Wind turbine extreme load estimation is especially difficult because turbulent inflow drives nonlinear turbine physics and control strategies; thus there can be huge differences in turbine response to essentially equivalent environmental conditions. The two main current approaches, extrapolation and Monte Carlo sampling, are both unsatisfying: extrapolation-based methods are dangerous because by definition they make predictions outside the range of available data, but Monte Carlo methods converge too slowly to routinely reach the desired 50-year return period estimates. Thus a search for a better method is warranted. Here we introduce an adaptive stratified importance sampling approach that allows for treating the choice of environmental conditions at which to run simulations as a stochastic optimization problem that minimizes the variance of unbiased estimates of extreme loads. Furthermore, the framework, built on the traditional bin-based approach used in extrapolation methods, provides a close connection between sampling and extrapolation, and thus allows the solution of the stochastic optimization (i.e., the optimal distribution of simulations in different wind speed bins) to guide and recalibrate the extrapolation. Results show that indeed this is a promising approach, as the variance of both the Monte Carlo and extrapolation estimates are reduced quickly by the adaptive procedure. We conclude, however, that due to the extreme response variability in turbine loads to the same environmental conditions, our method and any similar method quickly reaches its fundamental limits, and that therefore our efforts going forward are best spent elucidating the underlying causes of the response variability.

Estimating extreme loads for wind turbines is made especially difficult by
the nonlinear nature of the wind turbine physics combined with the stochastic
nature of the wind resources driving the system. Extreme loads, such as those
experienced when a strong gust passes through the rotor or when a turbine has
to shut down for a grid emergency, can drive the design of the machine in
terms of the material needed to withstand the events. The material
requirements in turn drive wind turbine costs and overall wind plant cost of
energy. Thus, accurate modeling and simulation of extreme loads is crucial in
the wind turbine design process. This paper discusses the use of adaptive
importance sampling (IS) in estimation of such loads. IS

The essential task in wind turbine extreme load estimation is to evaluate
the probability of exceedance (POE) integral

Existing approaches fall generally into two main classes. The first is based on extrapolation: data are gathered in different wind speed bins, extreme value distributions are fit to the empirical distribution function for each bin, and these are then integrated. The second is based on Monte Carlo (MC) methods: exceedance probabilities are written as expectations of indicator functions, samples are drawn from an assumed wind distribution, and unbiased estimates are made by the usual MC summation. Unfortunately, to date neither of these approaches is satisfactory. The crux of the difficulty is that on the one hand too many samples are required for converged MC estimates, but on the other hand reliable extrapolation of nonlinear physics under uncertain forcing is extremely problematic, especially without knowledge of the form (e.g., quadratic) of the nonlinearity. Nevertheless, the computational expense of MC implies that except in rare cases, some sort of extrapolation will be necessary in order to reach the desired 50-year return period estimates. This paper is motivated by the intuition that perhaps we can at least use MC–IS to make sure extrapolations are accurate to the resolution of data we actually have, and to gather data in ways that accelerate their convergence.

The difficulty of estimating these “tail probabilities” of interest in extreme load estimation is one of timescales. We are trying to estimate loads seen roughly once in 50 years using a set of simulations whose total length is only a few hours. This large difference in timescales means that any uncertainty in the data is necessarily magnified by the extrapolation. Small variations in short-term data could lead to significant over- or underestimation of long-term extreme loads.

One of our main conclusions will be that while we may have reasonable
knowledge of the distribution of environmental conditions a turbine faces, we
have very little knowledge regarding the distribution of the response
of the turbine to its environment, and this response variability may in fact
be so large that our knowledge of the distribution of environmental
conditions is of limited use. A conceptual aid is provided by the inverse
first-order reliability method (IFORM)

Our goal is to develop methods that make unbiased estimates that minimize variance as a function of the number of samples and simulations, and to use these to dynamically update extrapolations. Our proposed method, adaptive stratified importance sampling (ASIS), is essentially a global stochastic optimization method in which the search variables are the number of samples from each wind speed bin we use, and the objective function is the variance of our MC estimates. The key tool here is IS, which allows us to continually produce unbiased estimates of exceedance probabilities even as the distribution of bins changes. These quasi-optimal samples are then used to make the best possible extrapolations. Results below show that this is indeed a promising approach.

The organization of this paper is as follows. First we present the necessary
background on the existing extrapolation method (e.g., as recommended in the
International Electrotechnical Commission (IEC) standard

Throughout this paper, we use FAST, NREL's aeroelastic simulation tool

The particular turbine on which we are testing these methods is the NREL
5 MW reference turbine, often used for such studies

For this study, we selected two output channels of interest, tower base side–side bending (“TwrBsMxt” in FAST nomenclature) and tower base fore–aft bending (“TwrBsMyt”), which provide contrast because the wind speeds at which their highest loads occur overlap differently with the typical wind speed distribution. The side–side moments grow with hub height wind speed, making their extremes hard to estimate with traditional MC sampling because they do not overlap well with typical wind distributions. The fore–aft counterparts do overlap quite closely with the typical wind distributions.

The current standard for estimating extreme loads relies on extrapolation. We
refer the reader to the relevant literature for a detailed exposition of the
extrapolation method

Run TurbSim–FAST

For each bin

For each bin

(Optional). For each bin, convert each fitted distribution to the desired timescale (see below regarding peak extraction and timescales).

Finally,

The distributions chosen to fit the bin-wise CDFs are the theoretically appropriate extreme value distributions (generalized extreme value (GEV), three-parameter Weibull, etc.). However, this does not mean they accurately represent the behavior of the particular FAST loads in a specific context. The optimality properties of extreme value distributions are asymptotic properties, but we are performing “intermediate asymptotics”: long-term – but not infinitely long-term – trends. Nevertheless, these distributions are the appropriate starting point.

In this paper we are using a three-parameter Weibull distribution, but this is
not meant as a claim that this choice is better than any other in the
literature (Gumbel, GEV, etc). We used the three-parameter Weibull because we
have used it with success in previous work

Regarding the fitting procedure, in light of the interest in extrapolation,
rather than just fitting, we have fit the empirical CDF of the data directly to the theoretical CDF of the distribution by nonlinear least squares. We have done
this separately for the data from each wind speed bin. Furthermore, in order
to emphasize the largest peaks (i.e., the lowest probability values) we do not
use all the data, just the

It is important to be clear regarding various time spans at play here. First,
there is the ultimate time of interest, typically in wind studies the 50-year return period. This does not mean that in 50 years the event in
question happens with a probability of 1. Sometimes it is loosely defined as an
event having a probability of

The various time spans come into play as we extract peaks and make estimates.
The rule that connects them is the simple AND rule of probability:
if

Box-and-whisker plots of the distribution of the raw response
(specifically, all 1 min maxima) of the combined TurbSim–FAST simulation as
a function of wind speed bin for side–side

In this paper we adopt the following setup: all our simulations are 11 min
long, for which we discard the first minute as a transient and retain the final
10 min for our studies (we will occasionally be loose with the terminology
and refer to these as “10 min simulations”). To gather peaks, we take the
maximum of each 1 min segment in our simulations. This provides exactly
10 peaks per simulation, which allows for building 1 min empirical
CDFs (i.e., probability of exceedance in 1 min)
in a consistent manner. The resulting 1 min empirical POEs are converted to
10 min POEs as described above (i.e.,

To acquire a sense of the basic variability in the response, we have run
20 independent sequences of the simulations described above for the
extrapolation method (six random seeds per bin). Figure

MC methods are widely used to estimate expectations of
quantities calculated using stochastic simulations

For us

Although the estimates above in Eqs. (

For our purposes, finally, note that Eq. (

The following section is not essential to understanding our adaptive
IS algorithm and may be considered optional. However, we
believe it provides useful conceptual context, especially to understand the
limits of statistical methods for systems whose response variability is
large. Here, keeping in mind the goal – minimal variance unbiased estimates of
extreme loads through IS, minimizing the use of
extrapolation – we summarize the IFORM and EC methods. IFORM was introduced by
Winterstein

In the general IFORM approach, the combined environmental and response
variable space is considered to be one joint probability distribution, and
the quantile corresponding to the desired return period is explored to find
the maximal response. In practice, the distribution of the environmental part
of this combined space is assumed known. For example, in this paper, the
environmental component is wind speed, which is assumed to have a Weibull
distribution with a shape and scale of 11.28 m s

The EC variant of IFORM explicitly separates environment from response. It works best if the response of interest is a completely deterministic function of environmental conditions that themselves have known probability. Then one can directly search the environmental contour (e.g., all wind–wave–turbulence combinations that occur on average once in 50 years) to find the highest load. Otherwise, a conditional distribution of response subject to environment can model the response variability away from its median; in this case, EC is then similar to IFORM in practice. Together, IFORM and EC solidify the important notion of response variability: the magnitude of the variation in the nonlinear stochastic response for fixed environmental conditions.

These notions help to explain why IFORM or EC applied to wind turbine extreme
loads estimation may not be much different than other extrapolation methods.
On the one hand, as noted above, though systematic and efficient, IFORM
relies on extrapolation. Just like the standard extrapolation method, it
relies on being able to extrapolate from easily observable quantiles (5th,
25th, 50th, 75th, 95th, etc.) to the very difficult to observe quantiles
corresponding to the 50-year return period. On the other hand, EC is not
applicable when the main driver of variation in a system is the response
variability, which (see Fig.

The present task, that of estimating extreme loads with wind speed as the
only environmental variable, is governed mostly by the response variation.
Therefore we will not estimate extreme loads by IFORM in this paper. However,
IFORM is critical for conceptual understanding: where possible, our goal
should be to convert response variation (intractable) to environmental
variation (tractable) through better understanding of its physical cause. It
is the extreme response variability (different random seeds for the same
environmental conditions can cause very different FAST output because they
cause very different turbulent inflow) that makes extreme load estimation a
difficult problem. We return to this subject in Sect.

Weibull three-parameter fits to the empirical CDF for the wind speed bin
centered at 20 m s

When we perform the bin-wise simulations used in the extrapolation methods,
we are performing stratified sampling. Recognizing that these
samples can be described as a probability distribution provides a bridge to
using them in an IS context, as discussed in

Letting

To apply this formula to data from the binning method, we need the
appropriate

Bin-wise empirical cumulative distribution functions provide a
bridge from extrapolation, which builds POE from bin-wise fitted
distributions

Summarizing the convergence of tower base load estimates using
extrapolation and ASIS estimates over 100 independent runs. The top row is
side–side; the bottom is fore–aft. The

Behavior of ASIS and iterative updating of extrapolation for
side–side tower base bending load over 20 separate runs as a function of

Behavior of ASIS and iterative updating of extrapolation for
fore–aft tower base load over 20 separate runs. The

Thus we have a “bridge” between fitting and sampling. Bin-wise empirical CDFs can be directly compared with fitted distributions (in fact, they are what we fit to). But the estimate over all wind speeds can then be expressed generically in the IS language suited to comparison with MC and IS estimates. IS–MC methods do not provide any bin-wise information (there are no bins); thus it is otherwise impossible to “debug” their divergence from extrapolation. This formulation allows us to see, first, that error accrues from lack of convergence of empirical CDFs, for both methods. Additionally though, for extrapolation, the error is compounded by lack of fit between the chosen extreme value distribution and the empirical CDFs, which is the price we pay for being able to extrapolate to arbitrarily low POEs with small numbers of samples.

The discussion above indicates that the samples from the bin-based methods
can alternatively be used to make empirical estimates via their implied
importance distributions. This orientation suggests, also, that there is no
barrier to changing the distribution of samples as we go. Thus we can think of
the estimation procedure as an optimization problem: find the distribution of
bins (number of samples per bin) that results in unbiased estimates with
minimal variance. In

Our algorithm begins by running the standard six seeds per bin from the
extrapolation method (i.e.,

Compute

Allocate a target number of new samples (e.g., 20 per iteration) to bins in two ways:

allocate some percentage of the new samples in proportion to

recognize this is a global optimization problem, and allocate the rest to other bins randomly.

Run TurbSim–FAST for the new batch.

Append the new peak data to the existing data and update our empirical estimates of POEs and our
extrapolation estimates using the cumulative data according to
Eqs. (

Note the algorithm as stated does not explicitly recognize the stochasticity
of the underlying quantity

A slight complication comes from the need for the

Next, as stated

Another issue, even for a single load type, is how many of its peaks

Finally, because we are still refining the mechanics of the algorithm, ASIS
as stated does not include a stopping criteria. Since what ASIS minimizes is
the variance of our load estimates, stopping should be based on driving the
variance below a user-defined threshold. The difficulty, of course, is that
unlike a deterministic gradient descent procedure, our only access to the
actual variance is through further statistical estimates. In the results
below we simply repeat the stochastic optimization procedure 100 times and
compute the variance of the estimated loads directly. A less computationally
expensive approach is bootstrapping

In this section we demonstrate the basic mechanics of the algorithm in the
context of a study of
the effect of the number of peaks,

As an exercise, we examine the sensitivity of the ASIS results to

For each of the 100 independent tests, we ran extrapolation and ASIS for
25 iterations (an iteration of extrapolation is simply re-performing the
extrapolation procedure with the current ASIS bin data), which adds a varying
number of new samples to each bin at each iteration. The most obvious
observation is that indeed the variance of the estimates decreases
quickly as a function of iteration. ASIS reliably drives the variance of the
estimates of POE down, and simply recalculating the extrapolations to keep
up with ASIS drives the variance of the extrapolation estimates down as
well. There is a slight dependence on

Thus our adaptive extrapolation approach appears capable of reducing variance somewhat dramatically with minimal additional computation. The two approaches maintain correspondence even while adapting the bin distribution, which allows for leveraging the variance reduction of the empirical ASIS estimate to reduce the variance of the extrapolation estimate. And the latter is the estimate of real importance because that is what will be used in practice. Note that ASIS could as well drive IFORM estimates instead of the traditional extrapolation estimates. In both cases ASIS optimizes the distribution of samples (wind speed bins in this case) that are then used to fit statistical distributions, which are then used to extrapolate to desired return periods.

In this paper we have built a bridge between bin-based extrapolative methods and sample-based IS MC methods. With this, we proposed an adaptive stratified importance sampling (ASIS) algorithm that is both more efficient than existing MC approaches and maintains contact with the extrapolation methods and thereby allows for iteratively increasing the extrapolation accuracy. This is important because only the extrapolations are able to routinely make estimates of extremely long return period load exceedance probabilities.

The search for the optimal importance distribution is a stochastic optimization problem. As stated above, our algorithm is a convergent algorithm. But stochastic optimization is an active area of research, and more sophisticated algorithms may exist to improve our approach. We need to keep in mind, however, that the optimization problem is a means to an end. The real goal is minimal variance estimates with the smallest amount of effort. We want to use the optimal importance distribution at the same time as we are discovering it. In relation, we need to also keep in mind that we have the dual mission of both efficiently estimating the load POEs and accurately estimating their variance. We can use the peaks we sample to make unbiased estimates of variance just as we do expectation, but these are only estimates, and they themselves suffer from lack of convergence. The resampling method of bootstrapping described above offers a way to leverage a single dataset to estimate statistics and their variance, and in a practical setting this would be recommended (as opposed to the completely separate runs we have described above).

In principle there is no barrier to application of ASIS to higher-dimensional
problems. In particular, it is well known that turbulence intensity and
turbulence standard deviation have a large role in wind turbine extreme loads

This problem may be ripe for a machine learning approach: the physics is in the solver. To the extent it is possible, we should be able to learn from increasing numbers of data. For this, we need accurate variance estimation methods that can build a loss function for learning algorithms that examine data and decide how to process them to make the best next estimate, and to choose the best next places to sample; here we have presented a framework for extrapolating from such data that allows for learning the best extrapolation strategy from the variance minimization algorithm.

Conversely, we should realize there is a physical source of extreme response variation, which is the combination of turbulent inflow and nonlinear turbine response. By “opening up the black box”, i.e., circling back to the original physics, we hope to transfer what in the present setup is response variability into the realm of environmental variability, at which point we can use its probability distribution to hone in on just the loads of interest (i.e., the extreme loads) more quickly. Further studies into the root causes of extreme response variation in wind turbine loads and their ultimate incorporation into more efficient statistical extreme load estimation are ongoing.

The underlying data generated in the course of this study consist of hundreds of thousands of FAST output files amounting to hundreds of gigabytes of data. These data are not available in a public repository. Readers interested in the underlying data should contact the author.

The authors declare that they have no conflict of interest.

The authors would like to express their gratitude to professor Lance Manual for conversations on this topic, especially his insights into the IFORM and EC methods, as well as his extremely thoughtful reading of the paper. Edited by: Michael Muskulus Reviewed by: Lance Manuel, Lars Einar S. Stieng, Nikolay Dimitrov, and one anonymous referee