Predictions of the cycle-to-cycle aerodynamic loads on a yawed wind turbine blade under stalled conditions using a 3D empirical stochastic model

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Predictions of e cycle-to-cycle aerodynamic loads on a yawed wind turbine blade under stalled conditions using a empirical stochastic model To cite is article: MOUTAZ EGAMMI and TONIO SANT 6 J. Phys.: Conf. Ser. 7 Related content - amage assessment for wind turbine blades based on a multivariate statistical approach avid García, mitri Tcherniak and Irina Trendafilova - Fast Ice etection for Wind Turbine Blades via e angevin Equation Haijun Fang and inpeng Wang - Crack iagnosis of Wind Turbine Blades Based on EM Meod CUI Hong-yu, ING Ning and HONG Ming View e article online for updates and enhancements. This content was downloaded from IP address on //8 at :7

2 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// Predictions of e cycle-to-cycle aerodynamic loads on a yawed wind turbine blade under stalled conditions using a empirical stochastic model MOUTAZ EGAMMI and TONIO SANT epartment of Mechanical Engineering, University of Malta, Msida MS 8, Malta moutaz.elgammi.@um.edu.mt Abstract. This paper investigates a new approach to model e stochastic variations in e aerodynamic loads on yawed wind turbines experienced at high angles of attack. The meod applies e one-dimensional angevin equation in conjunction wi known mean and standard deviation values for e lift and drag data. The meod is validated using e experimental data from e NRE Phase VI rotor in which e mean and standard deviation values for e lift and drag are derived rough e combined use of blade pressure measurements and a free-wake vortex model. Given at direct blade pressure measurements are used, flow effects arising from e co-existence of dynamic stall and stall delay are taken into account. The model is an important step towards verification of several assumptions characterized as e estimated standard deviation, Gaussian white noise of e data and e estimated drift and diffusion coefficients of e angevin equation. The results using e proposed assumptions lead to a good agreement wi measurements over a wide range of operating conditions. This provides motivation to implement a general fully independent eoretical stochastic model wiin a rotor aerodynamics model, such as e free-wake vortex or blade-element momentum code, whereby e mean lift and drag coefficients can be estimated using aerofoil data wi correction models for dynamic stall and stall delay phenomena, while e corresponding standard derivations are estimated rough CF. Nomenclature C drag coefficient C lift coefficient C N normal force coefficient C T tangential force coefficient () drift coefficient () diffusion coefficient f s frequency (Hz) M conditional moment N sample size r local radial position (m) R tip radius (m) t time (sec) T bin leng of e time period at each bin V n normal flow velocity (m/s) tangential flow velocity (m/s) V t U X C G free stream wind speed (m/s) C, C., C N, C T sample data vector angle of attack (rad) mean mean of drag force coefficient mean of lift force coefficient standard deviation total time period (sec) time increment (sec) yaw angle (deg) blade azimu angle (deg) bound circulation (m /s) Gaussian white noise (angevin force) Content from is work may be used under e terms of e Creative Commons Attribution. licence. Any furer distribution of is work must maintain attribution to e auor(s) and e title of e work, journal citation and OI. Published under licence by td

3 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7//. Introduction Wind turbine design procedures still rely considerably on e use of models at are unable to simulate e real unsteady aerodynamic behaviour experienced at high angles of attack. Wind tunnel measurements conducted under controlled conditions have evidently shown at wind turbines in yaw experience a significant cycle-to-cycle variability in aerodynamic loads at high angles of attack []. Such behaviour was observed even under simple flow conditions involving a steady and uniform wind flow. The unsteady aerodynamic loads can sometimes also be observed under attached flow conditions characterized as amplitude and phase variations compared to quasi-steady flow conditions []. Various models have been developed to simulate e unsteady aerodynamic loads associated wi dynamic stall. Examples include e Beddeos and eishman model [], e Onera model [], Øye model [] and e Risø fgh model []. However, ese models still depend on aerofoil wind tunnel data and various semi-empirical models at usually cater for aerodynamic phenomena such as stall delay. These models are not able to simulate e stochastic cycle-to-cycle variations in e aerodynamic loads experienced by e rotor under yawed conditions. Several studies have been performed to evaluate e stochastic behaviour of e unsteady aerodynamic loads based on measured data from wind tunnels. Bertagnolio at et. [6] established a stochastic approach to model e unsteady effects on a blade exhibiting a static stall originating from e self-induced turbulent wake of an aerofoil. The input data to e model can be measured data or determined by a CF code for aerofoil sections at constant angles of attack. The model was also applied for wind turbines giving interesting insight on e characteristics of stall. uhur et al. [7] recently developed a stochastic model based on e angevin stochastic differential equation [8] to model e dynamic lift and drag force coefficients on an aerofoil under turbulent wind inflow conditions. The work aimed at replacing e typical lookup tables of e static lift and drags force coefficients by e dynamic response of e lift and drag forces. uhur et al. [9] furer extended e latter work for wind turbine applications. The stochastic model was integrated wi e blade element momentum (BEM) model in order to predict e dynamic forces on a rotating blade. Alough e model showed clear insight on e time history of e aerodynamic loads, applications of e model for a full scale wind turbine were not provided to demonstrate efficiency of e model. Stochastic models have been also used to simulate turbulent atmospheric flows and eir effects on e power performance of wind turbines as shown in work of Wächter et al. [] and Gottschall []. To e auors' knowledge, stochastic modelling of e aerodynamic loads to capture e unsteady effects of stalled conditions in yawed rotors is very limited. This work is a first attempt to develop a new approach to simulate such a stochastic behaviour. The analysis is based on e NRE Phase VI rotor at was tested in e NASA Ames wind tunnel []. The main scope is to use experimental data for aerodynamic loading for e yawed rotor to verify some assumptions important for developing a general fully independent stochastic model. The mean and standard deviation values for e lift and drag coefficients derived from such data are used as input in e stochastic model to predict e cycleto-cycle aerodynamic based on e principles of Gaussian white noise and e one-dimensional stochastic angevin equation [8]. Through e verification of e various assumptions in e present paper, it will be possible to evaluate e potential for developing a general fully independent stochastic model in wind turbine aerodynamic design codes such as ose based on e Blade-Element- Momentum (BEM) or free wake vortex (FWV) meods where e mean values of lift and drag coefficients are predicted via aerofoil data wi correction models for effects arising from dynamic stall and stall delay phenomena while e corresponding standard deviation values are determined using Computational Fluid ynamics (CF).. NASA Ames UAE Wind Tunnel ata For validation of e proposed empirical stochastic model, e two bladed NRE wind turbine rotor Phase VI is used []. This model is a.m diameter HAWT tested in e NASA Ames

4 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7//.8m 6.7m wind tunnel. The blade sections have e geometric profile of e S89 aerofoil. The case study presented here is for e upwind baseline configuration wi a -tip pitch angle, a -cone angle and a constant rotational speed of 7.6 rpm. Analysis in is study was restricted to a free wind speed of m/s and a yaw angle of. This paper only considered e experimental data collected wi e H configurations (see ref. []).. Meodology Overview This section presents e eoretical principles required to build e proposed empirical stochastic model. Overviews on ese principles are presented as follows:. ata Sampling The normal and tangential force coefficient data were binned per one-degree azimu angle. Therefore, each individual bin includes a total of data points, each corresponds to an individual rotor rotation. The analysis in e remaining sections of is paper was conducted at ese samples/bins. Figure illustrates some samples/bins for ese data at U = m/s at r/r=. at.. Bin.6.. Bin Bin Bin. C N (-). Bin Bin C T (-)... Bin. (deg) -. (deg) (a) Normal force coefficient (b) tangential force coefficient Figure : Variation of normal and tangential force coefficients wi blade azimu angle at U = m/s at r/r=. at Arrows indicate a selection of data bins for statistical analysis.. Normality Test for e Cycle-to-Cycle Aerodynamic oads This subsection presents normality tests carried out using e Shapiro-Wilk p-value (p-value>.) [] to identify e type of distribution of e aerodynamic loads along all radial locations and blade azimu angles over rotor rotations. Shapiro-Wilk s test analyzes e hypoesis of normality based on a random sample. It is defined as e ratio of e best estimator of e variance to e usual corrected sum of squares estimator of e variance. Furer details can be found in []. Figure shows results from e Shapiro-Wilk p-value test for e normal and tangential force coefficients at U =m/s and =. From is test, e percentage numbers of data points having non-normal distribution are.7% for C N and.% for C T. Hence, it could be reliably assumed at all data are normally distributed for engineering analysis.. Testing characteristics of White Noise of e Cycle-to-Cycle Aerodynamic oads The autocorrelation function of a continuous random signal can be defined as a condition in which e samples are uncorrelated and identically distributed wi variance equal to. A white noise process is, on e oer hand, a random process wi a flat power spectral density (PS). Power spectral density function illustrates how much power is included in each of e spectral component and describes e distribution of e power of a signal wi respect to its frequencies []. According to Wiener-Khintchine Theorem [], e Fourier Transform of auto-correlation function of a random process is used to calculate e power spectral density function of e random process. The measured C N and C T data was binned per one degree azimu angle (see Figure, Subsection. ). Figure shows e auto-correlation function and e power spectral density of C N and C T white noise (wi zero -. Bin

5 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// mean values) determined at wind speed m/s at yaw angle at r/r=. at bin given in Figure. In Figure, e auto-correlation function for C N and C T indicates at e C N and C T white noise data are zero everywhere except at time lag zero (variance equal to at zero time lag) meaning at samples/variables are statistically uncorrelated wi respect to each oer. This condition satisfies e defined white noise process wi variance ( ) equal to variance or power of e corresponding C N and C T white noise data at zero time lags. On e oer hand, e power spectral density (PS) of e C N and C T white noise data across rotations gives almost fixed power in all e frequency domain (- to + ). The same observation was also found for all oer azimu angle bins for e different radial locations. r/r r/r (a) C N (b) C T Figure : Normally and non-normally distributed C N and C T data at U =m/s and =. ark colour indicates e non-normally distributed data (p-value.), while e white colour indicates e normally distributed data (p-value >.). C N noise (-) Correlation PS (db/hz). -. x - White noise: = =.78 CN CN Time (s) Auto-correlation Function of white noise Time lags (s) Power spectral density of white noise -.. Normalized Frequency (a) C N Time (s) Auto-correlation Function of white noise (b) C T Figure : Power spectral density and auto-correlation function of C N and C T white noise at U =m/s and = at r/r=. at C T noise (-) Correlation PS (db/hz) - x - x -6 White noise: = =.6e- CT CT Time lags (s) Power spectral density of white noise Normalized Frequency

6 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7//. Gaussian White Noise (GWN) Process In order to model a Gaussian white noise process, e standard Gaussian random variable is a commonly used model for a random/stochastic process as given by: X μ σ.normal(,) () where Normal(, ) is a random process wi zero mean and standard deviation equal to unity, X denotes e lift or drag time series. is e mean of lift or drag force which can be determined directly from measurements or from aerofoil data and correction models for dynamic stall/stall delay implemented in a rotor aerodynamics code (such as FWV/BEM codes). is e standard deviation of e lift or drag force which can be eier measured data or predicted by a CF code capable of simulating e unsteady aerodynamic loads. To apply equation, e optimal sample size should be evaluated as follows: Optimal Number of Samples The optimal number of samples is of great importance for Monte Carlo simulation (MCS) as is stochastic technique is mainly based on e use of random numbers to analyze problems of interest. The number of samples cannot be randomly selected as is may lead to significant errors in e results. In is study, e number of samples is selected according to convergence of standard deviation of e investigated parameters; e lift and drag force coefficients using different sets of random numbers as input values to equation. Thus, e number of samples was gradually increased from up, and e corresponding converged standard deviations of e lift and drag force coefficients were determined as shown in Figure. From Figure, it is obvious at, samples are sufficient for successful Monte Carlo simulation (MCS) as it is guaranteed at e standard deviation of such samples is not affected by e sample size, while ose below samples are significantly impacted by e sample size. Hence, samples wi, random variables were mainly used for e current verifications and analysis in is study. Convergence of Number of samples for MCS x Convergence of C... Number of samples for MCS x Figure : Illustration for convergence of standard deviation of lift force coefficient to select e optimal number of samples for MCS at U =m/s and = at r/r=. at. One-imensional Stochastic angevin Equation The prediction of e aerodynamic loads when moving from one cycle/rotor rotation to anoer at each time step is achieved by applying a first-order stochastic differential equation based on drift and diffusion coefficients (e angevin equation [8]) as given by: dx( t) () () ( X ) ( X ) G ( t) () dt where X denotes e lift or drag force coefficient, and G (t) is Gaussian white noise named as angevin force [8] wi mean ( G (t) =) and variance ( G(t) =), where represents e average over time. () (X) is e drift term presenting e deterministic part of e angevin equation and it estimates e mean change or mean time derivative of e aerodynamic loads. () (X) is e diffusion term presenting e stochastic part which is characterized as e fluctuations in e

7 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// aerodynamic loads since it quantifies e random noisy fluctuations togeer wi G (t). The drift and diffusion coefficients can be reconstructed directly from measured data (Siegert et. al [], Gottschall and Peinke [6]) wi e use of: n ( n) ( X, ) lim M ( X, ) () n! where M (n) is e first (n=) and second (n=) conditional moment respectively as expressed by M ( n) n ( X, ) : X ( t ) X ( t) () X ( t) X ; The drift and diffusion coefficients are determined by dividing e lift and drag force coefficient time series in each sample by e number of bins. Equations and are en applied for X(t) =X in each bin. The drift coefficient () (X) is determined using n=, while e diffusion coefficient () (X) is calculated using n=. Since e angevin force term, G (t), in Equation involves a Gaussian white noise process (a random process), it cannot be assumed at e estimated new stochastic signal, representing e aerodynamic loads at each time step, is accurate. This means at several stochastic signals will be differ from one anoer as determined by e term G (t). In order to have a reliable estimate of e new stochastic signal of e aerodynamic loads at each time step, Monte Carlo simulation (MCS) is performed based on averaged converged standard deviation of different sets of estimated stochastic signals as shown in Figure. It is apparent at Monte Carlo simulation (MCS) requires at least stochastic signals to successfully provide accurate estimation of e aerodynamic loads at each time step. Convergence of Number of stochastic signals. Number of stochastic signals Figure : Illustration for convergence of averaged standard deviation of lift and drag force coefficient across different numbers of stochastic signals for MCS at U =m/s and = at r/r=. at. NRE Rotor ift and rag ata for Validation of Empirical Stochastic Model The proposed stochastic model requires input of bo e mean and standard deviation of e lift and drag coefficient data for e different rotor blade radial locations and azimu angles. Such data should take into consideration e effects associated wi bo dynamic stall and stall delay phenomena normally experienced by yawed rotors operating wi high angles of attack at e blades. To validate e proposed model, unsteady lift and drag coefficient data for e NRE rotor were used. These were derived directly using e iterative procedure developed by Sant et al. [8] which applies a free-wake vortex model to determine e angle of attack from e blade normal and tangential load coefficient data obtained from e blade pressure measurements. In e present study, e free-wake vortex model WInS developed by Sebastian [7] was adopted, allowing e estimation of e time history of e time-varying angle of attack for consecutive rotor rotations, hence accounting for cycle-to-cycle variability. A detailed description of is iterative approach may be found in [9]. The time history for was en used wi e measured data for e normal and tangential loads to extract e corresponding aerodynamic lift and drag coefficients over e rotor evaluations. Finally, e latter values were used to obtain e mean ( and C ) and standard deviation ( and C ) values to be used as input to e stochastic model. It should be noted at given at direct measured load data were used, en flow effects resulting from stall delay and dynamic stall phenomena, as well as ose associated wi e blade flow at e tip, are already taken into account. The standard deviations were extracted from e experimental data under various wind speeds and yaw angles as illustrated in Convergence of C.6.. 6

8 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// Figure 7. These were eventually used as lookup tables in e developed empirical stochastic model when modelling e cycle-to-cycle variations of e aerodynamic loads. Standard deviation (-) r/r=. r/r=.7 r/r=.6 r/r=.8 r/r=.9 Standard deviation C (-) r/r=. r/r=.7 r/r=.6 r/r=.8 r/r= Angle of attack (deg) Angle of attack (deg) Figure 6: Standard deviations of e lift and drag force coefficients derived from e NRE rotor phase VI at several radial locations, wind speeds and yaw angles.. Numerical Algorim of e Empirical Stochastic Model This section describes e numerical algorim of e stochastic model when implemented in e freewake vortex model WInS [7]. The aerodynamic quantities obtained from e NRE rotor blade pressure measurements described in Section above ( and C, and C ) were used as input to predict e cycle-to-cycle aerodynamic loads on a rotating blade of e NRE rotor. The following procedure was applied over successive 6 azimu angle steps (blade rotational increments of ):. For each new time step,, random variables (samples) are generated for each individual bin following Gaussian white noise (see Equation and Subsection.). These are added to e mean experimental aerodynamic lift and drag force coefficients, and C = determined from e first stage (Section ) as given by: μ σ.normal(,) () C initial initial Ψ Ψ μc σc.normal(,) (6) Ψ where and C are e standard deviation of e lift and drag force coefficients. These are computed from e lookup tables generated using e data plotted in Figure 6.. The drift coefficients, () (C initial, ) and () (C initial, ), and e diffusion coefficients, () (C initial, ) and () (C initial, ), are determined from e, samples generated in Step of e second stage and Equations and. The value of drift can be estimated by fitting a linear function, while e diffusion can be predicted by a second-order polynomial wi least square regression scheme applied for each case. Figure 7 illustrates sample results for e estimated () (C initial, ) and () (C initial, ) at bin in Figure at U = m/s and at r/r=... The angevin process (Equation ) is en applied to determine e lift and drag force coefficients at ( ) or (t=) over successive rotor rotations as given in e discrete form by: C C Ψ Ψ μ μ Ψ C Ψ τ τ () () Ψ () ( C,Ψ ) τ ( C,Ψ ) Normal(, ) initial () initial ( C,Ψ ) τ ( C,Ψ ) Normal(, ) initial initial where Normal (, ) is Gaussian white noise wi mean and variance. The time increment is equal to e inverse sampling rate. This may be replaced by e sampling (7) (8) 7

9 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// frequency of e actual values. Equations 7 and 8 are solved by Monte Carlo Simulation using stochastic random signals as explained in Subsection. and shown in Figure 8. R-square =.7; RMSE =.668 R-square =.; RMSE =.7.6. () (C ) data () (C ) data () (C ) fitted. (). (C ) fitted ()() ()() (a) ()(, ) (b) ()(, ) Figure 7: Estimated drift and diffusion coefficients of and C at bin in Figure at U = m/s and at.r.. Because e aerodynamic loads deviate around e mean and C values at each blade azimu angle across rotor rotations (see Figure 8), e model was extended to present e aerodynamic load oscillations for e current time step over rotor rotations as follows: σ cos( π f s ) σ C cos( π f s ) model C Ψ (9) C model C () Ψ where fs is e frequency of e wave ( f s /Tbin ), where Tbin is e leng of e time period at each azimu angle or bin presented in Figure. In Figure 8, e grey color represents e generated stochastic signals which have been already verified in Figure. The solid black markers denote e estimated averaged solution over e stochastic signals determined by Equations 9 and. Each individual marker represents e predicted value at each individual rotor rotation at each new time step and each individual bin. Stochastic signals Stochastic signals Model Measurement. CT (-) CN (-).6. Model Measurement Time (s) -. Time (s) Figure 8: Unsteady normal and tangential force coefficients predicted by e extended model (Equation 9 and ) at U = m/s, =, r/r=. at bin in Figure.. A necessary step to ensure at e predictions of e aerodynamic loads at each time step are reliable is to check for any predicted values outside e 99% confidence interval. It is crucial to make sure at 99% of e predicted aerodynamic loads always contain e true value. This is achieved to obtain e final unsteady cycle-to-cycle aerodynamic lift and drag force coefficients as follows: 8

10 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7//.8 C final () model N.8 C C final () model N where N is e sample size (N=,). 6. The bound circulations, B, are computed using e Kutta Joukowksi eorem using Equation and e cycle-to-cycle lift force coefficients C final. Γ B. Vrel.c () final 7. The free wake downstream for e current time step is generated to determine e unsteady induced velocities on e lifting line and all wake nodes via Biot-Savart law over rotor rotations. The angle of attack is en determined by: tan ( V n / V t ) () 8. The cycle-to-cycle normal and tangential force coefficients are en determined by Equations and 6. C C cos( ) C sin( ) () C N T final final C sin( ) C cos( ) (6) final Steps -8 are repeated for all time steps and radial locations over a whole rotor rotation (6 time steps). 6. Results Sample results from e empirical stochastic model are presented in Figure 9 for ree individual rotor rotations,, and, at wind speed m/s and a yaw angle at e % span location. Comparisons over rotor rotations are also achieved at U = m/s at at R and.8r as illustrated in Figure. It is observed at ere is in general very good agreement between e measurements and e predictions. Some slight under/over prediction can be observed in Figure 9 due to assumptions and estimations made in is study. It was previously assumed at all data follow a normal distribution for each blade radial and azimu position, but in fact some small regions do not really follow Gaussian distributions (Figure ). It is also expected at e determined drift and diffusion coefficients suffer from errors created by e strong measurement noise as well as e curve fitting procedure (Figure 7). espite ese, it is shown at e stochastic model is successfully capable of reproducing e cycle-to-cycle variability in e load coefficients experienced at blade azimu positions having a large angle of attack. Indeed, e averaged loads are already affected by e dynamic stall phenomenon, while e unstable/random process of e dynamic stall vortex shedding causes e variations around e mean values from cycle (revolution) to anoer. Hence, e model is able to capture e effects of e dynamic stall vortex shedding at high angles of attack relatively well. On e oer hand, e strategy of e binning process presented in Figure would simplify e complexity of e process used in e developed stochastic model when moving from a bin to anoer (i.e. when progressing wi time). Such binning process would typically reduce e correlation between e samples/bins given at e averaged values of C N and C T at each new bin are eier measurements or determined by a dynamic stall and stall delay models integrated wi a rotor aerodynamics code (as for example FWV or BEM codes). The results shown in Figures 9 and indicate at e used binning process is still valid to provide an acceptable level of accuracy. final 9

11 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// Cycle (NRE) Cycle (Predicted) 6 T Cycle (Predicted) N C (-) C (-) Cycle (NRE) Cycle (Predicted) (deg) Cycle (Predicted) Cycle (NRE) Cycle (Predicted) Cycle (NRE) Cycle (NRE) Cycle (NRE) Cycle (Predicted) (deg) Figure 9: A comparison between measured and predicted normal and tangential force coefficients at U = m/s at = at.r. These are results of, and individual rotor rotations. 6 Measurements (NRE) FWVM-( empirical stochastic model) Measurements (NRE) FWVM-( empirical stochastic model) CT (-) CN (-) (deg) (deg) (a).r Measurements (NRE) FWVM-( empirical stochastic model). Measurements (NRE) FWVM-( empirical stochastic model).6. CT (-) CN (-) (deg) (deg) (b).8r Figure : A comparison between measured and predicted normal and tangential force coefficients at U = m/s at = at.r and.8r. Results are shown for rotor rotations.

12 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// 7. Conclusions A empirical stochastic model has been developed and applied to predict e cycle-to-cycle aerodynamic loads on e blades of e NRE Phase VI rotor under yawed conditions. The results presented here show at e assumptions and approximations (e standard deviation, e normally distributed data, e estimated drift and diffusion coefficients and e white Gaussian noise principles) used wiin e empirical stochastic model should provide sufficient accuracy to develop a fully eoretical stochastic model for wind turbine design applications. The model presented in is paper was based on e input of e mean and standard deviation values of aerodynamic loads from e NRE wind tunnel measurements. Furer work is intended to apply e same technique wi e use of dynamic stall and stall delay models integrated in a FWV or BEM model to independently predict e mean values of e aerodynamic loads instead of using measurement data, while e standard deviation data can be obtained from a CF code. Furer work will also test e developed model wi different wind turbine rotors for which blade aerodynamic load distribution measurements are available over multiple rotations. References [] Hand M M, Simms A, Fingersh J, Jager W, Cortell J R, Schreck S and arwood S M Unsteady Aerodynamics Experiments Phase VI: Wind Tunnel Test Configurations and Available ata Campaigns; Technical Report NRE/TP--99; National Renewable Energy aboratory: Golden, CO, USA. [] eishman J G Challenges in Modelling e Unsteady Aerodynamics of Wind Turbines. Wind Energy, : 8. doi:./we.6. [] Peters A 98 Toward a Unified ift Model for Use in Rotor Blade Stability Analysis. Journal of American Helicopter Society; (): -. [] Øye S 99 ynamic stall simulated as time lag of separation, Technical report, epartment of Fluid Mechanics, Technical University of enmark, enmark. [] Rasmussen F 99 Engineering model for dynamic stall. Risø-M-8. Risø National aboratory. [6] Bertagnolio F Rasmussen F., Sørensen N.N., Johansen J., Madsen H.A, A stochastic model for e simulation of wind turbine blades in static stall, Wind Energy; (): -8. doi:./we.. [7] uhur M R, Peinke J, Schneemann J and Wächter M Stochastic modeling of lift and drag dynamics under turbulent wind inflow conditions, Wind Energy. doi:./we.699. [8] Risken H 996 The Fokker Planck Equation: Meods of Solution and Applications, nd ed.; Springer: Berlin, Germany; 8: -6. [9] uhur M R, Peinke J J, Kühn M M, Wächter M M Stochastic Model for Aerodynamic Force ynamics on Wind Turbine Blades in Unsteady Wind Inflow. ASME. J. Comput. Nonlinear ynam. ():--. doi:./.896. [] Wächter M, Milan P, Mücke T and Peinke J Power performance of wind energy converters characterized as stochastic process: applications of e angevin power curve. Wind Energy, : doi:./we.. [] Gottschall J 9 Modelling e variability of complex systems by means of angevin processes: On e application of a dynamical approach to experimental data, Ph esis, The Carl von Ossietzky University of Oldenburg, Germany. [] Shapiro S S and Wilk M B 96 An analysis of variance test for normality (complete samples). Biometrika; (-): 9 6. doi:.9/biomet/.-.9. JSTOR 79. MR 8. [] Brillinger R and John W Tukey's work on time series and spectrum analysis. The Annals of Statistics, (6): [] Chatfield C 989 The Analysis of Time Series-An Introduction, st edition, Chapman and Hall, ondon; 9 9. ISBN

13 The Science of Making Torque from Wind (TORQUE 6) Journal of Physics: Conference Series 7 (6) doi:.88/7-696/7// [] Siegert S, Friedrich R and Peinke J 998 Analysis of data sets of stochastic systems. In: Physics etters A, 7-8. [6] Gottschall J and Peinke J 8 On e definition and handling of different drift and diffusion estimates. New Journal of Physics, 8. [7] Sebastian T The Aerodynamic and Near Wake of an Offshore Floating Horizontal Axis Wind Turbine. Ph.. Thesis, University of Massachusetts, Amherst, MA, USA. [8] Sant T, van Kuik G A M and van Bussel G J W 8 Estimating e Angle of Attack from Blade Pressure Measurements on e NRE Phase VI Rotor using a Free-Wake Vortex Model: Yawed Conditions. Wind Energy; (): -. doi:./we.8. [9] Elgammi, M. and Sant, T., Combining Unsteady Blade Pressure Measurements and a Free- Wake Vortex Model to Investigate e Cycle-to-Cycle Variations in Wind Turbine Aerodynamic Blade oads in Yaw. Energies 6, 9(6), 6, doi:.9/en966.