment, which has
magnitudes as a function of mean wind speed and wind tur-
an energy jump of about a 100 on the power spectral density
bulence.
(PSD) scale. This is due to significant aerodynamic damping
that is not considered in Eq. (4). Therefore the bandwidth of

Wind Energ. Sci., 7, 1171–1181, 2022 https://doi.org/10.5194/wes-7-1171-2022

A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation 1179

Figure 9. (a) Comparison of the 10 min tower base fore–aft damage equivalent moment with measurements at 7 m s−1 using the Gaussian
process method without Vanmarcke correction. (b) Comparison of the power spectrum of the tower base moment showing the small increase
in energy at its first peak with the corresponding much larger peak in the blade root flap moment.

Figure 10. Comparison of the 10 min tower base fore–aft bending damage equivalent moments between the Gaussian process method and
measurement data on the C-08 turbine.

the spectrum needs to be provided in the expression for level on similar turbines. Figure 11 displays the extrapolated 1-
crossings, which is exactly what the Vanmarcke correction year DEL values for the blade root and tower base with the
applies. GPA as compared to measurements and the three-parameter
The Vanmarcke correction in the limit that α → 1 provides Weibull CDF, where it can be seen that the extrapolation us-
the equivalent bandwidth for a highly damped system, and ing GPA shows similar trends to using a parametric Weibull
herein α is taken as 0.99 to model the peak crossings of the fit to data. The empirical DEL using measurements reaches
tower base moment. Figure 10 provides the same compari- a 1-year probability of exceedance up to 10 m s−1 , beyond
son for the tower base fore–aft damage equivalent moment, which the measurements have fewer data. Since the results of
with the Vanmarcke correction, and now a very good match GPA tally well with measured DELs for the blade and tower,
between the DELs computed with this methodology and the this method can also be used to detect anomalous wind tur-
measured DELs is seen. bine operation, such as if DEL values that are significantly
The validation of the simulated DELs with the measure- different from the predictions by GPA are measured. This
ments shown in Figs. 8 and 10 over a year is sufficient justifi- can happen, for example, if there is tower resonance with the
cation for considering that the underlying stochastic process rotor rotational speed or if there is a shutdown of the turbine
is Gaussian. The matching comparisons in Figs. 8 and 10 under high turbulence or other uncommon events.
imply that the results from the GPA can be sampled to also Based on this method, the extrapolated simulated DEL
perform a stochastic extrapolation of the DEL magnitudes magnitudes (also using simulated standard deviations)
to obtain a 1-year DEL value or values of even higher re- should display higher DEL values for the same probability of
turn periods. Considering the computation speed of the Gaus- exceedance as compared to the extrapolated measured DEL
sian process analysis, it is also possible to directly simulate values, in which case the structural integrity of the turbine
multi-year damage equivalent moments if the correspond- structure is not compromised. This allows a direct quantifica-
ing standard deviation, minimum, and maximum values of tion of the life consumption of the turbine structures in a farm
the loads are known, for example from past measurements if the certification design loads of the turbines in question are

https://doi.org/10.5194/wes-7-1171-2022 Wind Energ. Sci., 7, 1171–1181, 2022

1180 A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation

Figure 11. Comparison of the 10 min (a) extrapolated blade root flap damage equivalent moment and (b) tower base fore–aft damage equiv-
alent moment, using the Gaussian process method and measurement data on the C-08 turbine between 7 and 13 m s−1 . Blade: measurement
1-year DEL – between 1.9 and 2.3, Weibull CDF 1-year DEL – between 2 and 2.7, and GPA 1-year DEL – between 1.9 and 3.1. Tower:
measurement 1-year DEL – between 3.3 and 4.4, Weibull CDF 1-year DEL – between 2.9 and 4.4, and GPA 1-year DEL – between 2.8 and
4.4.

available so that the relative difference in the DEL magni- ment wind farm control methods that either reduce loads or
tudes with the actual inflow conditions is obtained. Without increase power production, based on the need.
such an extrapolation, the probability of obtaining DEL mag-
nitudes higher than the design DEL magnitudes is not known,
and therefore the extrapolation of DEL is a necessary proce- Data availability. Measurement data from the wind turbine and
dure. the wind farm used in this work are not publicly available due to a
non-disclosure agreement between the Technical University of Den-
mark and the providers of the data.
5 Conclusions
Competing interests. The author has declared that there are no
Methodologies for computing DELs over multiple years and competing interests.
determining the probability of exceedance of DEL mag-
nitudes were developed and validated using measurements
from the Lillgrund wind farm. The synthesis of DELs using Disclaimer. Publisher’s note: Copernicus Publications remains
available mean and standard deviation of the loads was pre- neutral with regard to jurisdictional claims in published maps and
sented and validated for the blade root flap moment and tower institutional affiliations.
base fore–aft moment. This provides a fast methodology to
simulate the DELs for long durations without loss of accu-
racy. Different approaches for narrow-band processes (blade Acknowledgements. The author is grateful to Gunner Larsen
flap) and very narrow band processes (tower base fore–aft) from DTU Wind Energy for his inputs.
were delineated and shown to also be usable as data sets
for stochastic extrapolation to determine probabilities of ex-
ceedance. A suitable indicator to verify structural integrity Financial support. This research has been supported by the Eu-
of the turbine structure was proposed as the magnitude of the ropean Union’s Horizon 2020 Framework Programme, H2020 Soci-
DEL at the 1-year probability of exceedance, based on past etal Challenges – Secure, Clean and Efficient Energy, TotalControl
measurements and compared to the corresponding DEL mag- Project (grant no. 727680).
nitude in the design basis. The process of structural integrity
verification was shown and quantified through the compari-
Review statement. This paper was edited by Amir R. Nejad and
son of extrapolated DELs from measurements obtained from
reviewed by three anonymous referees.
a single turbine with the corresponding extrapolated DEL
magnitudes using simulation results. The capability to syn-
thesize DELs from 10 min load statistics also allows ease of
storage of multi-year data, without the need for time-series
analysis. The combined methods of synthesis of DELs and
stochastic extrapolation allow forecasting damage into the
future and can be used as a decision-making tool to imple-

Wind Energ. Sci., 7, 1171–1181, 2022 https://doi.org/10.5194/wes-7-1171-2022

A. Natarajan: Damage equivalent load synthesis and stochastic extrapolation for fatigue life validation 1181

References Hübler, C., Weijtjens, W., Rolfes, R., and Devriendt, C.: Reliability
analysis of fatigue damage extrapolations of wind turbines using
Barber, S., Mechler, S., and Nitschke, M.: The effect of wakes on offshore strain measurements, J. Phys. Conf. Ser., 1037, 032035,
the fatigue damage of wind turbine components over their entire https://doi.org/10.1088/1742-6596/1037/3/032035 2018b.
lifetime using short-term load measurements, J. Phys. Conf. Ser., IEC: International Standard IEC61400-1: Wind Turbines – Part 1:
753, 072022, https://doi.org/10.1088/1742-6596/753/7/072022, Design Guidelines, ISBN 978-2-8322-6253-5, 2019.
2016. Larsen, T. J. and Hansen, A. M.: How to HAWC2, the user’s man-
Cramer, H. and Leadbetter, M.: Stationary and Related ual, Tech. Rep. R-1597, Technical University of Denmark, De-
Stochastic Processes, John Wiley & Sons Inc, New York, partment of Wind Energy, ISBN 978-87-550-3583-6, 2012.
ISBN 9780471183907, 1967. Madsen, H. O., Krenk, S., and Lind, N.: Methods of Structural
Dimitrov, N. and Natarajan, A.: Wind farm set point optimiza- Safety, Prentice Hall, Englewood Cliffs, NJ, ISBN 0-13-579475-
tion with surrogate models for load and power output targets, 7, 1986.
J. Phys. Conf. Ser., 2018, 012013, https://doi.org/10.1088/1742- Meirovitch, L.: Fundamentals of Vibrations, McGraw Hill, New
6596/2018/1/012013, 2021. York, ISBN 978-1577666912, 2001.
Dimitrov, N. K., Natarajan, A., and Mann, J.: Effects of Miner, M. A.: Cumulative damage in fatigue, J. Appl. Mech., 12,
normal and extreme turbulence spectral parameters on 159–164, 1945.
wind turbine loads, Renew. Energ., 101, 1180–1193, Moriarty, P. J., Holley, W. E., and Butterfield, S. P.: Extrapolation of
https://doi.org/10.1016/j.renene.2016.10.001, 2017. Extreme and Fatigue Loads Using Probabilistic Methods, Tech.
Gallinos, C., Dimitrov, N., Larsen, T., Natarajan, A., and Hansen, Rep. NREL/TP-500-34421, https://doi.org/10.2172/15011693,
K.: Mapping Wind Farm Loads and Power Production – A 2004.
Case Study on Horns Rev 1, J. Phys. Conf. Ser., 753, 032010, Munters, W. and Meyers, J.: Dynamic Strategies for Yaw and In-
https://doi.org/10.1088/1742-6596/753/3/032010, 2016. duction Control of Wind Farms Based on Large-Eddy Simulation
Hoole, J., Sartor, P., Booker, J., Cooper, J., Gogouvitis, X. V., and and Optimization, Energies, 11, 177–190, 2018.
Schmidt, R. K.: Systematic statistical characterisation of stress- Natarajan, A. and Verelst, D. R.: Outlier robustness for wind turbine
life datasets using 3-Parameter distributions, Int. J. Fatigue, 129, extrapolated extreme loads, Wind Energy, 15, 679–697, 2012.
105216, https://doi.org/10.1016/j.ijfatigue.2019.105216, 2019. Ragan, P. and Manuel, L.: Comparing Estimates of Wind Turbine
Hübler, C., Gebhardt, C. G., and Rolfes, R.: Methodologies for fa- Fatigue Loads using Time-Domain and Spectral Methods, Wind
tigue assessment of offshore wind turbines considering scattering Eng., 31, 83–99, 2007.
environmental conditions and the uncertainty due to finite sam- Vanmarcke, E.: On the distribution of the first-passage time for nor-
pling, Wind Energy, 21, 1092–1105, 2018a. mal stationary random processes, J. Appl. Mech., 42, 215–220,
1975.

https://doi.org/10.5194/wes-7-1171-2022 Wind Energ. Sci., 7, 1171–1181, 2022