Fatigue assessment of wind turbines involves three main sources of uncertainty: material resistance, load, and the damage accumulation model. Many studies focus on increasing the accuracy of fatigue load assessment to improve the fatigue reliability. Probabilistic modeling of the wind's turbulence standard deviation is an example of an approach used for this purpose.

Editions 3 and 4 of the IEC standard for the design of wind energy generation systems (IEC 61400-1) suggest different probability distributions as alternatives for the representative turbulence in the normal turbulence model (NTM) of edition 1. There are debates on whether the suggested distributions provide conservative reliability levels, as the established design safety factors are calibrated based on the representative turbulence approach. The current study addresses the debate by comparing annual reliability based on different scenarios of NTM using a probabilistic approach. More importantly, it elaborates on the relative importance of load assessment accuracy in defining the fatigue reliability.

Using the DTU 10 MW reference wind turbine and the first-order reliability method (FORM), we study the changes in the annual reliability level and its sensitivity to the three main random inputs. We perform the study considering the blade root flapwise and the tower base fore–aft moments, assuming different fatigue exponents in each load channel.

The results show that integration over distributions of turbulence in each mean wind speed results in less conservative annual reliability levels than representative turbulence. The difference in the reliability levels varies according to turbulence distribution and the fatigue exponent. In the case of the tower base, the difference in the annual reliability index after 20 years can be up to 50 %. However, the model and material uncertainty have much higher effects on the reliability levels compared to load uncertainty. Knowledge about such differences in the reliability levels due to the choice of turbulence distribution is especially important, as it impacts the extent of lifetime extension through reliability reassessments.

Fatigue reliability of a structure is its ability to withstand cyclic loading during the design life. Fatigue life is a highly sensitive and uncertain variable

The IEC standard

Some studies investigate the performance of different NTM approaches in describing the site turbulence conditions. For example,

In addition, there are some studies on the loads corresponding to each turbulence characterization approach. The results of these studies vary from each other. For example, some studies

As a second matter, the effect of uncertainty in the material properties on fatigue reliability assessment is also covered in many studies. According to previous research

All in all, there are many studies on the accuracy and performance of the representative turbulence, and they all show the need for transition from such a model to lognormal and Weibull distributions, especially in the case of offshore wind farms with overall lower turbulence levels. However, they have not compared different scenarios to each other in general design conditions. It is still debated whether the two distributions always provide lower reliability levels for different load channels compared to the representative value approach. The current work addresses this gap and such debate. Knowing the difference between the reliability levels when following different NTM approaches is especially important because the established safety factors for the semi-deterministic approach in the IEC design standard are calibrated based on the representative turbulence approach. Thus, if reliability levels are assessed using the same safety factors while characterizing the standard deviation turbulence by distribution, the semi-deterministic approach and probabilistic approach do not meet at the same reliability level at the end of the design life. It is important to ensure the two alternative distributions are never underconservative. Furthermore, knowing the differences in reliability levels can be an asset for fast initial estimation of the possible extension of a lifetime when the considered approach in the design phase is known. Such information about the design assumptions is also crucial for more accurate lifetime extension assessments.

In addition, considering the results of the previous studies on the importance of material uncertainty in fatigue reliability, it is valuable to investigate how all the efforts for accurate modeling of the turbulence will transfer to a more robust reliability assessment. In the present study, we reveal how different approaches in the IEC standard for a general case can change the distribution of the fatigue loads. We also study the sensitivity of the reliability to the change in the fatigue load compared to its sensitivity to variations in other random inputs.

The results of the current study cover blade flapwise and tower base fore–aft load channels in a large wind turbine (10 MW) from IEC class 1A. We study the difference in distributions of the damage equivalent load (DEL) considering different NTM representations using many aeroelastic simulations and bootstrapping techniques. In addition, the results show the overall importance of DEL variation due to the turbulence model by revealing the relative effects of load uncertainty on the reliability. Knowing the extent to which the various sources of uncertainty affect reliability can help designers and researchers focus on effective areas to get robust reliability levels. The fatigue exponent is the exponent to which the load is powered in the damage models commonly used (as in the current study). Thus, it directly changes the share of the loads in the fatigue damage, and it also changes the distribution of DEL

We provide information about the wind turbine case study and the aeroelastic simulations in Sect.

We use 10 min aeroelastic simulations to obtain the load time series and estimate the fatigue reliability based on them. Figure

Flowchart of the procedure, main random inputs (

Section

Our case study is the DTU 10 MW reference wind turbine

We perform three groups of aeroelastic simulations in HAWC2

HAWC2 is an aeroelastic code for calculating wind turbine response in the time domain and was developed in the DTU Wind Energy department between the years 2003–2007.

softwareThe current work only covers normal operating conditions and does not consider the fatigue damage occurring in the fault, idling, start-up, or shutdown events. Thus, we perform the simulations based on the IEC standard design load case (DLC) 1.2

The mean wind speed varies from 4 m s

We use the Mann turbulence model for modeling the wind field (see

Table

Specifications of wind modeling in three groups of HAWC2 simulations corresponding to three study cases.

The study considers the flapwise bending moment in the blade and the fore–aft bending moment in the tower base as the main outputs of the simulations and the input for fatigue assessment of the blade and tower.

In the following, we present the mathematical background and relations we use for post-processing simulation load outputs and estimating the corresponding fatigue damage and reliability.

Wind as a random process is mostly described by its mean value and standard deviation (turbulence) at each point in time and space. The IEC standard

In Eq. (

The statistical parameters of the wind are correlated. In other words, the standard deviation of the wind (turbulence) changes with a change in the mean level. However, since the IEC design standard suggests binning of the wind speeds (as we do in our simulations), one can use the marginal distribution of turbulence in each wind speed bin. The first option is to consider the constant representative turbulence for each wind speed bin, equal to the 90 % quantile of the distribution, instead of the marginal distribution. The other option is to consider the whole distribution domain in each wind speed bin. The third edition of the IEC standard

The current study investigates the impact of NTM turbulence characterization choice on design fatigue reliability, as it has not been studied before. For this purpose, we define three cases for the different turbulence characterization approaches: case 1 covers the 90 % quantile turbulence value, and cases 2 and 3 refer to the lognormal and Weibull distributions, respectively. Equations (

Equations (

Equations (

In Eqs. (

Lognormal and Weibull distribution of turbulence at a mean wind speed (MWS) of 8 m s

The lognormal distribution is generally heavy-tailed; however, as Fig.

The Weibull distribution represents higher probabilities in the lower turbulence levels and covers more data from lower values considering the same number of observations. Therefore, we generally expect the use of the Weibull distribution to represent turbulence occurrences to be a less conservative approach. In the following sections, we study this expectation and its effects on the DEL estimations in each case.

Material fatigue resistance tests are often performed under constant-amplitude cyclic loading. The number of cycles that the test specimen can endure in each stress amplitude before failing is collected within

In Eq. (

For variable loading, as in the case of wind turbines, one should use a suitable model to relate the constant amplitude data to the accumulated damage. The Palmgren–Miner (Miner's) rule

In Eq. (

The blade's root cross section in the case study wind turbine is nearly circular, and the current study assumes the same. In addition, the pitch angle in the blade root is zero. Thus, the moments along the

In Eq. (

Blade root and tower base section parameters of the DTU 10 MW model in HAWC2.

Using Eq. (

The parameter

DEL is a tool to compare the damage caused by different variable amplitude loading scenarios.

In Eq. (

In Eq. (

In Eq. (

In Eq. (

In case study 1 (constant turbulence), the probability of the representative turbulence value is assumed to be unity. The following section presents the definition, mathematical relations, and procedure for estimating fatigue reliability.

Structural reliability is the ability of a structure to fulfill the structural design request for a defined period

In Eq. (

modeling of limit state equation

quantification of uncertainties and modeling by the stochastic variable

applying reliability methods to estimate the probability of failure (first-order reliability method in the current case).

The limit state function in assessing safety within time

In Eq. (

Miner's rule does not consider the load sequence effect in variable loading and thus leads to errors in fatigue damage prediction. There are some studies

In Eq. (

The limit state function in a specific time can be shown via expressions other than the common form of Eq. (

Combining Eqs. (

The parameters

To find the probability of failure, after defining the limit state function, the integration in Eq. (

As stated in the previous subsection, the FORM is one of the ways to estimate the solution of the integration in Eq. (

Equation (

The operator

To apply FORM analysis, we first fit distributions to the estimations of

Since the DTU 10 MW turbine is not designed against fatigue, we observed low reliability levels in the blade root and the tower base (failure occurrence in the tower base). To lower the errors in FORM in the case of the blade (high fatigue exponent and thus high nonlinearity), we calibrate the material strength towards low probabilities of failure for the sake of accuracy. Thus, in the current study, we increase the material fatigue strength proportional to the high fatigue loads while keeping the corresponding coefficient of variation (CoV) the same as for real material to avoid effects on the sensitivity analysis. These changes will affect our reliability levels. However, the main interest in the current study is the sensitivities and changes, not the values.

Table

Characteristics of the material and model variables.

We would like to see the sensitivity of fatigue reliability to changes in fatigue loads in addition to material strength. Different materials (used in different components) have different fatigue exponents, and thus the effect of change in their loading on reliability can be different (the higher the fatigue exponent, the higher the effects of loads in the overall damage and reliability). We want to take this fact into account while being consistent in the idea of the variability of

After specifying all the distributions and probabilistic parameters of each random variable in Eq. (

In transforming non-normal continuous distributions to standard normal, since the design points (

In Eqs. (

This method also provides information regarding the relative importance of each random variable or, in other words, the sensitivity of the output (reliability) to each input. A vector

The problem is solved in an

In Eq. (

The failure probability at each point in time (year in this case) depends on the survival at the previous point (1 year before). Thus, the annual reliability is useful for assessing the probability of failure at the end of each year. The annual probability of failure in time

Using Eq. (

To fit the distributions to DEL data in different case studies, we need to sample from the turbulence distribution in each wind speed bin to account for turbulence probability (see Eq.

Sampling points of turbulence within intervals of a size of 0.05 in the corresponding distribution in case 2 (lognormal distribution) and case 3 (Weibull distribution) at five different mean wind speeds.

Figure

The following section provides the results of the study.

The current section presents the results of the study in two steps. First, we introduce the distributions of DEL in different turbulence modeling cases in Sect.

Wind speed fluctuation is one of the main causes of fatigue damage, especially in the load channels like the blade flapwise and tower base fore–aft. A change in the wind standard deviation (turbulence) in each mean wind speed level directly changes the estimated fatigue damage. In the current section, we look into the change in the distribution of DEL with the change in the turbulence characterization approach in the IEC NTM. In case 1, each realization of DEL

Figure

DEL

The bar plots of Fig.

Another observation from Fig.

Using the 200 samples in each wind bin (consisting of constant turbulence and constant mean wind speed), we calculate the DEL

Probability density function (PDF) of normalized DEL estimates using 1000 bootstrapped samples of size 6 in different turbulence study cases in

Figure

Based on Fig.

Comparing the distributions of DEL

Although the Weibull distribution of turbulence generally results in lower DEL estimations, there is an overlap between distributions of all cases in the blade root. There is also an overlap between cases 2 and 3 in the tower base. This means if the designer uses only six realizations, there is a chance that the DEL

The general conservatism of using a 90 % turbulence level in the DEL evaluations can show itself in the fatigue reliability estimations. In addition, since changing the method of modeling the turbulence changes the standard deviation of the DEL realizations, the sensitivity of the reliability levels to the DEL changes can also vary from one case to another. We study the extent of such an effect in different fatigue exponents and in different load channels in the next subsections.

In the reliability assessments, we use the

Best distribution fits to

Best distribution fits to

Using the distributions of

Reliability index through the lifetime considering

The results in Fig.

In general, the tower base shows a very fast reduction in reliability over time. One possible reason is the high mean value of the

In the case of constant turbulence in the tower base, the change in the fatigue exponent (

Notice that the difference in the results in the tower base is more visible because of the higher initial variability in the DEL in this load channel. In the case of the blade, the same trends occur, but they are less visible.

We study the sensitivity of the reliability level at year 20 to each random input into the limit state function. The importance rank of each of the inputs is derived from FORM analysis (see Eq.

Importance factors of different random inputs in assessing the annual reliability level at year 20 in the blade's flapwise load channel in cases 1, 2, and 3 considering different Wöhler exponents.

Importance factors of different random inputs in assessing the annual reliability level at year 20 in the tower base fore–aft load channel in cases 1, 2, and 3 considering different Wöhler exponents.

Figure

The share of the logarithm of load in the overall reliability increases with the increase in fatigue exponent, as we expect. However, the importance of the load uncertainty compared to the other two parameters is negligible in all cases.

Figure

The effects of the fatigue loads on the reliability in the tower base are relatively higher than in the case of the blade root (see Fig.

As seen in the previous load channel (blade root flapwise), we observe an increase in the importance of the loads with an increase in the fatigue exponent. In addition, cases 2 and 3 turbulence modeling decrease the CoV of

According to both Fig.

The thickness of the tail in the lognormal distribution is dependent on its standard deviation. The standard deviation of the distribution in different cases of NTM is a function of the reference turbulence level (see Eqs.

Different distributions of DEL

In forming the distribution of the wind speed (using the Weibull distribution shown in Eq.

DEL

Figure

According to Fig.

DEL

As Fig.

To investigate the effects of the design class on the annual reliability, two of the cases for the blade with

Annual reliability index of blade,

The results presented in Fig.

The results (especially in Fig.

In addition, the results of reliability estimations show that the small overlaps seen in the DEL distributions are not very important when coming into the reliability framework. This has been made clearer in the sensitivity analysis, where the effects of load variations are shown to be relatively negligible.

All in all, although the choice of the turbulence characterization approach impacts the reliability and the sensitivity of the reliability to different uncertainty sources, these effects are not notable when compared to the high sensitivity of the reliability to other sources in every case. In other words, accurate modeling of the damage accumulation or more accurate characterization of the material properties (especially in the case of composites) can impact the reliability levels to a relatively higher extent. There is a 50 % change in the reliability level at year 20 in the case of the tower base with

There are also some limitations and potential for extension of the study. The most important assumptions that can limit the generality of the results are as follows:

The assessment of the remaining fatigue lifetime hinges upon reanalyzing the reliability. In performing such an analysis per the IEC standards, a designer can choose to follow different recommendations regarding probabilistic modeling of the turbulence. The ramifications of those choices are currently unclear. For example, as the present study shows, using six realizations for the estimation of DEL, in the case of the tower with a fatigue exponent of 3, the estimations based on the Weibull distribution of turbulence can differ by 40 % from the representative turbulence approach. Regarding fatigue reliability at the end of the design life (20 years), the differences are up to 50 % in the case of the blade root and up to 200 % in the case of the tower base. Such a high difference can change the possible scenarios at the lifetime extension stage and must be considered. The difference in reliability levels varies with a change in the design class. Consideration of the difference between classes is important in the case of any further calibration of the safety factors in future versions of the standard.

The results presented in this paper are applicable to the wind turbine design stage. The study informs the designer about the extent to which following different editions of the IEC standard can change the expected value and uncertainty of the fatigue damage evaluations based on turbulence input. It also shows how the annual fatigue reliability and sensitivity of reliability change in load channels of interest and different fatigue exponents.

Furthermore, the reliability estimation is based on a simplified linearized limit state function, making the complete separation of the loads from material properties possible. If all the random variables are lognormally distributed (highly possible), this formulation results in a very simple and fast reliability analysis at the design level.

The importance ranks of the variables reveal that although the change in the turbulence characterization changes the distribution of the fatigue loads and the fatigue reliability, focusing on decreasing the material or models' uncertainty is more effective. This is due to the relatively high uncertainty in the material properties and linear damage accumulation rule.

Using Monte Carlo simulations, considering other sources of uncertainty in the load, testing for other wind turbine designs and classes, using other aeroelastic tools, considering offshore cases, and – finally – using different approaches for the design of experiments are some suggestions for future studies.

The parameters of the best-fitted distributions to

For low probabilities of failure, such as in structural components of the wind turbines, a lot of simulations are needed for MC. In a Monte Carlo analysis with

All data used for the DTU 10 MW case study, in addition to the codes, can be found at

SM, PV, and JR were responsible for the overall conceptualization of the study. SM wrote all the computer codes and performed all the data analyses. SM, PV, and KD were involved in the writing and editing of the paper.

At least one of the (co-)authors is a member of the editorial board of

The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work was authored in part by the National Renewable Laboratory (NREL), operated by the Alliance for Sustainable Energy, LLC, for the US Department of Energy (DOE) under contract no. DE-AC36-08GO28308.

The research has been funded by Technical University of Denmark (DTU).

This paper was edited by Raimund Rolfes and reviewed by two anonymous referees.