The Turbulent Nature of the Atmospheric Boundary Layer and itsImpact on the Wind Energy Conversion Process

Matthias Wachter, Hendrik Heißelmann, Michael Holling, Allan Morales,Patrick Milan, Tanja Mucke, Joachim Peinke, Nico Reinke, Philip Rinn∗

Accepted for publication in the Journal of Turbulence on May 17, 2012.

AbstractWind turbines operate in the atmospheric boundary layer,where they are exposed to the turbulent atmospheric flows.As the response time of wind turbine is typically in therange of seconds, they are affected by the small scale in-termittent properties of the turbulent wind. Consequently,basic features which are known for small-scale homogeneousisotropic turbulence, and in particular the well-known inter-mittency problem, have an important impact on the windenergy conversion process. We report on basic researchresults concerning the small-scale intermittent properties ofatmospheric flows and their impact on the wind energy con-version process. The analysis of wind data shows stronglyintermittent statistics of wind fluctuations. To achievenumerical modeling a data-driven superposition model isproposed. For the experimental reproduction and adjust-ment of intermittent flows a so-called active grid setup ispresented. Its ability is shown to generate reproducibleproperties of atmospheric flows on the smaller scales of thelaboratory conditions of a wind tunnel. As an applicationexample the response dynamics of different anemometertypes are tested. To achieve a proper understanding of theimpact of intermittent turbulent inflow properties on windturbines we present methods of numerical and stochasticmodeling, and compare the results to measurement data.As a summarizing result we find that atmospheric turbu-lence imposes its intermittent features on the completewind energy conversion process. Intermittent turbulencefeatures are not only present in atmospheric wind, but arealso dominant in the loads on the turbine, i.e. rotor torqueand thrust, and in the electrical power output signal. Weconclude that profound knowledge of turbulent statisticsand the application of suitable numerical as well as ex-perimental methods are necessary to grasp these uniquefeatures and quantify their effects on all stages of windenergy conversion.

1 IntroductionWind Energy is one of the promising renewable energiesof the presence, see e.g. Ref. [1]. It is therefore of primaryimportance for future sustainable energy supply systems.During the past decades, considerable advances in sci-ence and engineering have lead to wind energy converters(WECs) reaching efficiencies and sizes which appearedimpossible only a few years ago.

∗ForWind Center for Wind Energy Research, Institute of Physics,Carl von Ossietzky University, 26111 Oldenburg, Germany, email:matthias.waechter@uni-oldenburg.de

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Figure 1: PDF of measured wind speed incrementsδuτ (t) = u(t + τ) − u(t) for time lag τ = 3s (blue line)compared to a Gaussian PDF with same standard devia-tion (black line). At 7σ both differ by a factor of 107. Forthe mean wind speed of 6.6 m/s, τ = 3s corresponds to aspatial separation of about 20 m, which is a relevant scalefor, e.g., rotor blades of current wind turbines. Note thelogarithmic scale of the vertical axis. For details of themeasurement please see Ref. [2].

In contrast to the amazing progress in design and manu-facturing of WECs, the understanding of the physics of theenergy resource, i.e. the turbulent atmospheric boundarylayer, and its interaction with the WECs leaves many openquestions and remains insufficient. To give one example, inthe context of wind energy commonly Gaussian statisticsof fluctuations of wind fields are assumed, which is also re-flected in the current industry standard IEC 61400 [3]. It iswell-known that in turbulence highly intermittent (i.e. non-Gaussian) statistics are found, especially for probabilitydensity functions (PDFs) of velocity increments

δuτ (t) = u(t+ τ)− u(t) . (1)

This effect is even stronger in atmospheric flows [4–11].Figure 1 shows that for a 7σ event of δuτ the GaussianPDF underestimates the probability by a factor of 107,compared to the measured statistics. This means if theGaussian statistics predicts the event to occur every 1250years, in reality it is expected to happen once every hour.

Wind velocity increments as a function of the time scale τare chosen here because we are interested in the dynamicalresponse of wind energy systems to turbulent excitationby atmospheric flows. Based on a rough estimation usingTaylorâĂŹs hypothesis of frozen turbulence (1 s corrspondsto about 10–20 m) it is evident that the aerodynamicforces typically show dynamics on time scales even below

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1 s. Mechanical, electrical, and electronical componentsof a WEC present dynamic behavior with time constantswhich range up to 1 min due to, e.g., yaw control. Froma mathematical point of view the PDFs p(δuτ ) representtwo-point statistics including its higher order moments,as will be explained in Sec. 2.1. These are of particularrelevance for the nonlinear dynamics of a WEC.

The focus of this paper is twofold. First, we aim at adeeper understanding of key features of atmospheric tur-bulence from the perspective of a WEC system, i.e., ontemporal scales up to 1 min. We present results on howthe wind resource can be properly characterized, modeled,and experimentally reproduced on these scales. Second,we are interested in the effect of these turbulent featureson WEC systems, and how their properties are reflected inthe generated loads and electrical power output. For theinvestigation of both aspects, we have developed methodswhich are presented in this paper. The aim is to show inwhich aspects advanced turbulence research can contributeessentially to wind energy research. We claim that animproved understanding of turbulence is inevitable for newinsights in the response dynamics of WEC systems, in-cluding loads, extreme values, and power output variation.Rather than giving a comprehensive overview of existingmethods, we exemplarily present a number of approacheswhich are focused on the mentioned intermittent featuresof atmospheric turbulence. The point of our work is notclaiming to present the best methods of turbulence inves-tigations, but to show the necessity of further advancedturbulence research for wind energy systems.

The structure of the paper is as follows. Section 2 isinvestigating the typical small-scale intermittency of at-mospheric wind fields. Methods for the proper analysisand characterization are presented in Sec. 2.1, especiallyfocusing on the connection between homogeneous isotropicturbulence (HIT) and atmospheric turbulence. In Sec. 2.2we show methods for the proper experimental reproduc-tion of atmospheric flow conditions in a wind tunnel setup.In the second part of this section we discuss the impactof turbulent inflow conditions on a simple experimentaldynamic response system. Results are discussed on thebackground of HIT features in atmosperic turbulence. Theimpact of atmospheric turbulence on WEC systems is inves-tigated in Sec. 2.3, showing that the typical intermittencyfeatures affect all components of a WEC system as well asforces, loads, and power output. Section 3 is devoted tothe problem of how to quantify the impact of atmosphericturbulence on aspects of the wind energy system. We treatthe dynamics of power conversion by WECs as a result ofthe interaction between machines and atmospheric turbu-lence in terms of stochastic response systems. Section 4finally gives an outlook to future work towards a multiscaledescription of atmospheric turbulence in terms of arbitraryn-scale joint probabilities.

2 Small-Scale Intermittency

2.1 Analysis and CharacterizationAtmospheric flows constitute the resource or, loosely spo-ken, the “fuel” of wind energy systems. Detailed knowledgeof their properties and behavior is therefore essential forany improvement in wind energy conversion.

Wind fields are complex structures, fluctuating in timeas well as in space. The industry standard IEC 61400 [3]defines quantities and procedures for considering the influ-ence of atmospheric turbulence on WECs. As a practicalapproach to wind field characterisation, the ten minutemean value of the horizontal wind speed, u = 〈u(t)〉10min,is used together with the standard deviation σ with respectto the same time interval. These quantities are one-pointstatistics and thus can not reflect the succession of velocityvalues. Moreover, the dynamics of current WECs is notgrasped by ten-minute statistics. Fast load changes on therotor, the dynamical respons of the WEC, and resultingpower output dynamics rather take place at a time scaleof the order of 1 s.

The IEC standard proposes for a more detailed charac-terization of wind fields universal power spectral densitiesof the horizontal wind speed. These are also commonlyused for the synthetic generation of wind inflow data [12].An important limitation of these spectral models are thepurely Gaussian statistics of the resulting wind fields, espe-cially in the wind speed increments, which does not reflectexperimental results, as shown in Fig. 1.

While necessity and usefulness of standards are beyonddoubt, in the recent years growing demand for a more com-prehensive and more detailed characterization has becomeevident. To characterize variations of u in a general waywe consider wind speed increments

δuτ (t) = u(t+ τ)− u(t) (2)

as already introduced in Sec. 1. The moments 〈(δuτ )k〉of these increments and their dependence on the scale τare also called structure functions and reflect the variationstatistics of order k. It is straightforward to see that thesecond-order structure function 〈(δuτ )2〉 is directly relatedto the auto-correlation function and thus also to the powerspectral density. As 〈(δuτ )2〉 grasps only the width ofthe PDF p(δuτ ) it becomes clear that, taking only thepower spectra into account, Gaussian statistics of p(δuτ )are implied. Alternatively to the moments, also the τ -dependence of the PDFs p(δuτ ) can be investigated, whichconsequently reflect the variation statistics of all orders.

For practical applications, the high complexity of atmo-spheric flows has to be further disentangled and a set ofcrucial parameters and methods has to be defined. A num-ber of advances could already be achieved in this direction.In [5, 6] a model for atmospheric wind speed incrementPDFs was presented, reproducing characteristic propertiesof different locations, such as offshore sites or onshore sitesin different terrain. Based on earlier work of Castaing [13]the PDFs are considered as multiple superpositions of Gaus-sian distributions, combining the intermittency at a givenmean wind speed and the typical Weibull distribution of10 min mean wind speeds. An application example for anoffshore site located in the Baltic Sea is presented in Fig. 2.Especially the intermittency and the evolution of the shapeof the PDFs with the scale τ is well reproduced by themodel. Nevertheless, slight deviations remain. For details,including the derivation of the parameters, please referto Ref. [6]. This work is currently continued using newfindings from extended data analysis of other offshore mea-surement campaings. In a further step, this concept wasembedded into a hierarchical framework for the statisticalcharacterization of atmospheric turbulence in Ref. [14]. An

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Figure 2: Modeling of atmospheric probability densityfunctions for an offshore site, reproduced after [6]. Symbolsrepresent the normalized PDFs of atmospheric data sets insemi-logarithmic presentation. Solid lines correspond to amodel according to Eq. (20) of [6]. All graphs are verticallyshifted against each other for clarity of presentation. Fromtop to bottom τ takes the values of (0.2, 10, 20, 200, 2000) s.

advanced procedure was proposed for the disentangling ofatmospheric wind measurements into subsets which behavesimilar to homogeneous, isotropic turbulence. It could beshown that for the mentioned subsets the dependence ofthe intermittency parameter λ2 (for a definition please referto [14]) on the time scale τ corresponds with Kolmogorov’srefined scaling law for homogeneous isotropic turbulenceof 1962 [15],

〈(δuτ )n〉 ∝ τ n3−µ

n(n−3)18 . (3)

Here, µ characterizes the evolution of intermittency in thescaling behavior of structure functions. The dependenceλ2(τ) = λ2

0 − 19µ log τ is covered by (3) for n = 4, see

appendix of [14]. For different atmospheric data sets thevalue of µ = 0.26± 0.4 reported in the literature [15] couldbe confirmed.

In this section we present a similar model for PDFsof atmospheric wind speed fluctuations u′(t) = u(t) − uwith u = 〈u(t)〉10 min, which is a common quantity in thewind energy field. As u′ constitutes one-point statisticsit should nevertheless be clearly distinguished from thetwo-point statistics of increments δuτ . A first approachbased on cup anemometer data was included in [14]. Inour extended analysis here we use one month of ultrasonicdata measured at the offshore research platform FINOI, where the wind speed u is taken in the direction ofu. Ultrasonic anemometers offer higher quality data interms of higher sampling rates (10 Hz in our case) thancup anemometers and avoid effects of rotational inertia. Incontrast to laboratory turbulence, the pdf p(u′) is stronglyintermittent, see figure 3(a). Even after conditioning thestatistics of u′ on a certain value of u the intermittencypersists, see p(u′|u) in figure 3(b). Thus, the source ofintermittency is not only the non-stationarity of the meanwind speed. As shown in [14] in fact the statistics of u′ seemto follow a Gaussian distribution within single 10-minutetime spans. Therefore, the effect of the atmosphere can beunderstood as mixing these clean Gaussian statistics bythe randomization of the ten-minute standard deviationsσT over periods longer than 10 minutes. The distributionp(σT ) can in most cases at least approximately be describedby a log-normal distribution [16], compare Figs. 3(c) and

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Figure 3: Statistics of fluctuations u′ of atmospheric windspeeds, measured at the German offshore research platformFINO I which is located in the German Bight about 40 kmnorth of Borkum island. Measurements were taken in100 m height, where the yearly average wind speed is closeto 10 m/s. Parts (a) and (b) present empirical PDFs offluctuations u′ (symbols), model results according to Eq. (4)(solid lines) and Gaussian PDFs for comparison (dashedlines). Part (a) shows p(u′) for all data and (b) p(u′|u)conditioned on u = (10± 1) m/s. In parts (c) and (d) weshow distributions of logarithms of the standard deviationsσT . Part (c) again represents p(log σT ) for the full dataset while (d) shows p(log σT |u) conditioned as in (b).

(d).Inspired by the works of Castaing [13] and Boettcher [6]

we can now set up an improved model the PDFs of u′as superpositions of Gaussian distributions with differentstandard deviations, where the PDF of these standarddeviations itself is log-normal:

p(u′) =∫ ∞0

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where σm = exp {〈log σT 〉} and α2 = 〈(log σT )2〉 −〈log σT 〉2. In this model, α2 plays a similar role as λ2

in the statistics of increments, and quantifies the amountof intermittency in the PDF. This compact model repro-duces the empirical values quite well, as presented in figures3(a) and (b) for the unconditioned and conditioned PDFsof atmospheric wind speed fluctuations, respectively. Asall parameters of (4) can be derived from the basic 10-minute statistics which are commonly measured duringfield campaigns for wind energy applications, our modelgives access to the high-frequency statistics of u′ even ifno high-frequency data have been recorded. The windspeed fluctuations u′(t) = u(t)− u as discussed here haveto be clearly distinguished from increments δuτ , see (2)

3

and Fig. 2. Considering together the results for incrementsshown in Fig. 2 and fluctuations in Fig. 3, now models ofboth the one-point and two-point statistics for wind dataexist. Their validity was shown using examples of offshoredata. Both the wind speed fluctuations u′ and incrementsδuτ are relevant for WECs. Using these models, completeand straightforward characterizations of both quantitiesare possible. Besides the models described here, there existother possibilities, such as special parametric distributions,but a systematic comparison is beyond the scope of thispaper. We believe that a strength of our approach is itsmotivation by the physics of turbulence and in the easyaccess to the necessary parameters.

A proper characterization forms the basis for an im-proved understanding of atmospheric turbulence, whichin term leads to improved experiments, models, syntheticwind fields, and load estimations, to name a few examples.

2.2 Experimental ReproductionBesides an improved characterization and description ofatmospheric wind fields, it is of special relevance for experi-mental investigations to properly reproduce these statisticalfeatures in the wind tunnel. This includes in particular arescaling of spatial correlations (e.g., gusts) to dimensionsof the wind tunnel and the models in use, respectively.Up to today there exists no method for the generation ofturbulence that allows for a complete control of spatialcorrelations nor statistical properties of the resulting flowon all scales due to the natural decay of laboratory turbu-lence. The general aim of our setup is to generate and toreproduce intermittency features in a controlled way and ina defined response time window. To the best of our knowl-edge, this has not been addressed before by turbulencegenerating setups. Classically, atmospheric wind tunnelsare used for the generation of atmospheric like wind fields,where the flow is evolving over passive roughness elementsplaced over several meters in the wind tunnel. This methodis very limited when it comes down to the generation ofwind fields with defined intermittency. Furthermore, sev-eral approaches exist which combine different turbulencegenerating methods, e.g. [17–19]. This subsection describescurrent results concerning options and limits of artificiallycreated turbulence by an active grid with respect to spatialstructures and reproducibility.

Active grid

To overcome the problem of fixed solidity and fixed ef-fective mesh size of classical grids for the generation ofturbulence, a more flexible and powerful set-up was builtup at the University of Oldenburg. The so-called activegrid, cf. [20,21], allows for the generation of turbulent flowsby actively rotating nine vertical and seven horizontal axeswith square-shaped vanes mounted to them (see Fig. 4).Each of these axes is controlled by a stepper motor andcan be accessed individually with a maximum resolution of51200 steps per rotation. This grid can be used in passivemode with statically adjusted axes, as well as in activemode with moving axes. In active mode each axis can bedriven by e.g. sinusoidal or randomized signal resultingin a time-dependent solidity. This allows for a flexibleexcitation of the flow. Figure 5 shows a comparison of

Figure 4: Active grid mounted to the wind tunnel outlet atthe University of Oldenburg. Visible are the horizontal andvertical axes with mounted vanes and individual steppermotors.

the PDFs for different time lags measured in passive modewith all vanes oriented parallel to the inflow direction re-sulting in the minimum achievable solidity (a) and thePDFs measured in active mode with a randomized signal(b). It can clearly be seen that in the static configurationonly for the smallest time scales intermittency is observed.For the active mode, on the other hand, a pronouncedintermittency is generated for a wider range of time scales,as it was also implemented in the excitation signal [22].This shift in the turbulent cascade is also reflected in theReynolds number, which is Rλ = 188 for the passive modeand Rλ = 2, 243 for the active mode. The direct couplingof the motion of the individual axes to the resulting char-acteristics of the generated flow field therefore allows for adefined manipulation of the flow.

Reproducibility

To investigate the reproducibility of turbulent wind fieldswith different spatial structures created by the active grid,a control signal for the axes as shown in Fig. 6(a) wasrealized. The complete protocol consists of several of thesesequences in which the angular position of the axis changesfrom 90◦ to 72◦, 54◦, 36◦, 18◦ and 0◦. In the setup 90◦corresponds to a vertical orientation of the mounted vaneswith respect to the incoming wind velocity. Here, thetime for each complete pulse was chosen to be 0.2 seconds.One pulse is defined as the time from the beginning ofa plateau until the end of the following flank. With amaximum angular velocity of 9◦ per 10ms, a pulse lengthof 0.2 seconds results in 0.1 seconds, where the flaps arein the position of 0◦ and was chosen to be the shortestreasonable pulse. In order to keep the overall solidityof the grid constant for the whole sequence the protocolfor the outer eight axes has been programmed with anoffset of 90◦. For the characterization of the resultingflow, velocity measurements on the centerline behind thegrid have been carried out using hot-wire anemometrywith a sampling frequency of 40 kHz for different distancesd to the grid varying from 10 cm to 180 cm with ∆d =10 cm and an inflow velocity of 16 m/s. Figure 6(b) showsthe resulting velocity of the implemented sequence withincreasing distance to the grid. It can clearly be seenthat the different angles of the axes result in pulses with

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Figure 5: (a) PDFs of velocity increments δu = u(t +τ)− u(t) for passive mode of the active grid, reproducedafter [22]. Time lags τ from top to bottom are 2.5 · 10−4 s,5 · 10−4 s, 2.5 · 10−3 s, . . . , 5 · 10−1 s. (b) PDFs of velocityincrements for the active grid with stochastic control ofvane movement. Time lags τ are as in (a). The graphs areshifted vertically for clarity of presentation.

increasing velocities. It can also be seen that the measuredflow velocities change with increasing distance to the activegrid, which indicates a decay of the generated velocitypulses and spatial structures, respectively.

To investigate the decaying process of these structures inmore detail, three excitation protocols with several identi-cal sequences like presented in Fig. 6(a) have been realizedwhere just the length of the pulses has been changed. Se-quences with pulses of 0.2, 0.5, and 1 second durationhave been implemented and used for the excitation of thegrid’s axes. In order to determine where the uncorrelateddecay of natural turbulence starts to dominate, each excita-tion sequence has been identified in the measured velocitytime series and separated. To filter out spatial structuresfrom the flow, each of these velocity sequences has beenhigh-pass filtered with different filter frequencies startingfrom 0.14 Hz up to 133 Hz. Figure 7(a) shows an exampleof measured velocities for one excitation sequence with apulse length of 0.2 seconds at distance d = 40 cm togetherwith the high-pass filtered signal for filter frequencies of5 Hz (green) and 40 Hz (red). The signals are shifted verti-cally for clarity of presentation. It can clearly be seen thathigher filter frequencies result in damping of the impressedpulses.

Furthermore, correlation coefficients have been deter-mined for different realizations of measured velocities usingidentical filter frequencies. Figure 7(b) shows the averageof the resulting correlation coefficients for sequences of allthree pulse durations plotted over the filter frequency ofthe applied high-pass filter. The velocity sequences have

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been measured at a distance of d = 40 cm to the grid. Thedashed lines represent the corresponding standard devia-tion for each graph. It can clearly be seen that the filterfrequency at which the correlation coefficient drops offincreases with shorter pulse durations. Applying Taylor’sHypothesis, this is also in agreement with the equivalent inspatial terms, where shorter excitation pulses correspondto smaller spatial structures and therefore need higher filterfrequencies to be filtered out.

To investigate the stability of the generated spatial struc-tures with increasing distance to the grid, the measuredvelocity sequences have been split into isolated single pulses.Conditioned on pulses with identical motion of the vanes,this procedure allows for a similar analysis as already ap-plied to the whole sequence. Based on the split sequencesa plot according to Fig. 7(b) can be calculated for eachdistance d to the grid, conditioned on pulses 1, 2, 3, 4,and 5, respectively, for all three pulse durations. The filterfrequency for which the correlation coefficient drops offto 90% of its respective maximum was defined to be thecutoff frequency. It defines the smallest time, and thus thesmallest spatial structure, of the generated flow that is notdominated by natural decay and therefore can reproduciblybe generated by the active grid. The plots in figure 8 showthe cutoff frequency over the distance d to the grid for (a)conditioned on the pulse with the largest resulting velocitydifference and (b) conditioned on the pulse with the secondlargest velocity difference for all pulse durations. The re-sults show that the cutoff frequency for the pulse with thelargest velocity difference is nearly constant over the wholedistance and the standard deviation for the shortest pulseduration increases with distance (Fig. 8(a)). In Fig. 8(b) a

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sudden decrease in the cutoff frequency is visible at a dis-tance d = 130 cm for the pulse with the shortest durationof 0.2 s, whereas the cutoff frequencies for the pulses withlonger pulse duration stay constant. This indicates, thatthe spatial structure created by the 0.2 second pulse is notstable over the whole measurement distance, which has tobe accounted for in the design of excitation protocols.

Test of standard anemometers

The ability of the active grid to generate turbulent flowsituations with reproducible temporal and spatial features,as shown in the previous subsection, provides a power-ful tool not only for basic turbulence investigations, butalso for the characterization of airfoils and model windturbines exposed to intermittent wind fields. As an appli-cation example, comparative measurements of the sameflow situation have been carried out with two standardanemometers for wind energy applications: a Thies FirstClass Advanced cup anemometer and a Gill WindmasterPro 3D ultrasonic anemometer. The signal recorded witha standard 1D hot-wire probe served as a reference, sincethe hot-wire provides the highest spatial and temporalresolution.

While 1D hot-wire and cup anemometer lack any di-rectional information, the data from the 3D ultrasonicanemometer includes the horizontal wind speed magnitude

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Figure 8: Cutoff frequency over distance d to the grid.The cutoff frequency describes the filter frequency of theapplied high-pass filter, where the averaged correlationcoefficient of sequences conditioned on (a) the pulse withthe highest velocity difference and (b) the pulse with thesecond highest velocity difference decreased to 90% of itsrespective maximum. The curves belong to pulses withduration of 0.2s (black), 0.5s (blue) and 1s (red). Thedashed lines indicate the corresponding error.

and direction. However, the alignment of transducers andsupporting structures of the sonic anemometer can causeflow disturbances under certain orientations [23]. The usedGill Windmaster Pro anemometer features three supportrods with an angular distance of 120◦, where the positionof the sensor’s north spar coincides with the position ofone supporting rod. Thus, the northward orientation (0◦)of the anemometer implies a support structure being up-stream the measurement volume, while a southward (180◦)orientation implies free inflow to the measurement volume.

All sensors were calibrated before the measurement andhave been successively placed at a distance d =130 cmdownstream of the active grid with the center of theirrespective measurement volumes on the centerline of thewind tunnel test section. The excitation protocol of theactive grid was chosen in such a way, that the resultingmodulated wind flow obeys non-Gaussian statistics as theycan be found in atmospheric wind fields.

For each mean wind speed, 30 minutes of data wererecorded with hot-wire, cup anemometer and ultrasonicanemometer in northward and southward orientation. Thehot-wire data was sampled at 20 kHz and smoothened us-ing an average over 20 samples. The lower-resolving cupanemometer and sonic anemometer were sampled at 4 Hzand 32 Hz, respectively. Figure 9(a) shows a 30 secondexcerpt of the time series recorded with hot-wire (black),

6

(a)

(b)

Figure 9: (a) 30-seconds-excerpt of the wind speed datameasured with 1D hot-wire anemometer (black), 3D ul-trasonic anemometer (red) in southward orientation andcup anemometer (blue). The general evolution of the tur-bulent flow is covered by all anemometers, but the cupanemometer cannot resolve the fast fluctuations with smallamplitudes. (b) PDF of the wind speed fluctuations u′for the down-sampled (4 Hz) time series from (a) withGaussian fit (dashed gray).

ultrasonic anemometer (red) in southward orientation andcup anemometer (blue). A good agreement of the time se-ries for the hot-wire and the sonic anemometer is observedand the fact that all anemometers capture the strong fluc-tuations at e.g. t =2 s or t =13.5 s gives evidence aboutthe robustness and reproducibility of the used excitationprotocol. However, the cup anemometer can only resolvethe slower and more pronounced gust events. The fasterfluctuations of the wind speed (e.g. t = 22 s to t = 25 s)cannot be resolved, which on the one hand is due to itslow temporal resolution and on the other hand is a resultof the spatial extent of the anemometer itself.

A comparison of the basic statistical quantities com-monly used in wind energy, i.e. mean wind speed u, stan-dard deviation σu and turbulence intensity TI = u/σu, forthe time series shown in Fig. 9(a) is given in Tab. 1. Only aslight deviation from the hot-wire reference is observed forthe ultrasonic anemometer’s data, while all quantities of the

Table 1: Measured mean wind speed u, standard deviationσu and turbulence intensity TI for the time series fromFig. 9(a).

sensor u σu TI

hot-wire 17.1 m/s 3.8 m/s 22 %sonic south 17.6 m/s 3.5 m/s 20 %cup 15.5 m/s 2.3 m/s 15 %

cup anemometer are significantly lower than the reference.A closer look at the statistics for all sensors is provided inthe PDFs of the measured wind speed fluctuations u′ inFig. 9(b). The higher resolved hot-wire data and ultrasonicanemometer data have been down-sampled to the resolu-tion of the cup anemometer at 4 Hz, in order to allow for acomparison of the PDFs. Even the down-sampled hot-wirereference (black) shows a clearly non-Gaussian PDF withhigher probabilities for strong wind speed decreases, whichis also observed in the PDF of the ultrasonic anemometer(red). In contrast, the PDF of the fluctuations covered bythe cup anemometer (blue) obeys a Gaussian distributionas fitted in the plot (dashed gray).

Besides the comparison of the wind speed measurements,it is also worth comparing the wind directions measuredwith the 3D ultrasonic anemometer in different orienta-tions. For this purpose the PDFs of the wind directionfluctuations φ′(t) = φ(t)− 〈φ(t)〉 are shown in Fig. 10 foran inflow velocity of about 10 m/s (a) and about 20 m/s(b). The comparison of the two different orientations forboth wind speeds gives evidence of the increased probabil-ity for the occurrence of larger direction fluctuations forthe northward alignment (0◦) of the anemometer. Thisresult is expected, since a supporting rod is located up-stream the measuring volume and thus vortex sheddingcauses wake effects. The self-induced effect is more pro-nounced for the higher velocity in Fig. 10(b). Additionally,the measured mean wind speed of 16.2 m/s for northwardalignment is reduced by approximately 1.4 m/s comparedto the southward orientation. Considering the geometry ofthe ultrasonic anemometer an increasing systematic errorin the measurements can be concluded for inflow from 0◦,120◦ and 240◦, respectively.

To conclude this subsection, the active grid offers thegeneration of turbulent flows with similarly intermittentstatistics to atmospheric wind as was shown in [22]. Espe-cially promising in this context is also the generation ofreproducible flow structures in a deterministic sense, as ithas been shown in this subsection. Besides anemometertesting and characterization, also the interaction of typicalstructures like airfoils or model wind turbines with repro-ducible flow fields shall be investigated in the future. Itshould be noted that the active grid offers also further ap-plications for wind energy research, e.g., generating shearflows [17] or special structures like low level jets and largescale intermittency [24].

2.3 Impact on loads and power generationof wind energy converters

The turbulent nature of the wind with its characteristicintermittency can be expected to have a major impacton wind energy converters. These machines operate inthe turbulent atmospheric boundary layer and are thus

7

(a)

(b)

Figure 10: (a) PDF of the wind direction fluctuationsmeasured with the 3D ultrasonic anemometer in northward(blue) and southward (red) orientation at about 10 m/sinflow velocity. (b) PDF of the wind direction fluctuationsmeasured with the 3D ultrasonic anemometer in northward(blue) and southward (red) orientation at about 20 m/sinflow velocity.

permanently exposed to turbulent wind fields. Among themany aspects of a complete WEC system, we choose herethe torque and thrust at the rotor as central force andload quantities to investigate the influence of the turbulentinflow characteristics on WECs.

Numerical studies have been performed using the FASTcode by NREL [26] together with the WindPACT 1.5 MWvirtual turbine [27]. The unique wind field measurementof the German GROWIAN campaign [25,28] was used asinflow condition. Additionally the Kaimal model of theTurbSim software [29] and a newly developed model basedon stochastic processes [30,31] were used for comparison.In a first analysis, the statistics of rotor torque incrementswere derived and compared [32]. It was found that thetypical intermittency of the wind transfers to a similarintermittency in the rotor torque when the measured windfield is used as input, see Fig. 12(a). The Gaussian in-crement statistics of the Kaimal wind field, on the otherhand, lead to Gaussian torque statistics, supporting thestatements above on the implicit Gaussian assumptionsin the IEC standard [3]. The intermittent synthetic windfield results in much more realistic torque statistics.

Here we present additional results of the rotor thrustforces from the identical numerical setup as above. InFig. 12(b) thrust increments obtained under the sameinflow conditions as in part (a) are compared at τ = 1 s.We find that the measured wind field leads to a highlyintermittent torque PDF, which is well reproduced bythe stochastic wind field model. The Kaimal wind field

(a) (b)

Figure 11: (a) GROWIAN test site of the 1980’s [25] and(b) layout of the wind measurement setup, compared tothe position of the rotor.

leads to purely Gaussian thrust increment PDFs and thusunderestimates extreme thrust loads.

Both torque and thrust form a complete decompositionof forces at the nacelle of a WEC. With these results wecan therefore conclude that intermittency is a general prop-erty of the wind-generated forces at the WEC. Moreover,thrust forces appear at all types of buildings and structureswhich are exposed to the wind. This emphasizes again therelevance of our results.

As we observe intermittent statistics of the rotor forcesat WECs, it can be expected that the generated electricalpower signal is also influenced by this property. In Ref. [33]it could be demonstrated that actually the electrical powersignal possesses similarly intermittent dynamics as thewind. These results are confirmed and extended in thispaper, see Sec. 3.

From our results we see that the use of Gaussian windfields in WEC simulation and design must lead to a dra-matic under-estimation of extreme torque, thrust, andthus load changes, which can be comparable to Fig. 1.Most interestingly these load features are not grasped bythe characterization based on standard rain flow countingmethods. The intermittent statistics of wind and loadsare then transferred into an evenly intermittent signal ofthe electrical power output. With a growing share of windenergy fed into the electrcal grid, the high probabilities ofextreme changes will have a more and more significant im-pact on the grid. Another important aspect also becomesclear from these results. Short time fluctuations are notsmoothed out by, e.g., some possible spatial averaging orrotational inertia of the rotor. In contrast the nature ofthe turbulent wind amplifies significantly the effects of theshort time fluctuations. For further developed WECs of thefuture, e.g., the 10 MW class, these turbulent propertieshave to be considered even more seriously.

To conclude, with the statistics of rotor torque andthrust increments we achieve a complete characterizationof the forces at the rotor of WECs. We find that theintermittency of the wind is transferred to the generatorby the torque, as well as to the complete structure of theWEC by the thrust forces. The resulting electrical poweroutput of these intermittent input forces will be subject ofthe following section.

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Figure 12: PDFs of (a) rotor torque increments, δQτ =Q(t + τ) − Q(t), and (b) rotor thrust increments, δTτ =T (t + τ) − T (t), derived from simulations of the WInd-PACT 1.5 MW virtual turbine. Wind inflow data wereused from the GROWIAN wind field measurement (circles,top), as generated by a stochastic intermittent wind fieldmodel (squares, middle), and from the output of the Turb-Sim software using the Kaimal model (diamonds, bottom).Gaussian PDFs are shown as dashed lines for comparison.Time scale τ is 1 s for all presented PDFs. Results of (a)have been published in [32].

3 Characterisation and model-ing of wind-generated electricalpower

Throughout the previous sections, it could be demonstratedthat the intermittent nature of atmospheric turbulence in-fluences all stages of the wind energy conversion process.The wind imposes its intermittent character upon all rele-vant quantities, such as wind speeds, mechanical forces andtorques, and electrical power output of WEC. To achieveprogress in wind power analysis and modeling, alterna-tive methods to those defined in the current standards [3]are needed. Here we present a method adapted to theintermittent character of atmospheric wind. The highlydynamical character of the power conversion process andthe intermittent properties of the electrical power outputhave been shown before [33–35]. They are confirmed bythe results of this section, cf. Fig. 15(b).

Future sustainable energy supply systems will have tomanage large shares of fluctuating energy sources suchas wind energy. This constitutes a demand of properlymodeling the high frequency dynamics of these energy con-version systems, including their nonlinear and intermittentproperties. As an alternative to highly detailed, complex(and thus computationally expensive) models the usage ofstochastic models has recently found a wide range of ap-plications [36, 37]. Generally spoken, for complex systemsthis approach tries to separate only few crucial variables1

from a larger number of less important ones, which arethen replaced by simple noise terms. By doing so, in manycases simply structured models are obtained which never-theless cover the essential dynamics of the system underinvestigation. Their parameters can be derived directlyfrom measurement data, cf. [37].

Recently the Langevin Power Curve (LPC) has beendeveloped as a dynamical power characteristic of WECs[34,35,39]. By this method, for each wind speed the stable(or attractive) fixed points of the energy conversion dynam-ics are obtained. Together they constitute the LPC of the

1so-called order parameters, following [38]

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Figure 13: Drift fields D(1) derived from WEC measure-ments, represented qualitatively using arrows. The localvalue of the drift D(1) is represented by the direction andlength of each arrow, while the color also indicates thedirection of the drift. The black dots represent the stablefixed points where D(1) = 0. (a) Drift field D(1)(P ;u) ofelectrical power output vs. wind speed. Here the blackdots repesent the Langevin Power Curve [40]. (b) Driftfield D(1)(Q;ω) of rotor torque vs. angular velocity. Forthis analysis the torque was obtained from available mea-surements of angular velocity and electrical power output.Power and torque are normalized by their rated values Prand Qr, respectively.

WEC, which provides important information, complemen-tary to the current industry standard IEC 61400-12-1 [3].

Here we extend this concept and propose a stochasticmodel of the electrical power output of WEC, driven bythe inflowing wind. The structure of the model aims to-wards the high-frequency dynamics of the conversion fromwind speed to electrical power. The power output P (t) ismodeled by a stochastic differential equation, namely theLangevin equation [41]

d

dtP (t) = D(1)(P ;u) +

√D(2)(P ;u) · Γ(t) . (5)

Here D(1)(P ;u) and D(2)(P ;u) are the drift and diffusionmatrices, respectively, and Γ(t) is a Gaussian white noisewith mean value 〈Γ(t)〉 = 0 and variance 〈Γ2(t)〉 = 2. Thedrift and diffusion matrices D(1)(P ;u) and D(2)(P ;u) canbe estimated directly from measurement data (sampled atthe order of 1Hz), as follows [34,41]

D(n)(P ;u) =1n! lim

τ→0

1τ〈[P (t+ τ)− P (t)]n |P (t) = P ;u(t) = u〉 , (6)

where 〈·|·〉 denotes conditional ensemble averaging. Notethat u(t) is thus conditioned on a fixed value u. An il-lustration of D(1) is given in Fig. 13(a). The drift field(or matrix) D(1) represents the dynamical map of the con-version process u(t) → P (t). The power production isattracted towards the stable fixed points P (u), while theturbulent wind fluctuations continuously drive the systemaway. This corresponds to the structure of the Langevinequation (5), where the drift field D(1) represents the at-tractor displayed in Fig. 13(a), and where the diffusionfield D(2) quantifies additional, random fluctuations stem-ming from turbulence. A second example of a drift field ispresented in Fig. 13(b). Here the dynamics of the torquedepending on the rotational speed is displayed, showing

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Figure 14: Comparison of P (t) (black) and P ?(t) (red) for(a) 100 h and (b) 30 min at sampling frequency 2.5 Hz.

that the method of the LPC is not limited to the dynamicsof power output. It rather defines a general mathematicaltool for analyzing the dynamics of complex, noisy, andnoise-driven systems.

While only the drift field with its fixed points as shownin Fig. 13 defines the LPC, the Langevin equation (5) usesboth the drift and diffusion matrices. Once these havebeen estimated from a dataset, they can be used in theLangevin equation to simulate the power output P (t) forany given wind speed u(t).

As a first hint for the validity of the stochastic model2,a visual comparison of P and P ? is given in Fig. 14. Astatistical comparison is performed based on power spectraldensities in Fig. 15(a). The overall shape of the spectrumis reproduced by the stochastic model. The most strikingdeviation is observed at frequency f ≈ 1 Hz, which corre-sponds to a 1 Hz oscillation in the measured time seriesP (t). We believe this is due to the interaction of the towerwith the rotating blades of the WEC3, which happens inthe same frequency range. As our model is based on a first-order stochastic differential equation (5) it can in principlenot reproduce a single-frequency oscillation like this. As afuture improvement, a suitable oscillation could be addedto the model to compensate for this limitation.4 Figure

2The results of the stochastic model are presented here for adataset obtained from a numerical model, see details in Ref. [42].Similar results were obtained for various measurement datasets, forwhich the results were surprisingly good, even though the noise bythe turbulent inflow will not fulfill the conditions of a Langevin noise.

3It should be noted that not only the tower interaction can gener-ate periodic fluctuations in the order of 1 Hz. Additional contributionsfrom the rotating rotor are not identified here.

4Here it should be noted also that in a typical measurement setupwith 1 Hz sampling frequency this oscillation would not be visible,making the validation of such a model extension impossible in mostcases.

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Figure 15: (a) Power spectral density S(f) of P (black)and P ? (red). S(f) is normalized by the variance σ2 of thetime series. (b) PDF of power increments δPτ for variousscales τ = (0.4, 0.8, 1.6, 6.4, 25.6, 26214.4) s from bottomto top, curves are shifted vertically for clarity. The solidred lines are estimated from P ? and the black dashed linesfrom P . The vertical axis is represented using a logarithmicscale.

15(b) illustrates the ability of the model to reproduce theintermittency of the original power signal. For example,changes in power production up to ±100 kW within 0.8 soccur. This aspect is especially important on shorter timescales when fast wind gusts are converted into fast powergusts. Moderate deviations are observed on the shortertime scales, where the model behaves less intermittent thandesired. Overall, the stochastic model obtains valuable re-sults as spectral properties and intermittency are mostlywell reproduced. Its fast and flexible structure makes thestochastic method useful for a realistic modeling of sin-gle WECs, wind farms and wind farm clusters (as will beshown elsewhere).

It should nevertheless be noted that the purpose ofthis model is not to compete with current full-featuredmultibody models of WECs, such as FAST [26] or FLEX[43]. In contrast to these, the proposed model is highlysimplified and does not aim at a detailed model of theprocesses inside a WEC. It rather constitutes an effectivemodel of a WEC as a whole, considering only the wind asinput and a single output quantity, such as electrical poweror torque. While the descriptive power of our model in thissense is limited, it benefits from high numerical efficienyas well as easy and straightforward parameter retrieval.

We can conclude that the stochastic model is able tocapture consistently the dynamics of power or torque, de-pending on wind speed or rotational frequency, respectively.Once the parameters have been estimated from measure-ments, for simulations only wind speed time series are

10

required as input. In the results especially the typicalintermittency is reproduced.

4 Outlook: n-point statistics andforecasting

Up to this point our analysis, modeling, and experimentsconcerning atmospheric turbulence were concentrating onincrements and fluctuations of relevant signals, e.g., windspeeds, torque, and electrical power. These statistics canbe obtained in a straightforward way and it has beendemonstrated that they are of high relevance.

Based on recent results presented in Sec. 2.1 we arepresently working on the decomposition of atmosphericturbulence into components which behave similar to homo-geneous isotropic turbulence. For this case, in the recentyears a framework of stochastic analysis and character-ization has been developed, which grasps the completestatistical properties of turbulence in terms of arbitrary n-scale joint probabilities [36,37,44,45]. Moreover, a methodhas been presented for the multiscale reconstruction of realturbulent time series [46,47].

From the combination of these work topics in the futurewe hope to make adcvances towards a multipoint descrip-tion framework for atmospheric turbulence. This wouldmean also a proper characterization of multiscale jointprobabilities, which correspond to coherent structures inwind fields. As a consequence, also multiscale short-timeforecasting of wind events, such as gusts and clusteringeffects, could be possible with only little computationaleffort, compare Ref. [46].

5 ConclusionsIn this contribution, fundamental as well as applied aspectshave been presented for advanced statistics, modeling andexperiments concerning the unique small-scale intermit-tency features of atmospheric turbulence and their impacton the wind energy conversion process. Based on the factthat atmospheric turbulence exhibits these complex sta-tistical properties especially in wind speed increment andfluctuation PDFs, we proposed an advanced characteri-zation. Similar to the modeling and characterization ofwind data in [6], here we achieve a proper description andmodeling of atmospheric fluctuation PDFs. First successfulattempts have been made towards the decomposition ofatmospheric turbulence into subsets which share the prop-erties of homogeneous isotropic turbulence. Based on theseresults a more realistic, yet still practicable, characteriza-tion of atmospheric wind becomes possible, overcomingthe implicit assumption of Gaussianity in current methodsand standards.

As turbulence, in principle, has to be considered an openproblem of classical physics, experimental work is indis-pensable. For the reproduction of crucial properties ofatmospheric flows the active grid offers extreme flexibilityin the design and control of wind tunnel experiments. In-vestigations of the reproducibility of flow structures havebeen presented as well as the application to testing andcharacterization of anemometer properties.

The impact of atmospheric turbulence on wind energy

conversion is investigated in two separate aspects. Numer-ical modeling of turbines shows evidence that the inter-mittent nature of the wind leads to high probabilities ofextreme load changes in both torque and thrust. Theseresults completely characterize the forces induced by therotor to the drive train of a WEC. Accordingly intermittentstatistics are found in the electrical power output of bothnumerical models and real turbines, which constitutes theinput to the electrical grid. An extended approach canmodel the power output of a WEC based on wind measure-ment data, reproducing properly the statistical propertiesof the real power output signal.

The common assumption of Gaussian wind statisticsnecessarily neglects these intermittency effects. We proposethat more realistic statistical description and modeling ofatmospheric turbulence is necessary to advance not only inscience but also in engineering as well as in manufacturingand operation of WECs.

Stochastic methods are shown as appropriate tools tocharacterize the dynamics and intermittency of the windenergy conversion process in more details, and thus providevaluable, additional information to standard methods. Theresults presented show that these methods have a largepotential of application for further aspects of wind energy.

References[1] X. Lu, M.B. McElroy, and J. Kiviluoma, Global po-

tential for wind-generated electricity, Proceedings ofthe National Academy of Sciences 106 (2009), pp.10933–10938.

[2] M. Wachter, J. Gottschall, A. Rettenmeier, and J.Peinke, Dynamical power curve estimation using dif-ferent anemometer types, in Proceedings of DEWEK,Bremen, 26.–27.11.2008, 2008.

[3] IEC, International Standard 61400: Wind Turbines,International Standard 61400-12-1, International Elec-trotechnical Commission, 2005.

[4] M. Ragwitz, and H. Kantz, Indispensable finite timecorrections for Fokker-Planck equations from timeseries data, Physical Review Letters 87 (2001), p.254501.

[5] F. Bottcher, C. Renner, H.P. Waldl, and J. Peinke,On the statistics of wind gusts, Boundary-Layer Mete-orology 108 (2003), pp. 163–173.

[6] F. Bottcher, S. Barth, and J. Peinke, Small and largefluctuations in atmospheric wind speeds, Stochastic En-vironmental Research and Risk Assessment 21 (2007),pp. 299–308.

[7] J. Vindel, C. Yague, and J.M. Redondo, Structurefunction analysis and intermittency in the atmosphericboundary layer, Nonlinear Processes in Geophysics 15(2008), pp. 915–929.

[8] L. Liu, F. Hu, X.L. Cheng, and L.L. Song, ProbabilityDensity Functions of Velocity Increments in the Atmo-spheric Boundary Layer, Boundary-Layer Meteorology134 (2010), pp. 243–255 10.1007/s10546-009-9441-z.

11

[9] J.F.m.c. Muzy, R. Baıle, and P. Poggi, Intermittencyof surface-layer wind velocity series in the mesoscalerange, Phys. Rev. E 81 (2010), p. 056308.

[10] R.W. Stewart, J.R. Wilson, and R.W. Burling, Somestatistical properties of small scale turbulence in an at-mospheric boundary layer, Journal of Fluid Mechanics41 (1970), pp. 141–152.

[11] H. Tennekes, Intermittency of the small-scale structureof atmospheric turbulence, Boundary-Layer Meteorol-ogy 4 (1973), pp. 241–250.

[12] M. Nielsen, G. Larsen, J. Mann, S. Ott, K. Hansen,and B. Pedersen, Wind Simulation for Extreme andFatigue Loads, Riso-R-1437(EN), Risø DTU, 2003.

[13] B. Castaing, Y. Gagne, and E.J. Hopfinger, Velocityprobability density functions of high Reynolds numberturbulence, Physica D 46 (1990), pp. 177–200.

[14] A. Morales, M. Wachter, and J. Peinke, Character-ization of wind turbulence by higher-order statistics,Wind Energy (2011).

[15] U. Frisch Turbulence: the legacy of A. N. Kolmogorov,Cambridge University Press, 2001.

[16] K.S. Hansen, and G.C. Larsen, Characterising Tur-bulence Intensity for Fatigue Load Analysis of WindTurbines, Wind Engineering 29 (2005), pp. 319–329.

[17] H. Cekli, and W. van de Water, Tailoring turbulencewith an active grid, Experiments in Fluids 49 (2010),pp. 409–416 10.1007/s00348-009-0812-5.

[18] H.S. Kang, and C. Meneveau, Direct mechanical torquesensor for model wind turbines, Measurement Scienceand Technology 21 (2010), p. 105206.

[19] R.B. Cal, J. LebrÃşn, L. Castillo, H.S. Kang, andC. Meneveau, Experimental study of the horizontallyaveraged flow structure in a model wind-turbine arrayboundary layer, JOURNAL OF RENEWABLE ANDSUSTAINABLE ENERGY 2 (2010).

[20] H. Makita, Realization of a large-scale turbulence fieldin a small wind tunnel, Fluid Dynamics Research 8(1991), pp. 53 – 64.

[21] L. Mydlarski, and Z. Warhaft, On the onset of high-Reynolds-number grid-generated wind tunnel turbu-lence, Journal of Fluid Mechanics 320 (1996), pp.331–368.

[22] P. Knebel, A. Kittel, and J. Peinke, Atmospheric windfield conditions generated by active grids, Experimentsin Fluids (2011), pp. 1–11 10.1007/s00348-011-1056-8.

[23] A. Wiesner, F. Fiedler, and U. Corsmeier, The Influ-ence of the Sensor Design on Wind Measuremnets withSonic Anemometer Systems, Journal of Atmosphericand Oceanic Technology 18 (2001), pp. 1585–1608.

[24] G.H. Good, and Z. Warhaft, On the probability distri-bution function of the velocity field and its derivativein multi-scale turbulence, Physics of Fluids 23 (2011),p. 095106.

[25] H. Gunther, and B. Hennemuth Erste Aufbereitungvon flachenhaften Windmessdaten in Hohen bis 150 m,Deutscher Wetter Dienst, BMBF-Projekt 0329372A,1998.

[26] J. Jonkman, NWTC Design Code FAST, NREL/EL-500-38230, National Renewable Energy Laboratory,Golden, Colorado, USA, 2005.

[27] D.J. Malcolm, and A.C. Hansen WindPACT turbinerotor design study, National Renewable Energy Labo-ratory, Golden, Colorado, 2002 NREL/SR-500-32495.

[28] F. Korber, G. Besel, and H. Reinhold Meßpro-gramm an der 3MW-Windkraftanlage GROWIAN:Forderkennzeichen 03E-4512-A, Große Winden-ergieanlage Bau- u. Betriebsgesellschaft, Hamburg,1988 Bundesministerium fur Forschung und Technolo-gie.

[29] N. Kelley, and B. Jonkman, NWTC DesignCodes: TurbSim; http://wind.nrel.gov/designcodes/-preprocessors/turbsim/ accessed 17.6.2011.

[30] D. Kleinhans, and R. Friedrich, Simulation of inter-mittent wind fields: A new approach, in DEWEK 2006- Proceedings, Bremen (Germany), 2006.

[31] D. Kleinhans, Stochastische Modellierung komplexerSysteme, Universitat Munster, 2008.

[32] T. Mucke, D. Kleinhans, and J. Peinke, Atmosphericturbulence and its influence on the alternating loads onwind turbines, Wind Energy 14 (2011), pp. 301–316.

[33] J. Gottschall, and J. Peinke, Stochastic modelling ofa wind turbine’s power output with special respect toturbulent dynamics, Journal of Physics: ConferenceSeries 75 (2007), p. 012045.

[34] E. Anahua, S. Barth, and J. Peinke, Markovian powercurves for wind turbines, Wind Energy 11 (2008), pp.219–232.

[35] J. Gottschall, and J. Peinke, How to improve theestimation of power curves for wind turbines, Envi-ronmental Research Letters 3 (2008), p. 015005 (7pp).

[36] R. Friedrich, J. Peinke, and M.R.R. Tabar, 2009, Im-portance of Fluctuations: Complexity in the View ofStochastic Processes. in Encyclopedia of Complexityand System Science Springer, Berlin in print.

[37] R. Friedrich, J. Peinke, M. Sahimi, and M.R.R. Tabar,Approaching Complexity by Stochastic Methods: FromBiological Systems to Turbulence, Physics Reports(2011) (accepted).

[38] H. Haken Advanced synergetics: Instability hierarchiesof self-organizing systems and devices, Springer, 1983.

[39] M. Wachter, P. Milan, T. Mucke, and J. Peinke, Powerperformance of wind energy converters characterizedas stochastic process: applications of the Langevinpower curve, Wind Energy (2010) Published online,DOI: 10.1002/we.453.

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[40] P. Milan, T. Mucke, A. Morales, M. Wachter, and J.Peinke, Applications of the Langevin Power Curve, inProceedings of EWEC 2010, Warshaw, 2010.

[41] H. Risken The Fokker-Planck equation, Springer,Berlin, 1984.

[42] P. Milan, M. Wachter, and J. Peinke, Stochastic Mod-eling of Wind Power Production, in Proceedings ofEWEA 2011, Brussels, 2011.

[43] S. Øye, FLEX 5 User Manual, , Danske TechniskeHogskole, Lyngby, 1999.

[44] R. Friedrich, and J. Peinke, Description of a Turbu-lent Cascade by a Fokker-Planck Equation, PhysicalReview Letters 78 (1997), p. 863.

[45] C. Renner, J. Peinke, and R. Friedrich, Experimentalindications for Markov properties of small-scale tur-bulence, Journal of Fluid Mechanics 433 (2001), pp.383–409.

[46] A.P. Nawroth, and J. Peinke, Multiscale reconstructionof time series, Physics Letters A 360 (2006), pp. 234–237.

[47] R. Stresing, and J. Peinke, Towards a stochastic multi-point description of turbulence, New Journal of Physics12 (2010), p. 103046.

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