Uncertainty quantification (UQ) is a well-established category of methods to estimate the effect of parameter variations on a quantity of interest based on a solid mathematical foundation. In the wind energy field most UQ studies focus on the sensitivity of turbine loads. This article presents a framework, wrapped around a modern Python UQ library, to analyze the impact of uncertain turbine properties on aeroelastic stability. The UQ methodology applies a polynomial chaos expansion surrogate model. A comparison is made between different wind turbine simulation tools on the engineering model level (alaska/Wind, Bladed, HAWC2/HAWCStab2, and Simpack). Two case studies are used to demonstrate the effectiveness of the method to analyze the sensitivity of the aeroelastic damping of an unstable turbine mode to variations of structural blade cross-section parameters. The code-to-code comparison shows good agreement between the simulation tools for the reference model, but also significant differences in the sensitivities.

The size of wind turbines has been rapidly increasing over the last decades. As a consequence, current wind turbine blades are more slender and flexible than ever before

The required multidisciplinary models to numerically represent these phenomena are complex and have a significant computational cost. Conventional models used in the industry and research employ a multi-body description with beam models for the flexible bodies and blade element momentum (BEM) models, with semi-empirical unsteady extensions, for the aerodynamics. The linear stability behavior is commonly investigated by a linearization of the governing equations around a steady-state equilibrium of the nonlinear system. These models and solution routines depend on numerous parameters, which complicates the identification of the key factors that influence the observed stability behavior. Global uncertainty quantification (UQ) can help identify these crucial factors.

Uncertainty quantification has been a relevant topic in almost all scientific fields. In engineering, it is commonly used to understand physical systems, to improve the design robustness, as a preprocessing step towards model updating and model calibration, or as a component of optimization procedures

To fill this gap, the present article describes a comprehensive methodology for the uncertainty quantification of wind turbine stability analysis. The effect of uncertain beam properties in the elastic blade model on an edgewise whirl wind turbine instability is analyzed. Multiple aeroelastic simulation tools are used in a code-to-code comparison to investigate the influence of the simulation tools on the uncertainty prediction.

The procedure for this study and the corresponding structure of this article are visualized in Fig.

Paper overview.

This section will first describe the wind turbine reference model and the required model modifications to create an interesting instability phenomenon. The wind turbine simulation tools are introduced and the main verification results with this new model are presented. Finally, the critical unstable reference condition is analyzed in detail.

The IWT-7.5-164 open-source reference turbine is used as the baseline configuration

Different techniques can be used to introduce an instability for this baseline reference turbine. As presented by

In this research, a comparison is made between multiple state-of-the-art aeroelastic simulation tools. All tools are based on the same category of low-fidelity engineering models. The main model properties are summarized in Table

Overview of the features of the simulation tools used.

Two categories of simulation tools can be distinguished. Bladed (lin.) and HAWCStab2 provide linear models (at nonlinear equilibrium points), which will be used for standard linear stability analysis. On the other hand, alaska/Wind, Bladed, HAWC2, OpenFAST

OpenFAST will only be used for the model verification studies and is disregarded for the stability analysis comparison and uncertainty quantification because there were fundamental differences in the instability modes.

, and Simpack–AeroDyn will be used in this work for nonlinear time domain simulations. The post-processing methodology to analyze the stability properties of these simulations will be discussed in Sect.The simulation tools used in this work were previously compared for the same reference turbine but without the stiffness reductions by

The isolated, clamped blade eigenfrequencies are shown in Fig.

Comparison of isolated blade eigenfrequencies.

As a second comparison, a steady quintuple gravitational load is imposed on the clamped blades. The blades are positioned both with the suction side downwards, such that gravitational loading is in the flapwise direction, and with the leading edge downwards such that the gravitational loading is in the edgewise direction. The gravitational loading multiple is representative for nominal operational loads. The results for the flapwise loading are shown in the left column of Fig.

Comparison of isolated blade deflections under steady quintuple gravitational loads.

The final verification test shows the static aeroelastic equilibrium for a rotor system with rigid tower in a uniform, steady wind field with a velocity of 10 m s

Comparison of the aeroelastic steady-state loads and deflections of the full rotor.

To allow a stability assessment based on the time domain simulations, the damping of the system has to be determined from the resulting time signals. Multiple approaches can be used to achieve this.

In this work, a different approach based on the dynamic mode decomposition (DMD) method is used. The higher-order DMD formulation by

DMD damping determination settings.

For completeness, it is important to mention that a number of other methods to determine the damping from time signals were attempted without success. Computing the logarithmic decrement from subsequent oscillation peaks, exponential curve fitting on the oscillation peaks, and linear curve fitting on logarithmic data all share the assumption that the signal only has a single degree of freedom. Their accuracy is largely dependent on a smooth, linear signal, and they therefore did not produce robust results for the multi-degree of freedom, nonlinear results of the wind turbine simulation tools. Attempts with bandpass filtering of the signal for a specific frequency range did not lead to robustness improvements.

The final stability analysis of the reference condition is shown as a Campbell diagram in Fig.

Remark: it has to be mentioned that the damping of a Bladed linearization with multiple operating points can differ slightly from the damping of a linearization performed for a single operating point (as will be done for the UQ studies in Sect.

Exemplary snapshot selection for the DMD analysis. This exemplary signal is the torsional deformation of one of the blades at 50 m blade length.

Linearizations (colored lines in

Figure

Overall, clear correlations between the DMD post-processed time series and the linearizations can be found. The time domain simulations of almost all tools are unstable in the same wind speed range from 10 m s

A closer look at the DMD results of the individual tools shows the following peculiarities. The HAWC2 time domain results agree on the whole well with the HAWCStab2 linearization. Two small differences are the overall slightly lower frequency of the second edgewise BW mode and the higher damping of this same mode at 11 and 12 m s

Two case studies were performed on the influence of structural beam properties on the critical reference condition. Case study 1 is a basic study with a limited number of easily understandable uncertain parameters which serves as a demonstration and verification of the overall process. Case study 2 is a step closer towards engineering practice and serves as a mock-up case study for the analysis of the influence of blade manufacturing defects on the aeroelastic stability.

The case studies are limited to the operating point at 12 m s

A non-intrusive, global, variance-based uncertainty quantification based on a PCE surrogate model will be used in this work. This methodology does not require a modification of the simulation codes and by means of the surrogate model, the required number of simulations can be reduced significantly compared to a standard Monte Carlo simulation. The uncertainty quantification covers the full domain spanned by the uncertain parameter distributions and captures the potential interaction between these parameters.

The Python implementation of the preprocessor, post-processor, and interfaces to the tools is available open-source in the framework

Polynomial chaos expansion (PCE) with point collocation is used as a surrogate model. The applied PCE models had a fourth-order polynomial. The quasi-random Hammersley sampling scheme was used and the number of training data points for a given number of uncertain parameters were based on the best-practice findings by

Verification of the PCE model is necessary to ascertain that the surrogate is a good representation of the true model. As an initial verification measure, the surrogate model can be evaluated at the training data coordinates. However, this measure can be influenced by overfitting. Therefore, a leave-one-out test should be used as cross-validation. This test is done by the computation of an individual leave-one-out surrogate model for each of the training data points. This surrogate model does not contain that specific training data point. A comparison of this leave-one-out surrogate model evaluated at the coordinates of the training data point and the true model evaluation at those coordinates is then done. In general, the computation of the surrogate model is not expensive, but as shown by

Alternatively, additional random control points, which are not contained in the training dataset, could be used for verification. The leave-one-out error has the benefit over adding random control point computations that no additional simulations are required. The downside is that the leave-one-out error might introduce additional errors, especially at the edges of the input parameter space, because it assembles new surrogate models, missing the data point where the leave-one-out error is computed.

Based on the training data and the leave-one-out surrogate model, two error estimation metrics are defined, which are as follows.

Here,

The uncertainty quantification is done on the PCE surrogate model. It is common to use the Sobol indices as global uncertainty quantification metrics. They are a measure for the contribution of each uncertain input parameter to the variance of the output quantity of interest. Two Sobol indices will be used in this paper. The first-order Sobol index is given by

and represents the isolated contribution of an uncertain parameter to the total output variance.

The Sobol indices condense the uncertainty into single values. This is a clear metric for both the contribution of a parameter to the total variance in a model and the interaction between parameters, but it does not give insight into the way an uncertain parameter influences the quantity of interest. This detailed view can be given at a negligible computational cost by analyzing the polynomials of the PCE model along the different uncertain dimensions.

This case study applies three straightforward uncertain parameters: flapwise, edgewise, and torsional stiffness of the blades. These uncertain parameters are given identical uncertainty distributions. The radial discretization of the uncertainties is determined by a nonuniform rational B-spline (NURBS), similar to the methodology proposed by

Note that to guarantee a correct modification of the parameters in the different tools, said modifications are done directly on the reference

Case study 1: comparison between leave-one-out surrogate model evaluations and training data samples.

The simulation tools are sampled with the quasi-random Hammersley scheme to generate 72 training data points for the surrogate models. As described in Sect.

The first-order and total Sobol indices are shown in Fig.

Case study 1: first-order Sobol indices – isolated uncertainty contribution of the investigated parameters (top row), total Sobol indices – uncertainty contribution of the investigated parameters including interactions with other parameters (bottom row).

Two noticeable differences should be pointed out. Firstly, HAWC2 is the only model where the interaction between the uncertain parameters has a noteworthy contribution (4.12 %). Secondly, the sensitivity of the flapwise stiffness is significantly higher in Simpack and alaska/Wind compared to the other tools.

Figure

Case study 1: first-order effects of an isolated parameter (other parameters at reference value) .

This case study was proposed to demonstrate a possible use case in the actual wind turbine development process and to show the comparability of the tools for more intricate uncertain parameters.

The edgewise stiffness, principal axis orientation, position of the center of gravity, and position of the shear center along the chord are used as uncertain parameters. These beam properties are known to be sensitive to manufacturing defects and assumed to have a significant impact on the stability.

The leave-one-out verification and corresponding error metrics are shown in Fig.

Case study 2: comparison between leave-one-out surrogate model evaluations and training data samples.

The first-order and total Sobol indices for case study 2 are shown in Fig.

Case study 2: first-order Sobol indices – isolated uncertainty contribution of the investigated parameters (top row), total Sobol indices – uncertainty contribution of the investigated parameters including interactions with other parameters (bottom row).

Figure

Case study 2: first-order effects of an isolated parameter (other parameters at reference value) .

Models for multi-body simulations of wind turbines consist even in the described low-fidelity case of a huge number of parameters describing the degrees of freedom – especially for the blade. Evaluating and understanding the influence of each of these parameters on the aeroelastic stability of the model is complex. Sophisticated uncertainty quantification methods have to be used to assess the sensitivity in both a mathematically rigorous and computationally efficient manner.

In this work a code-to-code comparison between industry-relevant low-fidelity aeroelastic simulation tools (alaska/Wind, Bladed, HAWC2/HAWCStab2, and Simpack) has been done on the sensitivity of beam structural parameters to an edgewise whirling instability. The edgewise whirling instability was established on the IWT reference turbine through a reduction of the blade stiffness. To enable the comparison of time domain and linearization tool results, a dynamic mode decomposition (DMD) post-processing methodology for nonlinear time domain simulation has been introduced. This procedure was tuned for the identification of the unstable modes and should be developed further if it is to be applied to all operating conditions and all aeroelastic modes. The comparison of the reference condition showed overall satisfying agreement between the tools. An accurate match of the frequency of the edgewise modes of the aeroelastic system in the selected operating states could be found in almost all tools. The modal damping showed similar trends over the operating points but noticeable differences in the absolute values. A detailed study of the instability mechanism and the possible differences in the separate tools was out of the scope. Further investigation in this respect could help to understand the differences in damping observed in the Campbell diagrams and the differences in case study 2 of the uncertainty quantification. This study would require an analysis of the complex aeroelastic mode shapes and the net aerodynamic work introduced into or extracted from the system. This analysis is not trivial and would require its own dedicated study, especially considering the variations in capabilities and precision among the different tools.

A PCE surrogate model was used in the uncertainty quantification to reduce the computational cost. The PCE models were successfully verified by means of leave-one-out tests, which proves that these models are well suited to represent the full uncertainty domain. Case study 1 showed equivalent sensitivities in all tools with a dominating influence of the torsional stiffness compared to flapwise and edgewise stiffness. Major differences between the tools appeared in case study 2. The dominating uncertain parameter and the trend of the sensitivities were vastly different. This shows the complexity involved in the aeroelastic stability assessment. Even though the basic aeroelastic properties and the reference stability analysis of the different models appeared very similar, the parameters influencing said instability can still be significantly different for different tools. In both case studies, the interaction between the uncertain parameters was limited, which would imply that the uncertainty quantification could have been done at equal accuracy but significantly lower computational cost. It is important to note that the results of both case studies have to be understood within the assumptions of this work. The results will depend on the presented simulation models, the fixed operating conditions, the instability mechanism itself, and the selection and definition of the uncertain parameters and quantity of interest. The generalization of these results is difficult and should be made with caution. The case studies only covered a small number of uncertain parameters. In the future, it would be interesting to extend this to other, also non-structural, parameters. Furthermore, the input uncertainty distributions should be based on realistic deviations, e.g., known uncertainties due to manufacturing imperfections or degradation over its lifetime.

The code of the uncertainty quantification framework used for this study is available at

HV was the main developer for the framework and the methodology in collaboration with OH. HV did the overall verification and analysis of the results in close collaboration with all other authors. HV and OH wrote the first version of the paper; internal review was done by CB and JR. SM was responsible for the simulation and analysis of the alaska/Wind results. JDP was responsible for the simulation and analysis of OpenFAST and co-responsible for the simulation and analysis of HAWC2 and HAWCStab2. OS was co-responsible for the simulation and analysis of HAWC2 and HAWCStab2 and made valuable contributions to the uncertainty quantification framework. JR generated the HAWC2 model and was the main source of information for the preprocessing of the beam properties. OH and CB were the principal investigators at DLR and LUH and supervised the contributions of their groups. All authors contributed to this research through discussions and advice in their respective fields of expertise.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This work is a collaboration of three partners from research and industry in the framework of the German national research project QuexUS. This project was funded by the German Federal Ministry for Economic Affairs and Climate Action, grant no. 03EE3011A/B. Many thanks to the DTU HAWC2/HAWCStab2 development team for support and improvement during the active phase of simulations and to the DNV Bladed development team for their support and advice.

The article processing charges for this open-access publication were covered by the German Aerospace Center (DLR).

This paper was edited by Mingming Zhang and reviewed by Ozan Gozcu, Rad Haghi, and two anonymous referees.