The present paper characterizes the performance of non-intrusive uncertainty quantification methods for aeroservoelastic wind turbine analysis. Two different methods are considered, namely non-intrusive polynomial chaos expansion and Kriging. Aleatory uncertainties are associated with the wind inflow characteristics and the blade surface state, on account of soiling and/or erosion, and propagated throughout the aeroservoelastic model of a large conceptual offshore wind turbine.

Results are compared with a brute-force extensive Monte Carlo sampling, which is used as benchmark. Both methods require at least 1 order of magnitude less simulations than Monte Carlo, with a slight advantage of Kriging over polynomial chaos expansion. The analysis of the solution space clearly indicates the effects of uncertainties and their couplings, and highlights some possible shortcomings of current mostly deterministic approaches based on safety factors.

The analysis and design of complex engineering
systems are typically based on sophisticated numerical models. While in the
past these have been mostly based on deterministic formulations, more
recently probabilistic approaches have been gaining an increased attention
because of their ability to account for uncertainties in both the models and
their inputs. Although numerous applications of probabilistic methods can be
found in many areas of engineering, so far formal uncertainty quantification
has been applied to a lesser degree in the wind energy field. In fact,
probabilistic approaches have been used to estimate wind turbine extreme
loads, as reported by

The behavior of wind turbines and of the environment in which they operate is
profoundly affected by uncertainties. Therefore, time is ripe for
investigating rigorous mathematical formulations to evaluate the robustness
of designs and to establish confidence levels on outputs of interest. In the
literature, already a few authors have taken the first steps in this
direction. One of the first wind-energy-related publications in this field is
the paper by

Modern simulation and design frameworks are typically based on validated
comprehensive aeroservoelastic models. Drastic rewritings of such complex
codes to incorporate stochastic formulations are clearly undesirable. To
enable the use of legacy codes as black boxes within a probabilistic
approach, studies have been recently focusing on the augmentation of
aeroservoelastic solvers with non-intrusive uncertainty propagation methods.
In addition to enabling the reuse of existing software, non-intrusiveness
also allows one to rapidly reap the benefits of any modeling improvement, as
the problem of uncertainty quantification is essentially decoupled from the
details of the underlying simulation model. This approach is followed
by

The present study expands and refines the work presented in

The study is conducted with reference to a conceptual offshore 10 MW wind
turbine, which is representative of the most up-to-date technology. The
machine is modeled with the code

The paper is structured as follows. Section

Uncertainties are commonly categorized into two macro families: aleatory and epistemic uncertainties. The former source of uncertainty emerges from the underlying randomness of a process, as for example described by the probability distribution of the wind speed at a certain site. The latter, on the other hand, originates from a lack of knowledge and data. This work considers the effects of aleatory model parameters and inputs with established underlying probability distributions.

Wind turbines are subjected to several sources of uncertainty. In addition to
the inherently stochastic character of the wind, which varies in time and
space for a multitude of reasons, uncertainties are also present in the
aerodynamic characteristics of the machine; in the mechanical properties of
the materials, structures and foundations; and in the characteristics
and performance of many of the subsystems of a wind turbine. Not only the
nominal values of all such parameters are uncertain but additional sources
of uncertainty are also introduced by manufacturing processes and the status of
wear and tear of each individual machine or component. Additionally, one
should not forget that measurements are also uncertain

Due to its preliminary character, this study limits its attention to uncertainties affecting the wind inflow and the aerodynamics of the blades. These are typical and relevant examples of aspects of a turbine model that can often only be described in statistical terms, but also have a profound impact on the behavior and overall performance of the system. It should, however, be remarked that the methods analyzed here are general and in principle applicable to problems other than the ones considered for this work.

Wind is a natural phenomenon where air particles move
dynamically following three-dimensional paths as a result of a number of
driving effects. In general, such a complex process can only be measured and
described in terms of its statistics. International standards, such as

However, effects such as solar irradiation, seasonal and long-term climate
changes, vegetation growth, and complex terrain conditions play important
roles in increasing uncertainties in the characteristics of the wind

This work assumes that both TI and SE are uncertain. However, field data
often exhibit a correlation between SE and TI that, according to

Here and in the following all uncertain parameters are modeled with scaled
beta distributions. Such distributions are preferred to other possible
choices for two reasons: first, they are highly flexible in shaping the
probability density function on account of given statistical data and,
secondly, they generate bounded distributions with lower and upper limits.
This is a necessary feature when modeling parameters that cannot assume
negative values. It should be noted, however, that neither NIPCE nor Kriging
are bound to scaled beta distributions, and truncated Gaussian, log-normal,
uniform distributions, or others could also be readily used. The parameters of
the beta distribution for the uncertain factor

A second important source of uncertainty in wind turbine simulation and design lies in the aerodynamic characteristics of the rotor. Among other effects, the performance of the airfoils – measured in terms of the aerodynamic coefficients of lift, drag, and moment – is considered a possible major source of uncertainty.

The estimation of airfoil aerodynamic coefficients can be obtained by experimental and numerical techniques. Both approaches are challenging and lead to uncertainties of an aleatory and epistemic nature, especially in the stall and post-stall regimes. Although potentially very significant, such uncertainties are not considered further in this work, which focuses instead on blade surface conditions.

During operation, the surface of a blade may be contaminated by the deposition of dust, dirt, insects, and pollen. Additionally, the blade surface can also be altered due to erosion caused by sand and rain. All these effects are typically and particularly prominent at the leading edge, which has a fundamental role in dictating the behavior of airfoils. As a result, changes in surface conditions during operation may result in significant uncertainties in power capture and loading.

Several studies have quantified the impact of erosion and contamination on
aerodynamic performance

Uncertainties in the actual extension of surface degradation along the span
of the blade are modeled by introducing a second parameter, termed extent of
spanwise degradation (ESD). Parameter ESD is defined as the nondimensional
span length – measured from blade tip – where factor

Interpolation of the airfoil aerodynamic coefficients between the fully clean and fully rough conditions.

As anticipated in Sect.

In

Principal characteristics of the 10 MW AVATAR wind turbine.

Here uncertainties in the wind characteristics and in
the airfoil polars are propagated throughout the aeroservoelastic model of an
offshore wind turbine, with the goal of comparing the performance of the
uncertainty quantification methods and of establishing their main convergence
characteristics. First, Sect.

Spanwise positions of the airfoils.

The AVATAR wind turbine is considered in this work, as a representative case
of a large offshore wind turbine. This conceptual machine was developed by a
consortium of academic and industrial partners within the EU project AVATAR

For airfoils DU97-W-300 and DU91-W2-240, which occupy the outermost part of
the blade, surface conditions are specified by the two parameters

Fully clean and fully rough aerodynamic coefficients

Uncertainties are considered for

Probability density functions for turbulence intensity
factor

Turbulence intensity distribution for varying wind speed.

An extensive MC analysis is first performed to characterize the solution space. The
three uncertainties are propagated throughout the aeroservoelastic model in a
power production state at 12 different wind speeds from cut-in to cut-out,
considering six turbulent seeds. Eight outputs of interest are analyzed,
namely maximum blade tip deflection (MTD), ultimate and damage equivalent
load (DEL) of the thrust measured at the main shaft (ThS), ultimate and DEL
combined blade root moment (CBRM), ultimate and DEL combined tower base
moment (CTBM), and finally annual energy production (AEP). MTD and ultimate
ThS, CBRM, and CTBM are obtained by computing the maximum overall value
across all time steps and wind speeds. DELs and AEP are instead averaged via
the Weibull distribution corresponding to wind class 1A, which is
characterized by a shape factor of 2 and an average wind speed at hub height
of 10 m s

The MC analysis was stopped at 1100 evaluations, where the convergence of
mean and standard deviations for all quantities consistently returned
variations below 1 % of their average values. While convergence is rapidly obtained for the mean values of the eight outputs of interest, standard deviations require a significantly higher number of evaluations to reach convergence. The statistics of the outputs are reported in
Table

Main statistics of the eight outputs of interest for 1100 MC function evaluations. MTD: maximum tip deflection; ThS: thrust at main shaft; CBRM: combined blade root moment; CTBM: combined tower base moment; DEL: damage equivalent load; AEP: annual energy production.

Here, six seeds were used to limit the computational cost of the MC analysis,
following accepted international standards

Probability density functions (PDFs) of key output metrics for varying number of seeds. Each case is based on 1100 sampling points.

The convergence of the uncertainty propagation methods is studied first. The analysis considers mean and standard deviation of AEP, maximum tip displacement, thrust, combined blade root moment, combined tower base moment, and the corresponding damage equivalent loads.

Third-order NIPCE and UK, both as implemented in DAKOTA

Both NIPCE and UK appear to be capable of estimating the eight outputs of
interest at a much reduced number of function evaluations compared to MC. In
addition, UK consistently converges faster than the other two methods, with a
reduction of 1–2 orders of magnitude with respect to MC for the
estimation of the output mean and standard deviation. The plots reported in
Fig.

Convergence of mean and standard deviation for key output quantities. The gray area reflects the potential inexactness of the MC benchmark, and it represents the 95 % confidence intervals for 1100 sampling points.

Key outputs (in percent difference with respect to the mean value)
and corresponding probability density functions, for

The results obtained by UK with 40 function evaluations are
then subjected to a more detailed analysis. Response surfaces for the eight
outputs of interest and their corresponding probability density functions are
shown in Fig.

The contour plots visibly show nonlinearities. Additionally, they also show
that the condition corresponding to a fully clean rotor, namely ESD and

MTD provides an interesting example. International standards prescribe
MTD to be 30 % lower than tower clearance. The top left plots in
Fig.

In addition, the contour plots of MTD and AEP indicate a fairly linear
behavior of the solution space, where the two outputs show a maximum
variation along the 45

This work has reported on the first steps towards the development of a framework for the non-intrusive propagation of uncertainties throughout black-box aeroservoelastic wind turbine models. Non-intrusiveness is key to the reusability of legacy models and for rapidly reaping the benefits of modeling improvements without the need for a extensive rewriting of such complex codes.

NIPCE and UK were applied to a large state-of-the-art conceptual wind turbine, considering power capture, tip deflection, and some typical design-driving loads as performance indicators. Uncertainties were considered for both the wind inflow conditions and the roughness of the blades, on account of soiling and/or erosion. For both methods, comparisons to standard brute-force Monte Carlo predictions indicate a good performance in terms of quality at a significantly lower computational cost. Of the two, UK appears to consistently converge faster than NIPCE.

The analysis of the results indicates nonlinearities and couplings among the various sources of uncertainty. In addition, it was found that the deterministic conditions prescribed by international design standards generate maximum values of loads and power production, which, however, are typically associated with a very low probability of occurrence. Although the results obtained here are not comprehensive enough to draw any significant conclusions, they do suggest that the use of formal mathematical methods of uncertainty propagation may lead to a revision of typical safety factors in the interest of more cost-competitive – but still fully safe – designs.

The present study should be refined in several important aspects. To start,
the problem of turbulent realizations deserves specific attention. Here the
number of turbulent seeds typically recommended by design standards was used,
but appeared not to be always sufficient for guaranteeing convergence of the
statistics. If the number of seeds needs to be increased in a substantial
manner to ensure convergence, this might require a change in the
methodological approach, as the computational cost might become prohibitive.
In this sense, the use of surrogate models, instead of the high-fidelity ones
used here, might become attractive. An additional problem of interest is the
computation of extreme states, which populate the tails of the probability
distributions and often act as design drivers. Here, ad hoc sampling
strategies have been developed by the statistical research community, and
could be applied to the problem at hand

Data can be provided upon request. Please contact the corresponding author Carlo L. Bottasso (carlo.bottasso@tum.de).

All authors equally contributed to this work.

The authors declare that they have no conflict of interest.

The authors wish to acknowledge Alessandro Croce and Luca Sartori of the Department of Aerospace Science and Technology of Politecnico di Milano for providing the data of the 10 MW AVATAR wind turbine. Additionally, credit goes to Dominic von Terzi and Thierry Maeder of GE Global Research for the fruitful discussions and the partial financial support of this research.

This research has been supported by GE Global Research (grant title “UQ for wind turbine aeroelasticity”).This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Program.

This paper was edited by Michael Muskulus and reviewed by two anonymous referees.