Effects of finite sampling on fatigue damage estimation of wind turbine components: A statistical study

Abstract

The variability of the wind turbine loads complicates fatigue assessment in the design phase, as performing simulations covering the entire lifetime is computationally expensive. The current work provides important information for assessing the uncertainty in fatigue damage estimation due to finite data. We study the sample size effect on mean, variance, and skewness of damage in each wind bin, identify the important wind bins, and study the uncertainty propagation from each wind bin to the lifetime damage using 3600 aeroelastic simulations and bootstrapping. To achieve less than 1% error in the damage estimation across all load channels in the current case study, at least 100 turbulence seeds are needed. Damage in different wind bins follows a lognormal distribution when using the conventional approach of six seeds. The provided insights and information allow the designer to achieve a specific level of accuracy for a given computational cost using strategic bin sampling.

Keywords

Wind turbine fatigue finite-sampling error Gaussian fatigue damage central limit theorem fatigue damage variance

Introduction

High-cycle fatigue is a dominant damage mechanism in most wind turbine components (Sutherland, 2000). One of the main difficulties in fatigue damage estimation is the variability of the wind loads and the complex response of the wind turbine as a nonlinear system. There are many sources of uncertainty in estimating the damage, including model uncertainty (wind inflow, aeroelastic response, cycle counting methods, damage accumulation rule, etc.) as well as uncertainty in wind conditions, material properties, and more. Identifying the share of each source allows for a more robust reliability assessment. Specifically, in the design phase, a part of the error stems from using a limited set of wind conditions due to the computational costs of wind turbine simulations. The effect of using a finite sample set of wind conditions for short-term aeroelastic simulations has been studied before for offshore wind turbines (Zwick and Muskulus, 2015) and identified as an important source of error.

In fatigue reliability assessment of wind turbines, not only the estimated value (mean) of the damage but also its variance is a significant contributing factor. Thus, proper modeling of the variance is crucial. In the approach suggested by the IEC standard (IEC, 2019), damages in the short-term random realizations of wind are derived and averaged. According to the law of large numbers, by gathering more data, the mean value will converge to the “true” damage, and the variance will decrease. The main question is therefore how much data is needed to reach an acceptable bias and variance in the estimated fatigue damage. One common practice is using six samples of a 10-minute interval in many of the load cases to account for the variability of wind (Hansen et al., 2015). Zwick and Muskulus (2015) as well as Müller and Cheng (2016) have shown that the variability of wind is not fully captured in six 10-minute simulations in the case of an offshore wind turbine. They investigated the importance of simulation error and its value in each mean wind speed considering fatigue and ultimate load assessments of offshore jackets. Their results show that considering six samples in each load condition can overestimate the fatigue loads by up to $29 %$ in an offshore 5 MW turbine case study.

Veldkamp (2008) shows that load modeling uncertainties, including the number of turbulence seeds, the average wind speed, and turbulence intensity, account for up to $15 %$ of the total uncertainty in the fatigue assessment of a tubular tower. It is important to revisit and reassess the common approaches and recommendations as turbines continue to grow larger and as turbine development continues in new types of site and wind conditions—both on and offshore.

The 10-minute intervals of wind time series are often modeled as a quasi-steady Gaussian process in the case of onshore wind turbines. However, the load response of any structural component is not necessarily Gaussian because of the non-linearity of the wind turbine’s aerodynamics and structure (Natarajan and Holley, 2008). Studies on the behavior of loads in different components (Ragan and Manuel, 2007) show that the loads mainly driven by wind (e.g., blade’s flapwise and tower base fore-aft moments) behave as Gaussian processes to some extent, and thus, they can be modeled as such with some simplified assumptions. In the case of flapwise moments in the blade, some studies only model the response around the rotational frequency because of the high fatigue exponent resulting in the high energy focus around this area. Therefore, they use a Gaussian narrow-band process to model the flapwise bending moments. The Gaussian approximation is also applicable to similar wind-driven loads such as tower base fore-aft or tower top pitch moments when the non-linearity of the wind turbine is low. There are approaches in general fatigue assessment for modeling fatigue damage caused by narrow-band Gaussian loads applied on linear oscillators (Bendat, 1964; Crandall et al., 1962; Miles, 1954). These studies have been improved upon more general cases in recent years (Benasciutti and Marques, 2021; Low, 2012b). The mentioned models for the prediction of damage variance in the case of narrow-band Gaussian loads have good potential to be used in the fatigue assessment of wind turbine components due to the similar load behaviors discussed. However, we should study simplifications embedded in these approaches for wind turbine applications when using short-term data. For example, the distribution of the expected damage is assumed Gaussian in the final stage based on the central limit theorem (CLT; Low, 2012b) in such procedures. However, the number of the required short-term data for this theory to hold depends on the skewness embedded in the load cycle distribution. The skewness is a function of the fatigue exponent, the bandwidth of the load process, and the damping coefficient (Low, 2012a). Therefore, we should investigate the assumption of the normal distribution of the expected damage based on how the number of required 10-minute realizations for CLT to hold for each load channel in the wind turbine. If the assumption of Gaussian wind and thus Gaussian narrow-band loads does not hold, the results and behaviors will be different (Marques and Benasciutti, 2020, 2021; Shuang and Song, 2018).

For many components, the operational conditions significantly affect the loads. In several studies (Agarwal and Manuel, 2009; Moriarty, 2008; Veers, 2010), dominant fatigue loads in various components have shown higher variability when the turbine operates near rated wind speed. Therefore, some mean wind speeds can be more important than others in forming the fatigue damage variability. In addition, small changes in operating conditions can cause significant changes in loads, and thus, in fatigue damage. It is important to identify the important wind bins impacting not just wind turbine loads but also the damage and to make sure that these bins are adequately sampled in the finite set of wind simulations performed in the design process.

To summarize, there are several key gaps in the state of the art. First, although there are some studies devoted to the investigation of the variation of loads and damage equivalent loads with change in the number of 10-minute simulations (Liew and Larsen, 2022; Thomsen, 1999; Zwick and Muskulus, 2015), the variation of uncertainty in the lifetime damage showing the effects of the fatigue exponent is not studied. This is important because the connection between uncertainty in bins to the lifetime damage allows for selective sampling in the design phase when linked to the results of the current study. In addition, the work shows the importance of fatigue exponent in forming the statistical uncertainty of fatigue damage. Secondly, the change in the skewness of damage data with the change in sample size has never been considered before. The study of skewness is especially important for the case of high fatigue exponents informing researchers using the Gaussian damage assumption. This paper provides key information for turbine designers by addressing these issues via identifying the important wind bins, the change of statistical parameters of damage in bins with the change in sample size, and the propagation of the uncertainties to the lifetime damage. Such an approach has the potential for informing the IEC design standard requirements around the number of 10-minute wind realizations to run for aeroelastic simulations.

We conduct the current study as follows. “Methodology” includes the framework of simulations and the basic relations. Then, in the “Results” section, we investigate the damage-equivalent load in different components of DTU 10-MW wind turbine using bootstrapped data from a large number of 10-minute aeroelastic simulations. First, we present the detail of this procedure. Then, we investigate the effects of finite sampling on the uncertainty in lifetime DEL estimation plus shares of different mean wind speeds in lifetime damage. The best distribution fit for the damage data and change of the probabilistic behavior with change in the sample size are parts of the discussions in the ending part. The “Discussion” section contains the discussions around the overall results. Finally, the Section “Conclusion” presents the conclusions from the study and notes several areas for future extension of the work.

Methodology

In the current work, we perform aeroelastic simulations of the DTU 10-MW reference wind turbine in the HAWC2 software (Larsen and Hansen, 2007) and use the results for the statistical studies. HAWC2 software simplifies the structure using beam elements and models the wind with parameters including mean wind speed, wind shear, wind standard deviation, and direction. Different realizations of wind with specific parameters set by the user are randomly generated as time series named different turbulence seeds. We use a high number of realizations to study the statistical behavior of their corresponding fatigue damage. We study both the first and second moments as well as the distribution of the damage as a random parameter. The following subsections present the features of aeroelastic simulations, mathematical formulations of the problem, and the methods used for post-processing the simulation results, respectively.

Aeroelastic model

A total of 3600 10-minute aeroelastic simulations of the DTU 10-MW reference wind turbine (Bak et al., 2013) are performed in HAWC2 software. The turbine under study is an onshore turbine with a rotor diameter of 178.3 m and a hub height of 119 m. It is rated at 10 MW and a mean wind speed of 11.4 m/s. We set the wind direction constant and equal to zero. The mean wind speed varies from 4 m/s (cut-in) to 26 m/s (cut-out) in steps of size 2 m/s. We identified the transient part of the simulations by checking the side-side response time series of the tower base in one high mean wind speed (24 m/s) as the most chronic case for structural stability (due to low damping). According to that test, we consider the cut-off transient of the simulations to be 100 seconds. Thus, the initial duration of simulations is 700 seconds and we only consider the last 600 seconds of the response time series in the current study. We use the Normal wind model, suggested in the IEC standard for the DLC 1.2 load case. The table shows the set of constants for this model. The Mann turbulence model (Mann, 1998) is used for generating the turbulence boxes. Because of the constant wind direction, we assign more points of evaluation alongside the wind direction in the Mann turbulence box to gain more resolution and accuracy where the most load variation occurs. The number of points in the wind direction is 8192 and there are 32 points in each of the other two directions. We assume that 300 turbulence seeds (realizations of wind) are enough data to be close to the “true” mean value of the infinite data and in the primary convergence study, the extent of accuracy of such an assumption is investigated. A single wind direction, constant shear, constant turbulence level within each wind bin, 12 mean wind speed levels, and 300 turbulence seeds within each wind speed bin result in 3600 simulations in the current study. Although non-operation conditions (like DLC 6.4) are also relevant for the fatigue assessment of some components because of reduced aerodynamic damping, in the current study we only consider operating conditions. Table 1 shows the specifications of wind load modeling of the simulations in the current study.

Table 1.

Specifications of wind modeling in Hawc2 simulations.

Parameter	Assignment
Wind condition	Normal
Turbulence in each wind bin	90% quantile
Turbulence model	Mann
Reference turbulence intensity	0.1
Reference mean wind speed $(m / s)$	50
Wind shear	0.2
Average mean wind speed $(m / s)$	10
Cut-in mean wind speed $(m / s)$	4
Cut-out mean wind speed $(m / s)$	26
Rated wind speed $(m / s)$	11.4
Length of each wind speed bin $(m / s)$	2
Wind speed probability distribution	Rayleigh
Yaw angle (°)	0
Number of realizations per wind bin	300
Mann box grids along the wind direction	8192
Mann box grids along other two directions	32
Time steps in the simulations (seconds)	0.01

We use some mathematical relations to relate the simulation load results to the fatigue damage. The details of these relations are provided in Section “Mathematical relations.”

Mathematical relations

The simulation load results are transformed into damage to perform the statistical study. The current section presents the corresponding methods for this transformation and the models we use for the variance study.

Damage-equivalent load and its relation with random fatigue damage

In the current work, we use the concept of damage-equivalent load (DEL) as it is a useful tool in variable-amplitude loading for reporting the amount of material degradation due to fatigue. DEL is the magnitude of the constant-amplitude load or stress that causes the same fatigue damage as the actual loading at the same reference number of cycles $(N_{eq})$ (Burton et al., 2011). In addition, we use rainflow counting (Endo et al., 1967) and Palmgren–Miner (Miner’s) rule (Miner, 1945; Palmgren, 1924) to formulate and estimate DEL due to their common use in industry and research practice. Rainflow counting is a method to count cycles embedded in a random loading and allows using the S-N curve for fatigue assessment in such loading. In Miner’s rule, the accumulated fatigue damage of a random loading is a linear summation of the incremental damages in different amplitude levels of the loading, as shown in equation (1). Failure occurs when the magnitude of the theoretical damage reaches a certain level, often calculated as unity divided by standard safety factors.

D = \sum_{i = 1}^{N_{s}} \frac{n_{i}}{N_{i}}

(1)

In equation (1), parameter $D$ accounts for the magnitude of fatigue damage, $n_{i}$ is the number of cycles of the $i th$ load amplitude, and $N_{i}$ is the allowed number of cycles according to the S-N curve. In addition, $N_{s}$ accounts for the total number of rainflow-counted stress amplitude levels. Basquin formulation (Basquin, 1910) is a mathematical representation of the S-N curve fitting well for most of the materials. Equation (2) shows Basquin relation.

N = k S^{m}

(2)

In equation (2), $N$ is the number of allowed cycles with amplitude $S$ and $m$ is the fatigue exponent also known as Wöhler’s exponent. $k$ in equation (2) is the Basquin coefficient and it is constant. We use Basquin relationship and assume $m$ to be equal to 10 in the case of the blade and equal to 3 in the case of other components in the current study. The constant $k$ cancels out in the definition of DEL and thus, we do not set any value for it in the current work. It should be noted that we do not consider any mean corrections in the current study and the mean loading level is assumed to be zero for simplification.

Since DEL is a load, according to Basquin relationship and Miner’s rule (equations (1) and (2)), one can represent a factor of damage by simply using the term $DE L^{m}$ .

Thus, a mixture of equations (1) and (2) using the DEL concept can be shown as equation (3).

E [{DEL}_{s}^{m}] = \sum_{i = 1}^{N_{s}} (\frac{n_{i} M_{i}^{m}}{Neq})

(3)

In the above equation, $DE L_{s}$ is the DEL of a sample of 10-minute time series containing $N_{s}$ number of rainflow counted cycles. In this equation, $M_{i}$ is the loading amplitude in the $i$ th level within each sample, $n_{i}$ is the number of cycles within each moment range and we set it equal to one in the present work. In other words, we replace the half cycles with complete cycles—having the same amplitude—because whether they are half or full cycles is purely by chance of whether the largest peaks occur early or late in the sample. Thus, by considering them as full cycles, we will consider the large cycles important in fatigue assessment, especially in the case of high fatigue exponents. In addition, we assume $N_{eq}$ to be 951 cycles which approximately corresponds to the average number of cycles in a 10-minute interval based on the simulations of DTU 10MW turbine with DLC 1.2 load case conditions.

In a scenario where we have $SS$ realizations of wind, we can gather information of all the damages and calculate the mean value (equation (4)). This relation assumes that the probabilities of all sample DELs $(DE L_{s})$ are equal and uniform.

E [(DE L_{bin})^{m}] = \sum_{s = 1}^{SS} \frac{{(DE L_{s})}^{m}}{SS}

(4)

In general, the sample size $(SS)$ in equation (4) depends on data availability. Combining equations (3) and (4), the damage in each wind speed bin $({DEL}_{bin}^{m})$ is related to the load ranges in the sample time series by equation (5).

E [{DEL}_{bin}^{m}] = \sum_{j = 1}^{N_{bin}} (\frac{n_{j} M_{j}^{m}}{NeqSS})

(5)

In equation (5), $M_{j}$ is the loading amplitude in the $j th$ level within all samples and $n_{j}$ is its corresponding number of occurrences. The number of total cycles in $SS$ number of realizations is equal to the summation of all $Ns$ values shown as $N_{bin}$ in equation (5). With the assumption of single cycles within each load level $(n_{j} = 1)$ , equation (5) turns into equation (6):

E [{DEL}_{bin}^{m}] = \sum_{j = 1}^{N_{bin}} (\frac{M_{j}^{m}}{NeqSS})

(6)

Since $N_{eq}$ is the average number of cycles in each 10-minute sample (equal to 951), and thus $N_{eq} SS$ is the total number of cycles when combining all the time series.

As mentioned before, in the present work, ${DEL}_{bin}^{m}$ is used as a metric to quantify the damage in each wind bin since it is proportional to damage, and thus, they share the same probability distribution. Equation (6) shows the average of $M_{j}^{m}$ values. If we consider $M_{j}^{m}$ as a random variable with a specific distribution, then $E [{DEL}_{bin}^{m}]$ is proportional to the mean of the mentioned distribution according to equation (6). Based on the central limit theorem (Florescu and Tudor, 2013), in an adequately high level of $SS$ , one can expect the distribution of ${DEL}_{bin}^{m}$ to be Gaussian. Furthermore, the mean level of the corresponding distribution should converge to the “true” value of $E [{DEL}_{bin}^{m}]$ as the number of samples converges to infinity.

The above-mentioned theory holds for any level of $SS$ for a normally distributed $M_{j}^{m}$ . However, it takes a large number of data for the theorem to be true in the case of skewed parent distributions. The higher the skewness of the parent distribution, the larger the sample size must be.

The skewness of the distribution of $M_{j}^{m}$ relates to the distribution of $M_{j}$ and the magnitude of $m$ . The fatigue exponent is a positive integer that naturally increases the distance between the $M_{j}^{m}$ data compared to $M_{j}$ , and thus, it increases the skewness in the corresponding distribution. This effect is more intense in composite structures like blades because of their high fatigue exponent. More details are presented in the “Results” section.

Uncertainty propagation from damage in each wind bin to the lifetime damage

It is important to know how the uncertainty propagates from the damage in each wind bin to the overall damage estimations. The current part of the work presents some relations that facilitate such assessments. Considering that ${DEL}_{lifetime}^{m}$ represents lifetime damage, we can define its expectation in terms of ${DEL}_{bin}^{m}$ s and the probability of occurrence of each bin as equation (7).

E [{DEL}_{lifetime}^{m}] = \sum_{bin} E [(DE L_{bin})^{m}] \Pr (bin)

(7)

According to equation (7), we can calculate the damage share of each bin or in other words damage density as equation (8).

Damage density = \frac{E [(DE L_{bin})^{m}] \Pr (bin)}{E [{DEL}_{lifetime}^{m}]}

(8)

As shown in equation (7), ${DEL}_{lifetime}^{m}$ is a linear function of the random variables ${DEL}_{bin}^{m}$ . Using this linear dependence, equation (9) demonstrates the relationship between the corresponding variances including a parameter (G) to account for potential correlations between ${DEL}_{bin}^{m}$ s.

var ({DEL}_{lifetime}^{m}) = \sum_{bin} (\Pr (v_{bin}))^{2} var ((DE L_{bin})^{m}) + G

(9)

In equation (9), the operator $v ar$ indicates the variance of each random variable. The parameter $G$ is a function of co-variances of ${DEL}_{bin}^{m}$ s. $G$ is most likely to be zero because each $DE L_{bin}$ relates to random realizations of wind that are independent to some levels (scatter plots of some $DE L_{bin}$ s showing zero correlations are shown in the Appendix 1). Equation (9) shows the linear relationship between the variance of the lifetime damage ( ${DEL}_{lifetime}^{m}$ ) and the variability of the damage in each wind bin ( ${DEL}_{bin}^{m}$ ). Such a relation facilitates the estimation of the variation in the total damage based on the variation of the damage in each bin. In addition, it also elaborates on the sensitivity of the variance of ${DEL}_{lifetime}^{m}$ to the variations of damage in different bins. The “Results” section includes the study of both parameters (expectation and variance) based on data analysis from a large group of data and corresponding bootstrapped subsets. In the following, we present our proposed method for post-processing the results of the aeroelastic simulations.

Post-processing the simulation outputs

As mentioned before, the simulation results are post-processed to obtain damage values. Then we gather them in different sub-samples via bootstrapping to analyze mean values, variances, and probability distributions of damage data. The following section presents the general procedure of post-processes in addition to the steps taken to bootstrap from the parent database.

General workflow

As a primary step, we investigate the trends and behaviors in the turbine under study. Then, we make sure that the 300 number of data within each wind condition is large enough by investigating the extent of stabilization of the estimated $DEL$ when the number of data grows. The convergence study aims to investigate the change in the difference between the average of 300 seed data and the average of means of random subsets as the sample size grows. One thousand subsets are obtained in each bin and size level by bootstrapping. In addition, the second and third steps of the study use the same procedure to inspect the variance of data and the probability distributions, respectively. It has to be mentioned that in the case of the probability distribution study, we collect more subsets (3000 subsets in each level) to be able to fit more accurately and reach lower probabilities of exceedance in the empirical CDFs. The method used in the current study is the conventional scaling of short-term damage to longer times without probabilistic extrapolation. The general workflow for the studies is presented in Figure 1 and more details of bootstrapping procedure are provided in the last part of the current section.

Figure 1.

Flowchart of the study.

Bootstrapping procedure

As mentioned in the workflow the data sets are obtained by bootstrapping. The following contains the details of such a procedure which is a combination of the fifth, sixth, and seventh steps in Figure 1:

Randomly sample $SS$ 10-minute simulations from each wind bin.

Calculate the DEL of each random sample ( $DE L_{S}$ ) using equation (3).

Estimate the $DE L_{bin}$ from $SS$ number of $DE L_{S}$ values using equation (4).

Aggregate through different mean wind speeds to obtain $DE L_{lifetime}$ using equation (7) (for the case of convergence study of the $DE L_{lifetime}$ ).

Repeat the above procedure 1000 times for the studies in “Study of DEL estimations using different sample sizes” or 3000 times for the distribution study of $DE L_{bin}$ in “Distributions of the damage estimations”.

Thus, for each desired value of $SS$ , we produce 1000 random numbers in each mean wind speed bin for the convergence study and 3000 random numbers for the distribution study.

Results

The present section contains the results of the study in two parts. As a primary step before the main investigations, it is imperative to gain insight into the behavior of the $DE L_{bin}$ using all data of 300 seeds within each wind bin. Section “DEL trends” contains that insight. Then in the next section, we investigate the extent and form of the convergence of the mean of $DE L_{bin}$ estimates toward the mean of the parent database in each wind bin as a function of bootstrapped sample size. Then we identify the most important wind bins in the mean and variance of the $DE L_{lifetime}$ in each load channel. Finally, the last section contains the study of the probability distributions of ${DEL}_{bin}^{m}$ and ${DEL}_{lifetime}^{m}$ in all sample sizes. A brief description of the current section part is as below:

In DEL trends, using the full data set of 300 seeds, the fatigue critical point along the sections is identified. In addition, we study the trend of DEL change with a change in the mean wind speed.

In study of DEL estimations using different sample sizes, we investigate the convergence of the DELs estimated based on the bootstrapped data toward the “true” value. Furthermore, we identify the wind bins with the most significant shares in the lifetime damage for different components.

Finally, in “distributions of the damage estimations”, we present the best distribution for fitting the probability of the DEL data estimates from samples with different sizes.

DEL trends

As mentioned before in Table 1, we only consider zero wind direction and zero yaw angle in the current study. Thus, the load component along the wind direction is the most damaging load in all components under study. In other words, the possible fatigue failure for the blade-root, tower-base fore-aft, and tower-top pitch moments are governed by the flapwise, fore-aft, and fore-aft moments, respectively. We have checked the above fact before starting the processes in the current work and we consider these loads in the rest of the study.

DEL trend in the case study wind turbine

Having identified the critical point around the cross-section, we can now analyze the trends of $DE L_{bin}$ (DEL in each mean wind speed bin) before moving into the convergence study (Subsection “DEL trend in the case study wind turbine”). The following results track the trend of change in the mean values of $DE L_{bin}$ using all 300 seeds in different turbine components with the growth of the mean wind speed. This trend reveals the specific behavior of the estimated values using the primary data set in the case study wind turbine. Figure 2 shows the DEL within each wind speed bin $(DE L_{bin})$ estimated via all 3600 simulations for the blade root, tower base, tower top (yaw bearing), and shaft-considering torsion.

Figure 2.

$DE L_{bin}$ calculated using all 300 seeds versus mean wind speed in (a) blade root, (b) tower base, (c) tower top (yaw bearing), and (d) shaft (torsion).

As Figure 2 shows, DEL in the blade, tower-top, and shaft generally increases as the mean wind speed grows. This is because of higher turbulence standard deviation levels in higher mean wind speeds and, thus, more fluctuations in the load time series. Thus the DEL generally increases with an elevation of the mean wind speed. In the tower base, the DEL increases before dropping at 8 m/s and rising at wind speeds higher than 10 m/s. The reason for the high DEL levels at 6 and 8 m/s is that the natural frequency of the tower base is close to the third multiplication of the minimum rotational frequency of the blade in mean wind speeds between 6 and 8 m/s (especially at 7 m/s). The resonance around this frequency range causes an undershoot to rotational speeds lower than the design minimum and thus, higher DEL levels due to the high fluctuations in the load response.

Based on equation (7), the maximum $DE L_{bin}$ s are the candidates for having the most significant shares in $DE L_{lifetime}$ depending on their bin’s probability. Thus, the relative magnitudes of $DE L_{bin}$ s, shown in Figure 2, can help in the identification of the possible wind bins with the biggest shares in the estimation of $DE L_{lifetime}$ in each load channel when used alongside other information presented in following sections. The next sections contain more elaborations.

The following section contains the study of the behavior of the standard deviation of DELs and the importance of each bin in forming the standard deviation of $DE L_{lifetime}$ .

Study of DEL estimations using different sample sizes

The current section studies the effect of increasing the number of turbulence seeds (realizations of wind) on the estimations of the DEL in different components. As we assume the mean levels to converge to the values calculated based on the full dataset, the above investigations show the trend of change in the mean levels of the mean wind speed. The study in DEL trend in the case study wind turbine provides information about the behavior of the mean of the $DE L_{bin}$ in different mean wind speeds and different load channels. The present section includes the study of the standard deviation of $DE L_{bin}$ s. Thus, the two groups of results can provide some insight into the statistical behavior of DEL in general.

The following presents the convergence of the DEL estimate using the bootstrapping technique presented in bootstrapping procedure. Section “Variability of DEL estimation” analyses the convergence of the estimate of $DE L_{bin}$ and $DE L_{lifetime}$ as functions of the number of 10-minute simulations used in each wind speed. Then, Section “Important wind bins” studies the uncertainty propagation from DEL in each mean wind speed $(DE L_{bin})$ to the lifetime DEL $(DE L_{lifetime})$ based on the data and their distribution. The reasons behind the observed trends are explained using the mathematical relations presented in Aeroelastic model. The conventional scaling approach for DEL estimation is the primary approach in the present work. This approach includes the calculation of DEL using 10-minute data and considering the result to be constant over the lifetime.

Variability of DEL estimations

Using the method discussed in bootstrapping procedure, we randomly choose sample sets of different sizes from the full data set of 300 $DE L_{s}$ values. The procedure is repeated 1000 times with replacement for each sample size within different wind bins. In each set, a representative estimator of $DE L_{bin}$ is calculated via equation (4). The 1st, 5th, 50th, 95^th, and 99th percentiles of the 1000 representative values in each bin are presented to track the upper and lower 1% and the upper and lower 5% bounds of the normalized data. We normalize the data by their corresponding mean calculated based on the primary dataset (300 seeds).

Figure 3 shows the variability of $DE L_{bin}$ through different mean wind speeds in the blade’s root. These plots only present the blade as an example among all the components under study. Appendix 1 includes the trends of $DE L_{bin}$ s for the other components. As the lifetime values are of the utmost importance in the design phase, the plots of all components are presented for $DE L_{lifetime}$ later.

Figure 3.

The variation and convergence of the normalized $DE L_{bin}$ based on the blade root’s flapwise bending moment versus sample size—shown by black “*” lines: mean values of the 1000 estimates; blue “square” lines: upper and lower 5%; red lines: upper and lower 1%.

There are several interesting trends that can be observed in Figure 3. First, the variance in the $DE L_{bin}$ estimate is small even for small sample sizes at high wind speeds. However, there is a relatively larger variance of the parameter in the mean wind speeds below 8 m/s. This is due to the fact that in low mean wind speeds, very high load oscillations do not normally occur and thus, the accumulated fatigue damage is a result of many small oscillations. The results of the rainflow method can be very sensitive to the shape of the time series in such cases. Therefore, the estimation of the DEL can be very different for different samples within low mean wind speeds.

The second notable point in Figure 3 is that the skewness in the data at low wind speeds and low sample sizes reduces as the wind speeds or sample sizes grow. One reason for this is the thicker tail of load cycle data at higher wind speeds. The Section “Distributions of the damage estimations” provides more investigations regarding this point.

Finally, from Figure 3 it is generally clear that more than six simulations are required at each wind speed to get an accurate estimate of $DE L_{bin}$ . For example, if one follows the IEC standard of six simulations, the estimate of $DE L_{bin}$ has a 5% chance of being more than 10% off from the value based on 300 seeds at 4 m/s. To ensure a $DE L_{bin}$ estimate that is within 5% of the mean value based on the parent database with 99% accuracy, 50 simulations are needed at low wind speeds. Thus, one could consider a situation where more simulations are used at low wind speeds to achieve better convergence of $DE L_{bin}$ . However, the importance of the convergence of the different $DE L_{bin}$ values is dependent on their magnitude and on the probability of the wind speed bin, which are combined to yield the damage share of the bin. These damage shares are discussed further in Section “Important wind bins.”

Figure 4 represents the change in the lifetime damage-equivalent load $(DE L_{lifetime})$ estimations versus the sample size (number of wind visualizations) in different components. The $DE L_{lifetime}$ in Figure 4 is the aggregation of $DE L_{bin}$ s weighted by their corresponding bin probabilities (see equation (7)). As mentioned before, we normalize the data based on the average $DE L_{lifetime}$ calculated via 300 seeds.

Figure 4.

The variation and convergence of the normalized $DE L_{lifetime}$ versus sample size—shown by black “*” lines: mean values of the 1000 estimates; blue “square” lines: upper and lower 5%; red lines: upper and lower 1% for (a) blade root, (b) tower base, (c) tower top (yaw bearing), and (d) shaft (torsion).

Figure 4 shows the general decrease in the variability of the $DE L_{lifetime}$ data with an increase in sample size. This decrease is expected based on the law of large numbers (Révész, 2014). The mean level of the bootstrapped samples is very close to the mean value based on 300 seeds as shown by the black line converging to 1 from 20 seeds. This means that the mean is unbiased in the sample sizes above six seeds. In addition, the higher and lower bounds are within the 10% range in all the components when the sample size is 100. This trend shows that the realizations of DEL based on samples of 100 seeds or higher cannot be biased by more than 10%. Since the bounds are converging to the mean value at 250 seeds by less than 1%, we claim that the convergence is occurring at even 250 sample size by 1% error. Thus, our primary data set (300 seeds) is big enough.

The resonance within two wind speed bins in the tower base increases the variability in lifetime DEL realizations. In fact, the variability of the data within the impacted bins (6 and 8 m/s) has a high share in forming the variability of $DE L_{lifetime}$ because of their high wind bin probability (see equation (9)).

One of the reasons for the relatively low variability of $DE L_{lifetime}$ in the case of the blade is due to the high fatigue exponent (m = 10). The parameter $var (DE L_{bin})$ is less than unity in all wind bins and thus $var (DE L_{bin})^{m}$ has a much lower value. The summation of $\Pr (bin) var (DE L_{bin})^{m}$ s has a total lower value than each component (see equation (9)).

Part of the reason that the shaft torsion DEL has a larger normalized variability in Figure 4 is that its mean DEL is small compared to the other load channels. Another source of variations in the shaft torsion is the dependency on generator torque. The generator torque is proportional to the square of rotational speed in the mean wind speeds lower than the rated mean wind speed (Manwell et al., 2010). Thus, especially in low mean wind speeds, the variations from one simulation to another are considerable, which ultimately results in DELs with significant variation.

To study the trends in Figures 3 and 4 more precisely, we derive the coefficients of variation (CoV) of $DE L_{bin}$ and $DE L_{lifetime}$ data.

Figure 5 shows the coefficients of variation of $DE L_{bin}$ and $DE L_{lifetime}$ versus sample size within different components.

Figure 5.

The logarithm of coefficients of variation of $DE L_{lifetime}$ (black, “*” line) and $DE L_{bin}$ (colored lines) versus sample size in (a) blade, (b) tower base, (c) tower top (yaw bearing), and (d) shaft (torsion).

According to the plots in Figure 5, the variability of DELs in the mean wind speeds between 4 and and 8 m/s is mostly higher than the other mean wind speeds as observed previously in Figure 3. The second high level occurs in the range of 10–14 m/s and the rest of the bins show a lower CoV in their corresponding $DE L_{bin}$ data. This trend is a result of less growth in standard deviation (the numerator of CoV), which is positively correlated with DEL, in comparison to the growth of the mean value (the denominator of CoV) as the mean wind speed grows.

In the case of the tower base, the CoV of $DE L_{bin}$ in bins of 6 and 8 m/s is almost as high as in 4 m/s, this is because of the resonance and thus high DEL values are seen in Figure 2. High DEL levels turn to high standard deviation and thus, relatively high CoV values for the mentioned bins in the tower base. In the case of shaft torsion, the relatively slower growth of DEL concerning the growth of mean wind speed, as the mild slope in Figure 2(d) demonstrates, causes the CoV of some wind bins to be higher than 4 and 6 m/s.

In addition, the coefficient of variation of the $DE L_{lifetime}$ in the blade is lower than in other components with the same sample size. The spread of variance in different bins can be one reason. We will study this aspect in more detail in the next sections as we investigate damage shares of different wind bins.

In addition, in all of the cases, there is a fast decrease in the CoV with an increase in sample size from 6 to 20. Another interesting trend in the above plots is that in each component, the variation in the $DE L_{lifetime}$ is lower than the variation of $DE L_{bin}$ s in any sample size. This can be explained by equation (7) which defines the relationship between all realizations of $DE L_{lifetime}$ and $DE L_{bin}$ s. If we consider the DEL estimation based on 300 seeds close enough to the true mean value, Figure 2 shows a positive correlation between mean levels of $DE L_{bin}$ s. However, the corresponding variances are independent and thus uncorrelated since each realization of DEL is based on different realizations of wind which are completely random. Therefore, since different realizations of $DE L_{bin}$ s are also independent, we can consider equation (7), as a weighted sum of many independent variables. Considering having only one bin, the variance is a certain value. This value decreases by adding more data (data from the second bin) according to the law of large numbers. Thus, the weighted sum of all the ${DEL}_{bin}^{m}$ s has less variance than the variance of every single group of data (single bin). This fact implies that the bias in the estimation of $DE L_{bin}$ s will decrease as we estimate the $DE L_{lifetime}$ based on them. This effect can also be observed from the difference in the range of bounds in Figures 3 and 4(a).

The general conclusion is that although the higher turbulence standard deviation in the higher mean wind speeds, causes a higher level of mean fatigue damage at those mean wind speeds (see Figure 2), the variability of the DEL estimations is higher in low mean speeds. In other words, one must use relatively higher sample sizes within low mean wind speeds to obtain the same accuracy of DEL estimation as higher mean wind speeds. The necessity of this shift in sample size should be studied by identifying the important wind bins since the overall variability of $DE L_{lifetime}$ can be less than $DE L_{bin}$ s as seen in Figure 5. The next section contains such a study for the current wind turbine.

Important wind bins

To understand the share of each mean wind speed in the lifetime damage and thus identify the most important bins, we calculate the damage density by normalizing the damage in each mean wind speed by the lifetime damage. We calculate the damage density fraction via equation (8), considering ${DEL}_{bin}^{m}$ and ${DEL}_{lifetime}^{m}$ as the representatives of damage in each bin and lifetime damage, respectively. Thus, the relative share of bins depends on the magnitude of their corresponding DEL $(DE L_{bin})$ and the probability of their occurrence $(\Pr (bin))$ . In addition, the fatigue exponent $(m)$ plays a vital role in the trends.

Figure 6.visualizes the normalized damage shares of different wind bins for the four load channels of interest. It also reveals the contribution of each bin to the uncertainty of the $DE L_{lifetime}$ estimation.

Figure 6.

The normalized share of each wind speed bin (damage density) in overall damage (blue columns) in (a) blade, (b) tower base, (c) tower top (yaw bearing), and (d) shaft (torsion) plus probability density function of each bin (red dash-line).

According to Figure 6, in the case of the blade, the mean wind speeds of 4, 6, and 8 m/s have negligible shares in the lifetime damage. This is because of the low magnitude of both the $DE L_{bin}$ values and the corresponding wind bin probabilities. Furthermore, there is no apparent trend from 10 m/s up to the cut-off mean wind speed as the high $DE L_{bin}$ s cancel out with the low wind bin probabilities in equation (7). However, the most significant share of damage belongs to the 22 m/s wind bin. Although the probability of occurrence of the 22 m/s wind bin is very low (red dashed line in Figure 6), the very high magnitude of its corresponding $DE L_{b} in$ (see Figure 2) makes its share in the estimation of $DE L_{lifetime}$ relatively high.

In the tower base in Figure 6, as the $DE L_{bin}$ s have the highest values in 6 and 8 m/s (see DEL trend in the case study wind turbine) because of resonance, the combination with high probabilities of occurrence in these bins, makes the corresponding shares significantly high (see equation (7)). In this load channel, the probability of the wind bins and the magnitudes of the corresponding $DE L_{bin}$ s, and thus the damage shares are correlated.

Figure 6 shows that in the shaft the wind speed bins of 14 and 16 m/s take the highest shares in the lifetime damage because of the same combination of bin probability and magnitudes of the bins in this component (see equation (7)).

In the tower top, the peak values belong to the wind bins of 12 and 14 m/s for their higher probability. The distribution of the damage densities among different bins in this channel is more spread than all other channels except the blade root.

According to Figure 6, the damage in the blade is derived from eight different bins as their shares are not drastically apart. However, in the tower base and the shaft (especially the former), the damage is mainly a function of two or three bins. In general, the lower number of variables involved in a load channel results in higher variability/sensitivity in its $DE L_{lifetime}$ variability. It is important to remember that other factors, such as the nature of the loads, can also have an impact. However, the more components involved, the more robust the outcome (lifetime damage here) is. The present results explain one of the causes of the lower variability of the blade’s flapwise DEL compared to the other loads shown in Figures 4 and 5.

Comparing the peak of the probability density function of wind with different peaks of damage density in each load channel in Figure 6 clarifies the importance of checking damage densities instead of only considering the most occurring winds as is commonly seen in research areas such as, for example, controls. In the case of the blade, the correlation between important mean wind speeds and the wind speeds with a higher probability is almost zero. This is a good example of the extent of the error in following the current common practice in optimization problems when aiming for maximum energy outtake and/or minimum fatigue damage.

By identifying the wind bins with the highest damage density, the designer can consider a higher resolution in binning or a larger number of samples around these mean wind speeds. It is also possible to use a more robust method (conventional or extrapolation-based) to estimate the damage in these important bins. Such a procedure allows for decreasing the uncertainty in the lifetime damage assessments with minimum computational cost. One should notice that this is an optimal approach in turbines where most of the important wind bins are common among different load channels. However, such a condition may be rare.

Having identified the behavior of the expectations and variations of DEL in different load channels, different wind speeds, and different sample sizes sections “DEL trends” and “Study of DEL estimations using different sample sizes,” identifying the probability distribution of damages in the Section “Distributions of the damage estimations” completes the chain of tools needed for anticipating the uncertainty in fatigue damage estimation due use of limited data and controlling it.

Distributions of the damage estimations

In the current section, we study the statistical characteristics of damage in each wind bin $({DEL}_{bin}^{m})$ since they govern the statistical behavior of the accumulated damage ${DEL}_{bin}^{m}$ according to equations (7) and (9). We investigate the best probability distributions for modeling ${DEL}_{bin}^{m}$ s in different components and study the statistical characteristics based on the fittings.

We generate 3000 ${DEL}_{bin}^{m}$ data (within each sample size) using bootstrapping from 300 seeds in each wind bin. First, we study the assumption of Gaussian damage in different mean wind speeds in all components via the null hypothesis of the Kolmogorov–Smirnov (KS) test (Berger and Zhou, 2014) with a 95% confidence interval. Then, we utilize the probability plots to show the extent of deviation from Gaussian on one example. We use maximum likelihood to find out the best fit among candidate distributions.

Table 2 presents the results of the Gaussian assumption in different sample sizes for all components.

Table 2.

Investigations of Gaussian fit for the damage in bins $({DEL}_{bin}^{m})$ in different sample sizes and different components based on null hypothesis of KS test (5% significance level).

Number of seeds in all wind bins	Blade root (m = 10)	Tower base (m = 3)	Tower top (m = 3)	Shaft (m = 3)
6	Non-Gaussian	Non-Gaussian	Non-Gaussian	Non-Gaussian
20	Non-Gaussian	Non-Gaussian	Non-Gaussian	Non-Gaussian
50	Non-Gaussian	Non-Gaussian	Non-Gaussian	Non-Gaussian
100	Gaussian in most bins	Gaussian in most bins	Gaussian	Gaussian
200	Gaussian	Gaussian	Gaussian	Gaussian

According to Table 2, the assumption of Gaussian damage is not valid for small sample sizes which are often used and are recommended by the IEC current standard. The convergence rate of the DEL/damage distribution to Gaussian is relatively slower in the blade and the tower base. The results affirm that the higher fatigue exponent in the blade contributes to relatively high skewness in its corresponding damage distribution. According to Low (2012a), the higher damping level in the case of the tower base causes a high skewness in the loads and, consequently, the corresponding DEL. Therefore, based on the central limit theorem, the convergence of the distribution of the mean of the samples to the normal distribution is slower in the tower base.

Figure 7 shows the probability plots of the ${DEL}_{bin}^{m}$ data of different components in their most critical wind bins (see Figure 6).

Figure 7.

Probability plots of ${DEL}_{bin}^{m}$ data in (a) blade root (22 m/s), (b) tower base (8 m/s), (c) tower top (12 m/s) in sample sizes of 6, 20, and 50.

The plot in Figure 7 shows the probability plots of ${DEL}_{bin}^{m}$ data. A normal distribution would appear as a straight line. The blue dash lines represent the normal distribution fits, and the red dash lines show log-normal distributions. However, the divergence of the plots from the straight line shows that the data in different sample sizes are not following a Gaussian distribution. In addition, the plots show that with an increase in the sample size, the corresponding distribution converges to normal. In addition, in the case of the blade, the convex shape of the plots is more visible. This has compliance with the data in Table 2. In addition, the mean values are not far from each other in different sample sizes, but the curves are more distanced in the lower and upper tails. This means that the higher moments like skewness change with a change in the sample size.

The results presented in Figure 7 show the competence of lognormal distribution for describing the probability of damage data in the most important mean wind speeds of the case study load channels (for more details see the Appendix 1). According to the results of previous sections, 20 seeds are more desirable in the important mean wind speeds, making the goodness of fit acceptable. The next section contains more detailed discussions about each of the three studies presented in the current paper.

Discussion

The important wind bins in the estimation of the fatigue damage are not equivalent in different load channels. Since the wind turbine is an integrated dynamic system, the designer should identify the overall critical bins considering all channels to gain the maximum portion of the variance in the lifetime damage due to limited data. The overall information in the present case study shows acceptable convergence of the damage in the most important mean wind bins when using up to 50 seeds. For example, in the case of the blade, using six seeds in the wind speeds of 4–8 m/s, 50 seeds in the wind bins of 22, 18, and 14 m/s, and 20 seeds in the other bins, one can estimate the lifetime damage with acceptable accuracy without paying the computational cost of 50 seeds for all wind conditions. If the tower base resonance does not occur, it is likely that for the case study wind turbine, the designer can increase seeds in the mean wind speeds of 12, 14, and 16 m/s as overall critical wind bins for the whole system as they show a lower convergence rate.

The other main observation is the high variability of lower mean wind speeds compared to higher mean wind speeds in different components.

There is a general similarity among the tower’s fore-aft moment, the blade’s flapwise moments, and tower top pitch moment as they are all mainly driven by wind, and no periodic components dominate the load time series. The only exception in the tower base is because of the resonance and is case-dependent.

Lognormal distribution shows relatively good compatibility with the damage data CDF at each wind speed (see Appendix 1 for some examples). The fatigue exponent is enforcing some levels of skewness to the more symmetric distribution of DEL data in wind bins when calculating the damage. The higher skewness in some loads and in channels with higher fatigue exponents causes slower convergence of the distribution of the expected value to Gaussian. Thus, investigating the validity of the lognormal distribution for the damage data—as a skewed version of a normally distributed parameter (DEL) when powered to an exponent—is important to consider when using a lower number of samples. Based on equation (9) and provided insights on the statistical behavior of the damage, one can assess the level of statistical uncertainty due to limited data in each bin and its propagation from the bins to the lifetime estimations. This is possible when using the central limit theorem relationship for anticipation of the change in variance with a change in sample size (for sample sizes above 30).

The variance of the lifetime damage is positively correlated with the variance in each mean wind speed to power of “m” and with the probability of occurrence of each mean wind speed to the power of two, increasing the relative importance of highly variable loads with low probability in the overall variance (especially in the case of high fatigue exponents). As shown in the results, the mean wind speed of 22 m/s shows the most significant share of damage in the blade case which does not match the peak of the wind probability plot. This result shows that considering only high-probability wind bins in different tasks like energy production optimizations, controller designs, etc., can result in very high errors and wrong assessments.

The more bins are involved in the estimation of the lifetime damage, the more robust the estimation. The data showing relatively less variability in the case of the blade’s lifetime damage confirms this fact. In this case, the share of crucial mean wind speed (22 m/s) is not drastically higher than the rest of the bins. However, in the tower base, mean wind speeds of 6 and 8 m/s have the most significant shares in the estimation of the lifetime damage because of their high mean level value caused by resonance. Thus, the variability of the $DE L_{lifetime}$ estimations is higher in the case of the tower base. In such cases, more seeds are required for the same level of accuracy. The tower top’s most critical wind bins for the current wind turbine are 12 and 14 m/s. In the shaft, the wind bins of 14 and 16 m/s have the most significant shares in the damage calculations. The data relating to the tower base and the shaft show relatively higher variability because of relying on fewer bins. In other words, a slight change in one or two bins can directly transform into a higher variance of the lifetime DEL.

In the case of the blade’s flapwise moment, partly because of high fatigue exponents, most of the energy is focused around the first frequency, and a large portion of the fatigue damage occurs around it. As a result, this load is sometimes modeled with a narrow band process in this area as a simplification. Knowing the number of data needed for the Gaussian damage assumption to hold based on the current study results, one can use the simplified relations available in the literature for assessing fatigue damage caused by narrow-band processes. The same approaches for very narrow-band processes presented in some of the references for some mechanical components (mentioned in Introduction) can be used for the wind turbines utilizing the understanding and knowledge provided by the current study.

Conclusions

The current work investigates the effects of finite sampling on the variability of the estimation, variance, and skewness of the lifetime damage-equivalent load in various components based on the DTU 10-MW case study wind turbine. The investigations include the rate of convergence of the damage toward the ultimate mean value in each wind bin plus the importance of each bin based on damage shares. In addition, the lognormal distribution is fitted to the damage data in wind bins when using small sample sizes showing the skewness in damage data caused by fatigue exponent. The above information allows for tracking the uncertainty propagation from damage in wind bins to lifetime damage. The provided insights in the current study allow for better identification of the damage variance due to finite data which can be used in reliability assessments. The current work links between three elements: identification of the important bins, the change of their uncertainty with a change in sample size, and the share of the bins in the overall damage uncertainty. These elements form a framework for decreasing the error while keeping computational efficiency with a focus on more critical wind bins.

Current work studies a part of the statistical uncertainty of fatigue loads. Linking the results with other uncertainty sources can be useful. In addition, modeling the uncertainty associated with using Miner’s rule and rainflow counting models and with material properties, which have a more significant share in the damage’s uncertainty, can make the fatigue assessment more accurate. An inclusive framework including all sources of uncertainty would provide a more robust tool for assessing the fatigue reliability of the components. Furthermore, the current work is based on a conventional damage estimation which means scaling the average 10-minute damage to the lifetime period. Considering the same study, using statistical extrapolation of the loads and separating the portion of the uncertainty caused by limited data from the other sources of statistical uncertainty is valuable.

Footnotes

Appendix 1 Acknowledgements

This class file was developed by Sunrise Setting Ltd, Brixham, Devon, UK. Website: .

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Shadan Mozafari

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