Implicit Floquet analysis of wind turbines using tangent matrices of a non‐linear aeroelastic code

The aeroelastic code BHawC for calculation of the dynamic response of a wind turbine uses a non‐linear finite element formulation. Most wind turbine stability tools for calculation of the aeroelastic modes are, however, based on separate linearized models. This paper presents an approach to modal analysis where the linear structural model is extracted directly from BHawC using the tangent system matrices when the turbine is in a steady state. A purely structural modal analysis of the periodic system for an isotropic rotor operating at a stationary steady state was performed by eigenvalue analysis after describing the rotor degrees of freedom in the inertial frame with the Coleman transformation. For general anisotropic systems, implicit Floquet analysis, which is less computationally intensive than classical Floquet analysis, was used to extract the least damped modes. Both methods were applied to a model of a three‐bladed 2.3 MW Siemens wind turbine model. Frequencies matched individually and with a modal identification on time simulations with the non‐linear model. The implicit Floquet analysis performed for an anisotropic system in a periodic steady state showed that the response of a single mode contains multiple harmonic components differing in frequency by the rotor speed. Copyright © 2011 John Wiley & Sons, Ltd.     

Experimental identification of modal parameters of an industrial 2‐MW wind turbine

This contribution presents modal testing of a 2‐MW wind turbine on a 100‐m tubular tower with a 93‐m rotor developed by W2E Wind to Energy GmbH. This research is part of the DYNAWIND project of the University of Rostock and W2E. Beside classical modal analysis schemes, this contribution mainly focusses on the application of operational modal analysis techniques to a wind turbine. Specific problems are addressed, and hints for modal testing on wind turbines are given. Furthermore, an effective measurement setup is proposed for identification of the modal parameters of a wind turbine. The measurement campaign is divided in two parts. First, a measurement campaign using 8 sensor positions on a rotor blade was done while the rotor is lying on ground. Second, a detailed measurement campaign was done on the entire wind turbine with the rotor locked in Y position using 61 sensor positions on the tower, the mainframe, the gearbox, the generator, and the low‐voltage unit. While the rotor blade was tested by classical and operational modal analysis techniques, the entire wind turbine was tested by operational modal analysis techniques only. The mode shapes and eigenfrequencies of the wind turbine identified within the measurement campaigns are within the expected range of the design values of the wind turbine. But in contrast, the damping ratios differ strongly from those given in guidelines and literature. Furthermore, a strong influence of aerodynamic damping compared to structural damping is observed for the first tower mode even for a parked wind turbine.     

A study of modal damping for offshore wind turbines considering soil properties and foundation types

The modal damping ratio for each mode is crucial to characterize the dynamic behavior of offshore wind turbines and widely used by simulation software in wind turbine engineering, such as Bladed and FAST. In this study, modal damping ratios of offshore wind turbines are systematically studied for different soil properties and foundation types. Firstly, the modal damping ratios and modal frequencies for the first and second modes of a gravity foundation–supported offshore wind turbine are studied. An offshore wind turbine supported by a monopile foundation is then investigated to clarify the characteristics of modal damping ratios and modal frequencies for the monopile foundation. The soil parameters are identified by means of genetic algorithm (GA). Predicted modal damping ratios and modal frequencies as well as modal shapes show good agreement with the field measurements for both foundations. Finally, a sensitivity analysis study is carried out to investigate the effects of soil properties and foundation types on modal damping ratios. For the gravity foundation–supported offshore wind turbine, soil properties affect the modal damping ratio of the second mode largely, but affect that of the first mode little, while for the monopile‐supported offshore wind turbine, soil properties affect the modal damping ratios of the first and second modes significantly. Predicted natural periods and modal damping ratios of the first mode for both foundations by a pair of simple models agree well with those by numerical models.     

Modes of response of an offshore wind turbine with directional wind and waves

A coupled dynamic analysis of a floating wind turbine system has been performed to investigate effect of wave direction relative to wind on the system. Hydrodynamic loads are calculated by linear frequency domain approach and aerodynamic effect is taken into account by increasing hydrodynamic damping and restoring matrices with aerodynamic damping and gyroscopic stiffness.A modal analysis of the system was made to explain the calculated motions. It brings out the natural frequencies, natural modes and modal damping coefficients of the system. Excitation of natural modes, by waves explains the motion observed in the response amplitude operators, and the effect of wave direction relative to wind. This modal analysis helps to better understand the behaviour of floating wind turbine systems.     

The dynamics of an offshore wind turbine in parked conditions: a comparison between simulations and measurements

Offshore wind turbines are complex structures, and their dynamics can vary significantly because of changes in operating conditions, e.g., rotor‐speed, pitch angle or changes in the ambient conditions, e.g., wind speed, wave height or wave period. Especially in parked conditions, with reduced aerodynamic damping forces, the response due to wave actions with wave frequencies close to the first structural resonance frequencies can be high. Therefore, this paper will present numerical simulations using the HAWC2 code to study an offshore wind turbine in parked conditions. The model has been created according to best practice and current standards based on the design of an existing Vestas V90 offshore wind turbine on a monopile foundation in the Belgian North Sea. The damping value of the model's first fore‐aft mode has been tuned on the basis of measurements obtained from a long‐term ambient monitoring campaign on the same wind turbine. Using the updated model of the offshore wind turbine, the paper will present some of the effects of the different design parameters and the different ambient conditions on the dynamics of an offshore wind turbine. The results from the simulations will be compared with the processed data obtained from the real measurements. The accuracy of the model will be discussed in terms of resonance frequencies, mode shapes, damping value and acceleration levels, and the limitations of the simulations in modeling of an offshore wind turbine will be addressed. Copyright © 2014 John Wiley & Sons, Ltd.     

Open‐loop frequency response analysis of a wind turbine using a high‐order linear aeroelastic model

Wind turbine controllers are commonly designed on the basis of low‐order linear models to capture the aeroelastic wind turbine response due to control actions and disturbances. This paper characterizes the aeroelastic wind turbine dynamics that influence the open‐loop frequency response from generator torque and collective pitch control actions of a modern non‐floating wind turbine based on a high‐order linear model. The model is a linearization of a geometrically non‐linear finite beam element model coupled with an unsteady blade element momentum model of aerodynamic forces including effects of shed vorticity and dynamic stall. The main findings are that the lowest collective flap modes have limited influence on the response from generator torque to generator speed, due to large aerodynamic damping. The transfer function from collective pitch to generator speed is affected by two non‐minimum phase zeros below the frequency of the first drivetrain mode. To correctly predict the non‐minimum phase zeros, it is essential to include lateral tower and blade flap degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

Improved Modal Dynamics of Wind Turbines to Avoid Stall‐induced Vibrations

Stall‐induced edgewise blade vibrations have occasionally been observed on three‐bladed wind turbines over the last decade. Experiments and numerical simulations have shown that these blade vibrations are related to certain vibration modes of the turbines. A recent experiment with a 600 kW turbine has shown that a backward whirling mode associated with edgewise blade vibrations is less aerodynamically damped than the corresponding forward whirling mode. In this article the mode shapes of the particular turbine are analysed, based on a simplified turbine model described in a multi‐blade formulation. It is shown that the vibrations of the blades for the backward and forward edgewise whirling modes are different, which can explain the measured difference in aerodynamic damping. The modal dynamics of the entire turbine is important for stability assessments; blade‐only analysis can be misleading. In some cases the modal dynamics may even be improved to avoid stall‐induced vibrations. Copyright © 2003 John Wiley & Sons, Ltd.     

水平轴风力机塔架的频率和振型特性实验研究

为了研究风力机塔架的振动特性,文章利用动态信号采集分析系统,对水平轴风力机塔架进行了实验模态分析和运行模态分析测试,得到了塔架静止与振动两种工况下的固有频率与模态振型,分析了塔架的振动特性。通过对风力机振动信号的频谱分析发现,风速小于10 m/s时,只能激励起塔架挥舞方向与摆振方向的二阶模态;通过对风力机塔架的模态分析发现,风力机发生振动,塔架固有频率与模态振型发生小幅度改变;随着风速和振动烈度的增大,塔架模态参数的变化幅度随之增大。该研究可以为风力机塔架优化设计提供借鉴。     

海上风电结构横风向气动力阻尼特性研究

基于横风向气动力阻尼理论计算模型,以NREL-5 MW海上风电机组为例,对其运行过程中横风向气动力阻尼进行计算,并采用FAST软件对计算结果进行验证。之后,研究转速、叶片桨距角和运行方式对横风向气动力阻尼的影响。研究结果表明：NREL-5 MW海上风电机组结构运行状态下的横风向气动力阻尼在0%~0.8%范围内变化,其随风电机组运行转速及叶片桨距角的增大而增大;此外,海上风电机组不同运行方式对其横风向气动力阻尼也会产生较大影响。     

An analytical frequency-domain model of aerodynamic mass and damping of floating wind turbines

The fore-aft motion of the rotor-nacelle assembly (RNA) of a rotating floating wind turbine (FWT) can cause an oscillation in aerodynamic thrust, which may be equivalently treated as frequency-dependent aerodynamic mass and damping effects. In this study, an explicit frequency-domain analytical model is proposed to calculate the equivalent aerodynamic mass and damping of FWTs, with proper linearization of control system. Assuming that an FWT operates under steady wind conditions and a forced oscillation is exerted at the RNA along the wind direction, the thrust fluctuations are equivalently represented by the force and moment acting on the nacelle instead of pure aerodynamic loads. Based on the thrust oscillation expression, equivalent aerodynamic mass and damping are derived analytically. After verifying the model by numerical comparison, it is used to demonstrate equivalent aerodynamic mass and damping of three wind turbines (5–15 MW). Effects of wind turbine up-scaling and controller dynamics are addressed. Results show that equivalent aerodynamic mass and damping present a nonlinear characteristic with oscillation frequency in the below-rated region, while the relationship is close to linear for higher wind speeds. The effect of wind turbine up-scaling has a visible impact on equivalent aerodynamic mass and damping, especially at near-rated wind speed. Controller gains affect equivalent aerodynamic mass and damping and should be tuned reasonably in the controller design for FWTs. Outcomes of our study can be used to establish a frequency-domain coupled model of FWTs and are beneficial for conceptual design and parameter optimization of the platform of FWTs.     

Model‐independent periodic stability analysis of wind turbines

In this work, a new method is proposed for the stability analysis of wind turbines. The method uses input–output time histories obtained by conducting virtual excitation experiments with a suitable wind turbine simulation model. Next, a single‐input/single‐output periodic reduced model is identified from the recorded response and used for a stability analysis conducted according to the Floquet theory. Since only input–output sequences are used, the approach is model independent in the sense that it is applicable to wind turbine simulation models of arbitrary complexity. The use of the Floquet theory reveals a much richer picture than the one obtained by widespread classical approaches based on the use of the multi‐blade coordinate transformation of Coleman. In fact, it is shown here that, for each principal mode computed by the classical approach, there are in reality infinite super‐harmonics of varying strength fanning out from the principal one at multiples of the rotor speed. The relative strength of each harmonic in a fan provides for a way of measuring how periodically one specific fan of modes behaves. The notion of super‐harmonics allows one to justify the presence of peaks in the response spectra, peaks that cannot be explained by the classical time‐invariant analysis. The Campbell diagram, i.e., the plot of system frequencies vs. rotor speed, is in this work enriched by the presence of the super‐harmonics, revealing a much more complex pattern of possible resonant conditions with the per‐rev excitations than normally assumed. Copyright © 2014 John Wiley & Sons, Ltd.     

Prediction of seismic loadings on wind turbine support structures by response spectrum method considering equivalent modal damping of support structures and reliability level

In this study, a new damping correction factor is proposed to provide an accurate design response spectrum for megawatt (MW) class wind turbines. The predicted response spectra show good agreement with those by time history analysis (THA). An equivalent modal damping of wind turbine support structures is then proposed for response spectrum method (RSM) based on the modal decomposition method for non‐classically damped structures. Seismic loadings on wind turbine towers and footings by RSM with the proposed damping correction factor and equivalent modal damping are also investigated and compared with those by THA. Finally, a quantile value between 0.5 and 0.85 in the damping correction factor is calibrated to ensure the same reliability level as evaluated by THA currently used for estimation of seismic loadings on support structures.     

Impact of aerodynamic modelling on seakeeping performance of a floating vertical axis wind turbine

This study focuses on the impact of the aerodynamic model on the dynamic response of a floating vertical axis wind turbine (VAWT). It compares a state‐of‐the‐art quasi‐steady double multiple streamtube (DMS) solver, a prescribed vortex wake (PVW), and a free vortex wake (FVW) solver. The aerodynamic loads acting on a bottom‐fixed VAWT and computed with the three aerodynamic solvers are compared, then the dynamic responses of the floating turbine in irregular waves and turbulent wind with the different aerodynamic solvers are compared. Differences are observed, particularly in the mean motions of the platform. Eventually, the aerodynamic damping computed by the solvers are estimated with aerodynamic simulations on the turbine with imposed surge and pitch motions. The estimated damping can then be correlated with the dynamic response amplitude of the VAWT. Substantial discrepancies are observed between the three solvers at high tip speed ratio, when the rotor is highly loaded. It is shown that the quasi‐steady DMS solver seems to give greater amplitude of motions for the floating VAWT because of strong rotor/wake interaction that are not correctly accounted for.     

Assessment of passive instability suppression means on pitch‐regulated wind turbines

The significance of three types of design modifications in view of defining passive means to extend the stability bounds of modern wind turbines is assessed in this paper. The first concerns the use of optimized airfoil shapes on a fixed blade planform while the other two concern the increase of structural flexibility by either bringing closer the flap and lead‐lag mode frequencies or introducing a soft yaw connection. Such an exploration of the stability envelope aims at providing the necessary understanding of the mechanisms that control aeroelastic damping and therefore at identifying means for improving the stability behaviour of the lowest damped system modes. Stability calculations are performed in the context of linear eigenvalue analysis using a state‐of‐the‐art stability tool. The model accounts for the full wind turbine configuration and the eigenvalue problem is formulated with reference to the non‐rotating (ground‐fixed) frame of reference through the multi‐blade transformation of all the rotating degrees of freedom. Results are presented in reference to a commercial multi‐MW, pitch‐regulated, variable‐speed wind turbine. They indicate that the soft yaw concept offers more significant margins of improvement compared with the flap‐lag coincidence, while aerodynamic optimization could be a basis for improvement. Copyright © 2007 John Wiley & Sons, Ltd.     

Development of an aeroelastic code based on three‐dimensional viscous–inviscid method for wind turbine computations

Aerodynamic and structural dynamic performance analysis of modern wind turbines are routinely estimated in the wind energy field using computational tools known as aeroelastic codes. Most aeroelastic codes use the blade element momentum (BEM) technique to model the rotor aerodynamics and a modal, multi‐body or the finite‐element approach to model the turbine structural dynamics. The present work describes the development of a novel aeroelastic code that combines a three‐dimensional viscous–inviscid interactive method, method for interactive rotor aerodynamic simulations (MIRAS), with the structural dynamics model used in the aeroelastic code FLEX5. The new code, called MIRAS‐FLEX, is an improvement on standard aeroelastic codes because it uses a more advanced aerodynamic model than BEM. With the new aeroelastic code, more physical aerodynamic predictions than BEM can be obtained as BEM uses empirical relations, such as tip loss corrections, to determine the flow around a rotor. Although more costly than BEM, a small cluster is sufficient to run MIRAS‐FLEX in a fast and easy way. MIRAS‐FLEX is compared against the widely used FLEX5 and FAST, as well as the participant codes from the Offshore Code Comparison Collaboration Project. Simulation tests consist of steady wind inflow conditions with different combinations of yaw error, wind shear, tower shadow and turbine‐elastic modeling. Turbulent inflow created by using a Mann box is also considered. MIRAS‐FLEX results, such as blade tip deflections and root‐bending moments, are generally in good agreement with the other codes. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamic behavior of wind turbines influenced by aerodynamic damping and earthquake intensity

In the present paper the effects of aerodynamic damping and earthquake loads on the dynamic response of flexible‐based wind turbines are studied. A numerical analysis framework (NAF) is developed and applied. NAF is based on a user‐compiled module that is developed for the purposes of the present paper and is fully coupled with an open source tool. The accuracy of the developed NAF is validated through comparisons with predictions that are calculated with the use of different numerical analysis methods and tools. The results indicate that the presence of the aerodynamic loads due to the reduction of the maximum displacement of the tower attributed to the dissipation of earthquake excitation energy in fore‐aft direction. Emergency shutdown triggered by strong earthquakes results to a rapid change of aerodynamic damping, resulting to short‐term instability of the wind turbine. After shutdown of the wind turbine, enhanced dynamic response is observed. For the case where the wind turbine is parked, the maxima displacement and acceleration of tower‐top increase linearly with the peak ground acceleration. With the use of the least‐square method a dimensionless slope of tower‐top displacements is presented representing the seismic response coefficient of tower that can be used to estimate the tower‐top acceleration demand. Moreover, on the basis of the seismic response coefficient, an improved model for the evaluation of load design demand is proposed. This model can provide accurate predictions.     

An investigation on wind turbine resonant vibrations

Wind turbine resonant vibrations are investigated based on aeroelastic simulations both in frequency and time domain. The investigation focuses on three different aspects: the need of a precise modeling when a wind turbine is operating close to resonant conditions; the importance of estimating wind turbine loads also at low turbulence intensity wind conditions to identify the presence of resonances; and the wind turbine response because of external excitations. In the first analysis, three different wind turbine models are analysed with respect to the frequency and damping of the aeroelastic modes. Fatigue loads on the same models are then investigated with two different turbulence intensities to analyse the wind turbine response. In the second analysis, a wind turbine model is excited with an external force. This analysis helps in identifying the modes that might be excited, and therefore, the frequencies at which minimal excitation should be present during operations. The study shows that significant edgewise blade vibrations can occur on modern wind turbines even if the aeroelastic damping of the edgewise modes is positive. When operating close to resonant conditions, small differences in the modeling can have a large influence on the vibration level. The edgewise vibrations are less visible in high turbulent conditions. Using simulations with low‐level turbulence intensity will ease this identification and could avoid a redesign. Furthermore, depending on the external excitation, different aeroelastic modes can be excited. The investigation is performed using aeroelastic models corresponding to a 1.5 MW class wind turbine with slight variations in blade properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation model of wind turbine 3p torque oscillations due to wind shear and tower shadow

To determine the control structures and possible power quality issues, the dynamic torque generated by the blades of a wind turbine must be represented. This paper presents an analytical formulation of the generated aerodynamic torque of a three-bladed wind turbine including the effects of wind shear and tower shadow. The comprehensive model includes turbine-specific parameters such as radius, height, and tower dimensions, as well as the site-specific parameter, the wind shear exponent. The model proves the existence of a 3p pulsation due to wind shear and explains why it cannot be easily identified in field measurements. The proportionality constant between the torque and the wind speed is determined allowing direct aerodynamic torque calculation from an equivalent wind speed. It is shown that the tower shadow effect is more dominant than the wind shear effect in determining the dynamic torque, although there is a small dc reduction in the torque oscillation due to wind shear. The model is suitable for real-time wind turbine simulation or other time domain simulation of wind turbines in power systems.     

Large‐eddy simulation of stable boundary layer turbulence and estimation of associated wind turbine loads

Stochastic simulation of turbulent inflow fields commonly used in wind turbine load computations is unable to account for contrasting states of atmospheric stability. Flow fields in the stable boundary layer, for instance, have characteristics such as enhanced wind speed and directional shear; these effects can influence loads on utility‐scale wind turbines. To investigate these influences, we use large‐eddy simulation (LES) to generate an extensive database of high‐resolution ( ～ 10 m), four‐dimensional turbulent flow fields. Key atmospheric conditions (e.g., geostrophic wind) and surface conditions (e.g., aerodynamic roughness length) are systematically varied to generate a diverse range of physically realizable atmospheric stabilities. We show that turbine‐scale variables (e.g., hub height wind speed, standard deviation of the longitudinal wind speed, wind speed shear, wind directional shear and Richardson number) are strongly interrelated. Thus, we strongly advocate that these variables should not be prescribed as independent degrees of freedom in any synthetic turbulent inflow generator but rather that any turbulence generation procedure should be able to bring about realistic sets of such physically realizable sets of turbine‐scale flow variables. We demonstrate the utility of our LES‐generated database in estimation of loads on a 5‐MW wind turbine model. More importantly, we identify specific turbine‐scale flow variables that are responsible for large turbine loads—e.g., wind speed shear is found to have a greater influence on out‐of‐plane blade bending moments for the turbine studied compared with its influence on other loads such as the tower‐top yaw moment and the fore‐aft tower base moment. Overall, our study suggests that LES may be effectively used to model inflow fields, to study characteristics of flow fields under various atmospheric stability conditions and to assess turbine loads for conditions that are not typically examined in design standards. Copyright © 2013 John Wiley & Sons, Ltd.