The large amount of computational effort required for a full fatigue assessment of offshore wind turbine support structures under operational conditions can make these analyses prohibitive, especially for applications like design optimization, for which the analysis would have to be repeated for each iteration of the process. To combat this issue, we present a simple procedure for reducing the number of load cases required for an accurate fatigue assessment. After training on one full fatigue analysis of a base design, the method can be applied to establish a deterministic, reduced sampling set to be used for a family of related designs. The method is based on sorting the load cases by their severity, measured as the product of fatigue damage and probability of occurrence, and then calculating the relative error resulting from using only the most severe load cases to estimate the total fatigue damage. By assuming this error to be approximately constant, one can then estimate the fatigue damage of other designs using just these load cases. The method yields a maximum error of about 6 % when using around 30 load cases (out of 3647) and, for most cases, errors of less than 1 %–2 % can be expected for sample sizes in the range 15–60. One of the main points in favor of the method is its simplicity when compared to more advanced sampling-based approaches. Though there are possibilities for further improvements, the presented version of the method can be used without further modifications and is especially useful for design optimization and preliminary design. We end the paper by noting some possibilities for future work that extend or improve upon the method.

The large number of environmental states that need to be considered for
the design of offshore wind turbine support structures is a significant
challenge. A simulation is required for each such state, often referred to as
a load case, when analyzing the response of these structures to the offshore
environment. Each simulation of this kind, at least when carried out with
accurate aero-elastic software, is a nontrivial task in terms of
computational effort. Assessing the structural performance in the fatigue
limit states for operational conditions alone typically means thousands of
load cases when following relevant standards

Several previous studies in the area of simplifying fatigue assessment
through load case reduction have been carried out.

Multiple studies of load case reduction have also been conducted for floating
support structures.

While achieving various degrees of success in terms of accuracy and ability
to reduce the computational effort, a common trait in most of the cited
studies above is that their aims differ slightly from ours. These studies,
the one by

The method proposed in this study, like in many of the cited studies above, is based on the idea that there is a large amount of information about the total fatigue damage contained in a small subset of load cases. Furthermore, a fundamental assumption for this method is that the relative fatigue response to each load case remains approximately constant for an extended family of related support structure designs. This makes it possible to train the method on one full fatigue analysis, using the complete set of load cases, and then use the method to propose which load cases should be assessed for future analyses of designs that have been modified. The method itself is based on sorting the load cases by their contribution to the total fatigue damage and then obtaining the partial sum of their contributions, up to a certain, smaller number of load cases. The relative difference between this partial sum and the total fatigue damage is assumed to be constant when the underlying support structure design is modified. From the corresponding partial sum of any new design, multiplied by a scale factor derived from the original relative difference, the total fatigue damage of that design can then be obtained. Hence, using an approach relying simply on sorting and summation, an estimate for the total fatigue damage based on a significantly reduced set of load cases is readily available.

Even when restricting the area of study to operational loading conditions and
fatigue analysis for the support structure, there is a substantial amount of
work that has to be carried out in order to verify that the structure
satisfies design requirements. Keeping in accordance with the standards means
covering a lot of different environmental conditions

From Eq. (

Illustration of the model used in this study

Step-by-step summary of the estimation method.

If we only wanted to know the total fatigue damage at a single location in
the structure, Eq. (

By using one full fatigue assessment of a base design, we can then train our
method on these data. Sorting the load cases by the severity at each location
and then taking the union of the resulting sets, we obtain the sampling set

Normalized severity per load case at the tower bottom, with load cases
separated into different wind speed bins, with the 25 most severe load cases
specially marked

For the simulations used in this study we have used the fully integrated
aero-elastic software tool FEDEM Windpower

As noted previously, one of the main motivations for this study has been applications to design optimization. Hence, we
have found it pertinent to test our method in a setting that would resemble situations likely to be encountered during
an optimization loop. Starting with an initial support structure design on which the method is trained, how well would
the method perform in predicting the fatigue damage of the modified designs encountered during the optimization? In
other words, we want to see how the method performs for designs that correspond to configurations that might represent
intermediate steps, or even something close to a solution, of a design optimization problem. This prompts a few
different strategies for how to obtain these modified designs. First of all, the type of optimization framework we want
to investigate here is mass (or weight) optimization. In this framework, the diameters and thicknesses of various
elements are changed until the design is as light as possible, while satisfying certain constraints on structural
performance. To see how the method would perform during an optimization procedure of this type, we chose designs
for which the element diameters and thicknesses had been scaled either up or down compared to an original design. To represent
different types of scenarios, the scaling was done both systematically across the entire structure and randomly from
element to element. For each of these strategies and for two different magnitudes of scaling, the elements of the
structure were scaled once according to the given strategy, and a new design was thus obtained. In total, seven new
designs were generated. Their names (for easy reference later) and quick summaries of how each design was scaled
are given in Table

Relative errors,

The modified designs used in this study, with names and how they have been modified (scaled).

As an initial point of entry, we may ask which of the load cases are in fact
the most severe for the base design and hence which ones will be sampled by
the method. From the distribution shown in
Fig.

For each support structure design listed in Table

Relative errors,

In Fig.

Relative errors,

In Fig.

The relative errors,

Relative errors,

In Fig.

Values of

When we initially defined the method, it was based on a basic assumption:
that the relative error when using only the

As seen above, the proposed method is able to predict the total fatigue
damage of the modified designs with a high degree of accuracy. With the
exception of design MI10, all estimates eventually converge towards errors of
2 % or less (in some cases much less) and with drastic reductions in the
load case set (factors of 50–200 in most cases). Even for the case of MI10,
for which the error is about 4 %–6 % for all but the smallest sample
sizes, this result is quite convincing in terms of the level of accuracy that
can be expected for such an approach given the extent of the modifications to
the structural models. In fact, higher accuracy than that reported for design
MI10 might not even be required. A 5 % error in the prediction of total
fatigue damage represents a change in the lifetime of a support structure by
1 year if the real expected lifetime is 20 years. This is certainly within
the range of other types of errors one might expect in terms of uncertainties
in the modeling or the environmental conditions, both of which are usually
accounted for by multiplying the total fatigue damage by partial safety
factors of 2–3. In such a framework, errors on the order of 10 % might
even be acceptable, in which case a very large load case reduction is
possible for all models. Additionally, there seems to be a clear connection
between consistent changes to the size (mass) of the structural elements and
whether the estimates for the fatigue over- or underpredict the true value.
In fact, the two properties are directly correlated. Though in practice the
consequence of underpredicted fatigue damage is much more severe than
overpredicted fatigue damage, the fact that the correlation is as visible as
indicated in Figs.

One of the reasons the method is as efficient as it is when analyzing more
than one location in the structure is the behavior seen in
Fig.

The methodology has been shown to be quite effective for a range of different
support structure designs, but there is one limitation that should be noted:
the presented results were all obtained while using the same turbine model.
The turbine model will have a very important impact on global dynamics, e.g., 1P and 3P frequencies and the total system mass, damping and stiffness, and
it is hence likely that changing turbines would induce changes in the fatigue
distribution that could be challenging for the method to handle, for example,
if severe resonance effects are encountered. On the other hand, a support
structure design is usually constructed with a specific turbine model in mind
and the results have shown that the method can handle significant changes to
global dynamics to a certain extent (as seen for model MI10 in
Fig.

Another possible limitation, at least for some applications of the method, is
that the results here have been derived using only normal stress. On the one
hand, this is standard practice in the industry and therefore also for many
research applications. Furthermore, the methodology has for some time been
seen to give fairly accurate (often conservative) fatigue estimates for
applications in the oil and gas industry (see, e.g.,

One of the most discernible outcomes of the testing framework is the
indication that the method works best for designs that have been randomly
modified, as seen when comparing Fig.

The various design configurations that were used to test the method were chosen in an attempt to cover as many scenarios of interest as possible. However, not all types of scenarios could be accommodated and hence there are some configurations about which we cannot make strong conclusions. The most obvious of these is the fact that we have scaled both diameters and thicknesses by the same factor. Even for the randomized designs, the scale factor was only randomly sampled on an element-wise basis. A situation in which either diameters are increased and thicknesses decreased or vice versa could easily occur in practice during design optimization. On the other hand, based on our result, it seems that the most significant factor in determining the effectiveness of the method is whether or not there are global changes in eigenfrequency. Hence, though we are unable to explicitly confirm this based on our results, we expect that even in configurations like that described above (or other potential untested ones), the method should be viable under the same criteria: as long as there are no global changes that induce nonproportional changes in fatigue damage for only a certain subset of load cases, the method performs well.

Comparing the approach taken in this study with most previous work on load
case reduction, in particular the studies cited in the introduction of this
paper, one of the main advantages is the simplicity of the method. Because
most of the other studies (e.g.,

The simplicity of the method might also suggest the possibility of
improvements, at least in some of the scenarios shown. While some attempts at
applying sequence acceleration techniques were made, with little or no
positive effects (hence why this was not shown), it is certainly possible
that such approaches, or similar ideas, might decrease the error of the
estimates or at least decrease the number of samples needed to reach a
certain level. We additionally note that further ideas of how to apply the
method for specific applications could also be developed. For example, since
systematic design modifications of a certain size can impact the accuracy of
the method, as seen especially for design MI10 in Fig.

One limitation of the results obtained in this study is the fact that only
operational loading conditions (power production) were analyzed. Since many
other conditions are relevant for design, it would be pertinent to ask whether
the method could be extended to these cases as well. Based on the results
obtained here, it seems clear that the effectiveness of the method in these
other scenarios would depend on whether or not the fatigue damage also changes proportionally in
these cases when the design is modified. If this
property still holds, then most likely the error level when using only the

In this study we have presented a simple approach for reducing the number of load cases required for accurate fatigue assessment of an offshore wind turbine support structure under operational conditions. By making a simple assumption about the relative error incurred by only using the most severe load cases in the total fatigue sum, specifically that this error remains approximately constant as the design is modified, we are able to make accurate predictions for the fatigue damage of a set of seven modified designs. One key part of the method is that the ordering of the severity of each load case is slightly different from location to location. Hence, we have used the union of the reduced sets at each location to form a total sampling set that is used in the method. While slightly increasing the number of samples needed, this has a significant impact on the overall performance in terms of balancing the accuracy at each location in the structure. The overall results of the method are very promising, achieving errors of a few percent or less for sample sizes of 15–60, depending on how the designs have been modified. Only in one case, for which the increased dimensions of the design caused significant changes in the eigenfrequency and subsequent dynamic amplification for some wind speeds, were the errors a bit higher, though they were still less than 6 % in this case for comparable sample sizes. Considering that even a sample size of 100 means a reduction of the load case set (initially numbering 3647) of about a factor of 36, the method generally allows for very large savings in computational effort for fatigue assessment. The method is particularly effective for designs for which modifications have been made randomly from element to element, achieving errors of less than 1 % for reasonably small sample sizes. This, in particular, though also the overall performance, makes the method useful for applications to design optimization. The fact that the method seems to consistently under- or overpredict the fatigue damage based on whether the design has been consistently scaled up or down even makes it possible in some situations to further correct the estimates in order to ensure that the method is always conservative.

One clear advantage compared to state-of-the-art approaches for load case reduction, aside from the overall accuracy, is the simplicity of the method. Whereas the most common approaches rely on various types of sampling techniques that require some amount of statistical and computational complexity, our approach relies entirely on sorting, the union of small sets (combining and then discarding duplicates) and basic arithmetic. Aside from the overall attractiveness of such simplicity, this makes the method more useful for applications in industry for which complex methodologies can lead to unacceptable bottlenecks in the work flow. The simplicity of the method presented in this study (on both a conceptual and implementation level) could also be attractive for other scientists, who may not be as comfortable with advanced sampling methods.

While the method as is can readily be applied in many settings, some future developments can be envisioned. For example, one could study possibilities for improving the convergence of the estimates or investigate specific ways of applying the method to design optimization that adapts to regimes for which the estimates are expected to lose accuracy. A future study might also look into whether, or to what extent, the method could be extended for use within a probabilistic design or reliability framework. In practice, this would mean seeing whether the fundamental assumption of the method, the invariance of the relative fatigue estimation error when sampling only the most severe load cases, also holds when parameters other than those related to the structural dimensions are altered. Finally, the performances of the method for other support structure types (jackets, floating support structures, etc.), other turbine models and other loading scenarios (other than power production) are all open questions for future work.

The data used for plotting the figures, and corresponding Python scripts to make the plots, are available in the Supplement. The raw fatigue data are available upon request. The underlying raw simulation data are too large to distribute.

The supplement related to this article is available online at:

LESS conceived of the basic ideas, performed the analysis and wrote the paper. MM provided essential feedback throughout the whole process and helped to steer the direction of the analysis, helped with the presentation of the results and gave ideas for the paper.

The authors declare that they have no conflict of interest.

This work has been partly supported by NOWITECH FME (Research Council of Norway, contract no. 193823) and by the Danish Council for Strategic Research through the project “Advancing BeYond Shallow waterS (ABYSS) – Optimal design of offshore wind turbine support structures”. Edited by: Lars Pilgaard Mikkelsen Reviewed by: Jan Häfele and one anonymous referee