Stochastic Modeling of Wind Power Production

Patrick Milan∗ Matthias Wächter† Joachim Peinke‡

ForWind - Center for Wind Energy Researchof the Universities of Oldenburg, Hannover and Bremen

University of Oldenburg, Germany

Abstract

A stochastic model is proposed to reproducesynthetically the power production of a windenergy converter (WEC) from any given windmeasurement. The structure of the modelaims towards the high-frequency dynamicsof the conversion from wind speed to poweroutput. These dynamics appear to be a su-perposition of an attractive power curve plussome additional fluctuations stemming fromwind turbulence. These adversary dynamicscan be characterized through respectively adrift and a diffusion matrix. These two matri-ces can be inserted into a stochastic equationnamed the Langevin equation, which cangenerate a time series of power output froma given time series of wind speed. For thispaper, the computation is performed and thetime series of power output measured andmodeled are compared. Various statisticaltests are presented. It can be concluded thatthe stochastic model reproduces quantita-tively well various complex statistics observedon measurements. The tests performed in-volve ten-minute average values, ten-minutestandard deviations, as well as the spectrumand increment PDFs. Beyond the standardten-minute statistics usually considered, thefast gusts measured in high-frequency arealso modeled. Its fast and flexible structurecould make the stochastic method useful fora realistic modeling of the intermittent powerproduction of WECs.

Keywords: turbulence, gust, power pro-duction modeling, high-frequency dynamics,wind speed/power conversion

∗patrick.milan@forwind.de†matthias.waechter@forwind.de‡joachim.peinke@forwind.de

1 Introduction

The atmospheric wind in which wind energyconverters (WECs) operate is a complex pro-cess [1]. Turbulent structures are observedon short time scales in high-frequency windmeasurements. While these short-time fluc-tuations should not affect strongly the annualenergy production, they generate importantalternating loads on the entire structure ofevery WEC. Thus raises the question of thenegative impact of turbulence on downtimesand on the overall lifetime of a WEC. In par-allel, such turbulent wind makes any WECa fluctuating, intermittent source of energy.Wind gusts are transformed into intermittent,rapidly-changing “power gusts”. When mea-sured at high-frequency (≈ 1Hz ), the powerproduction fed into the electric grid by a sin-gle WEC is a complex, intermittent signal.

Wind turbulence cannot be controlled or re-duced at will. The site topography and sur-face roughness have a direct impact on theturbulence intensity, making some sites lessturbulent than others, mainly offshore sites.Our rising dependence on wind energy callsfor a better understanding of such phenom-ena, as well as reliable models. However, thestaggering level of complexity of turbulent ef-fects makes modeling a complex task. On theone hand, simple models typically neglect thehigh-order statistics that govern gusts. On theother hand, advanced CFD codes demand somuch computer resources that they cannotmodel entire WECs (yet).

In this paper, a stochastic model is pro-posed as an alternative. The model con-verts a high-frequency time series of windspeed u(t) into a high-frequency time se-ries of power output P (t), reproducing theconversion process performed by a WEC.

The stochastic theory used allows for a fastmodel that reproduces quantitatively well thestatistics observed on power output measure-ments. In a first step, the WEC must be char-acterized appropriately. In a second step, thischaracterization, in the form of the drift anddiffusion matrix D(1) and D(2) is inserted intoa stochastic Langevin equation. A synthetictime series of the power output can then begenerated by solving the Langevin equationiteratively.

2 Step 1: characterizing theWEC dynamics

2.1 Data requirements

In order to properly reproduce the power pro-duction of a WEC, the stochastic model mustbe parametrized. The necessary parameterscan be extracted from a measurement per-formed beforehand on the WEC of interest.The variables to be measured are the simul-taneous time signals of wind speed u(t) andof net power output P (t) of the WEC. Thetime series should have a sampling frequencyin the order of 1Hz , as displayed in figure1. The intermittent, gusty nature of the windis observed on short time scales. This jus-tifies the need for high-frequency measure-ments that contain detailed information aboutthe fast fluctuations, and that is unavailablewith the standard ten-minute averaging tech-nique used for measurement. Modeling suchcomplex statistics is one important goal of themethod presented.

In order to represent the dynamics of theWEC accurately, the original measurementshould be long enough to span all necessarywind conditions1. The wind speed u(t) shouldbe measured at hub height from a met mast,as described by the IEC power curve stan-dard [2]. Additional data requirements andcorrections described in the IEC norm maybe applied here for optimal results. Only the

1How long is enough for a good measurement de-pends on the wind conditions. A first guess for the nec-essary duration is in the order of two weeks. Unlikeprocedures based on ten-minute averaging, the use ofhigh frequency data allows for a faster convergence. Itis however expected that longer measurements yieldsbetter modeling results.

0.5 1.0 1.5 2.0 2.5 3.0

810

1214

t [hours]

u(t)

[m/s

]

0.5 1.0 1.5 2.0 2.5 3.0

500

1000

1500

t [hours]

P(t

) [k

W]

Figure 1: Example of a simultaneous mea-surement of wind speed u(t) and power out-put P (t) at frequency 1Hz . The measure-ment was performed on an operating multi-MW WEC. It is not the dataset used later on.

sampling frequency of the recorded data mustbe in the order of 1Hz instead of the ten-minute averaging applied in the IEC norm.The stochastic model is by nature related topower curve methods, as it aims to convertwind speed into power output. This intrin-sic connection makes it a flexible tool whichcan potentially be applied to any WEC designthat displays a clearly defined power curve. Ithas also been shown to be well applicable to-gether with different wind measurement prin-ciples, such as LIDAR [3] or nacelle anemom-etry measurements.The data set used contains 900 000 data-points, sampled at 2.5Hz. This correspondsto a measurement period of approximately 4days. The wind speed is an actual wind mea-surement on a met mast at the GROWIANsite in Germany [4, 5]. However, the time se-ries labeled as measurement time series ofpower P (t) used in this paper was obtainedfrom a computer model for a 1.5MW WEC.This other model is a mechanical WEC modelnamed FAST [6]. This model was already ac-cepted to model well the dynamics of a WEC[7]. The reason for using this data is that high-frequency measurements of wind speed andpower output are difficult to obtain. When-ever measurement power is mentioned in this

paper, it should be understood as output ofthe mechanical model FAST (and input forthe stochastic model). An extension to realWEC data is however straightforward when-ever measurement data is available, and theresult is expected to hold valid.

2.2 Characterizing the dynamics

The stochastic model follows the idea that thepower output P (t) can be approximated bya stochastic differential equation named theLangevin equation [8]

d

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

√D(2)(P ;u)·Γ(t) , (1)

where D(1)(P ;u) and D(2)(P ;u) are called re-spectively the drift and diffusion fields, andΓ(t) is a Gaussian white noise with meanvalue 〈Γ(t)〉 = 0 and variance 〈Γ2(t)〉 = 2.The stochastic approach consists in solvingthe Langevin equation, as further describedin section 3. Because Γ(t) is a set of ran-dom numbers, the term

√D(2)(P ;U)·Γ(t) is a

stochastic, i.e. random term, andD(1)(P ;u) isa deterministic term in the Langevin equation(1). The drift field D(1) represents the dynam-ical response of the WEC, while the diffusionfield D(2) quantifies random turbulent fluctua-tions together with Γ(t), as explained below.An essential aspect of the method is that thetwo fields D(1) and D(2) can be estimateddirectly from measurement data (sampled atthe order of 1Hz ), as follows [8, 9]

D(n)(P ;u) =1

n!limτ→0

1

τ

〈[P (t+ τ)− P (t)]n|P (t) = P ;u(t) = u〉 , (2)

where 〈 | 〉 denotes the conditional ensem-ble average. To this end, the calculation isconditioned over the two-dimensional statespace {P ;u}. In a practical sense, thefields D(n)(P ;u) are two-dimensional matri-ces computed in each discrete bin of the two-dimensional space {P ;u}. An illustration ofthe drift field D(1) is given in figure 2.The values of the drift field (matrix) D(1) areplotted using arrows in figure 2. The driftis plotted using a blue arrow when positive,i.e. D(1)(P ;u) > 0, indicating that in theseregions the power output P (t) increases (onaverage). The inverse situation happens for

u [m/s]

P /

Pr

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

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●

● ●

●

●

●

●

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●

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●

●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●

Figure 2: Drift field D(1)(P ;u) representedqualitatively using arrows. The local value ofdrift D(1)(P ;u) is represented by the directionand length of each arrow, while the color alsoindicates the direction of the drift. The blackdots represent the stable fixed points whereD(1)(P ;u) = 0, also called the Langevinpower curve [7, 10]. The binning chosen herewas 0.5m/s for u and 50 bins for P . The poweris normalized by rated power Pr = 1500kW .

the red arrows, where the drift is negative,and the power output decreases. Figure 2is a map that illustrates how the power pro-duction changes for each combination of windspeed and power. The WEC permanentlyadapts its power production to match with anever-changing turbulent wind. While the IECpower curve gives the average power corre-sponding to each wind speed, the Langevinpower curve defined as D(1) = 0 (black dotsin figure 2) represents the stable, attractivedynamics of the WEC2. For information, D(1)

and D(2) could be estimated only in the binswhere measurement data was recorded. Thewhite regions in figure 2 represent these re-gions where the data never went. This is not alimitation to the model because these regionsare not of interest, as the WEC never exploresthem.Although the Langevin and IEC power curves

2Additionally, D(1)(P ) must have a decreasing slopearound the stable fixed points. An increasing slopewould indicate an unstable fixed point, when only sta-ble fixed points are of interest here.

have a similar shape [11], the Langevin powercurve brings additional meaning as it is thedynamical attractor of the WEC in the two-dimensional space {P ;u}. Following this re-sult, the WEC dynamics can be understoodin a simple way. The power production is at-tracted towards the stable fixed points whereD(1)(P ;u) = 0, i.e. the Langevin powercurve [7, 10]. Under constant, laminar windconditions, the WEC would relax towards theLangevin power curve. However, the turbulentwind fluctuations endlessly drive the systemaway from the stable fixed points. The under-lying dynamics of the conversion from windspeed to power output can then be separatedinto an attractive power curve and the addi-tional turbulent fluctuations driving the systemaway from the stable power curve. This corre-sponds exactly to the structure of the stochas-tic model, where the drift field D(1) repre-sents the attractor displayed in figure 2, andwhere the diffusion field D(2) quantifies ad-ditional, random fluctuations stemming fromturbulence3. As the WEC dynamics are char-acterized, the estimated drift and diffusionfields (matrices) can be used in the Langevinequation to simulate the power output P (t) forany given wind speed u(t), as is introduced insection 3.

3 Step 2: modeling the WECpower production

The two matrices D(1)(P ;u) and D(2)(P ;u)carry the necessary information about theconversion from wind speed u(t) to poweroutput P (t) at high-frequency. It should benoted that the goal of such model is not toreach an exact reproduction of the power timeseries, but rather to model a time series thathas the same statistical properties. The useof a random noise Γ(t) makes an absolute,exact reproduction impossible. It is throughthe proper estimation of the D(n) fields thatthe correct statistics can be modeled. Theirproper estimation is detailed in section 2.2.The two fields (matrices) must be inserted

3The authors would like to note that the stochastictheory presented here was successfully applied to othersystems such as airfoil lift dynamics [12] or turbulentflows [13]. Numerous complex systems were studied inthe same manner in various scientific fields.

into the Langevin equation that becomes amodel for the power output P ?(t) of the WEC.For clarity of reading, the power signal orig-inally measured on the WEC will be furtherlabeled P (t), and the power modeled by theLangevin equation P ?(t). The Langevin equa-tion (1) introduced in section 2.2 is valid for acontinuous process. While such process ex-ists only in a mathematical sense, a concreteapplication requires to adapt the Langevinequation to a form discrete in time, i.e. de-fined at every discrete time step τ as follows

P ?(t+ τ) = P ?(t) + τ ·D(1)(P ?(t);u(t))

+√τ ·D(2)(P ?(t);u(t)) · w(t) . (3)

According to stochastic calculus, the noiseterm Γ(t) in (1) has to be integrated overthe time step τ , leading to the Wiener noise√τ · w(t), where w(t) is delta-correlated with

a variance of 2 (cf. [8], section 3.6). The sam-pling time τ should be chosen similar to thesampling time of the measurement time se-ries used in section 2.1 to estimate the fields.Additionally, the wind speed u(t) used in thediscrete Langevin equation (3) should also besampled with a similar sampling time. Thereason is that the dynamics represented bythe two fields is characteristic of the sam-pling rate of the measurement time seriesintroduced in section 2.1. Using the fieldson a very different sampling time makes lit-tle sense as they do not characterize the dy-namics on this time scale. For consistency, allsampling rates of all signals used should bethe same (or at least of similar order). For in-formation, the results displayed in this paperwere obtained using τ = 0.4s (for a samplingfrequency f = 1/τ = 2.5Hz).Γ(t) is a Gaussian white noise. This meansthat it is a series of independent random num-bers that are Gaussian distributed. Its meanvalue must be 〈Γ(t)〉 = 0 and its variance〈Γ2(t)〉 = 2 following [8]. Such random signalcan be generated fast and easily from mostscientific programming softwares. For exam-ple, 107 such data points are generated in lessthan 1s on a standard desktop computer bythe computing software R [14]. For informa-tion, all calculations were performed using Rin this project.Finally, the initial condition P ?(t = 0) influ-ences only slightly the future evolution as

P ?(t) will adjust rapidly to the correspondingwind speed u(t)4. The model can then be rununambiguously to simulate P ?(t).Because the model must first be constructedfrom a measurement of u(t) and P (t), theuser can always compare the measured sig-nal P (t) with the modeled signal P ?(t). To testthe validity of the model, a visual comparisonof P and P ? is given in figures 3 and 4.One can observe in figure 3 that the stochas-

Figure 3: Comparison of P (t) (black) andP ?(t) (red) for 100 hours at sampling fre-quency 2.5Hz.

tic model manages to reproduce the mea-surement signal, at least in a visually sat-isfying manner. Here, the time series con-tain so many data points that it is impossi-ble to really distinguish the two signals. How-ever, P ?(t) seemingly spans the same rangeat P (t), which already guarantees that themodel spans the correct values in a grossway. A detailed statistical comparison of thetwo time series is provided in section 4.

4 Statistical validation of themodel

4.1 Ten-minute statistics

The time series displayed in figure 3 are visu-ally similar, but this is not sufficient to validate

4An optimal initial value is P ?(t = 0) = PIEC(u(t =0) ), where PIEC (u) is the IEC power curve of the WEC.

the stochastic model. A statistical compari-son of the measured and modeled time seriesP (t) and P ?(t) is detailed in this section. Onecan decompose the series P (t) into

P (t) = 〈P 〉10min + Premain(t) , (4)

and similarly for P ?(t). 〈P 〉10min repre-sents the ten-minute average of P (t), andPremain(t) gives the remaining fluctuationsaround the ten-minute average. While〈P 〉10min gives the slow ten-minute trend,Premain(t) represents the fast fluctuations re-sulting from turbulence.Two comparisons are performed:

• the ten-minute average 〈P 〉 characterizesthe evolution on a long time scale (10minutes). While this paper focuses moredirectly on faster fluctuations, it is impor-tant to reproduce the ten-minute evolu-tion. The ratio 〈P ?〉10min/〈P 〉10min quan-tifies the quality of the model to repro-duce the power production on the longtime scale of 10 minutes;

• the ten-minute standard deviationsd(P )10min characterizes the magnitudeof the remaining fluctuations5. The ratiosd(P ?)10min/sd(P )10min quantifies thequality of the model to reproduce thefast, turbulent fluctuations in terms ofmagnitude.

Figure 4 illustrates visually the ability of thestochastic model to reproduce the power out-put time series on shorter time periods. Themodel manages to reproduce the power pro-duction of the WEC on both the slow trend,and also on the additional fluctuations. Onecan see in figure 4(b) that every ten minutes,the average and standard deviation of the twoseries are similar. Once the model was ex-ecuted, and the entire time series of P ?(t)was generated, one can calculate all the ten-minute ratios of averages and standard devia-tions. The total ratio is defined as the averageof all the ten-minute ratios. It is calculated forboth the average and the standard deviation.The final ratios obtained are displayed in table4.1.

5The standard deviation denoted sd(P )10min in thispaper is calculated following sd(P )10min =

⟨(P −

〈P 〉10min)2⟩10min

.

t [min]

P [k

W]

800

1000

1200

1400

1600

P(t)P*(t)

0 5 10 15 20 25 30

1000

1300

1600

t [min]

<P

> ±

sd(

P)

(a)

(b)

Figure 4: (a) Time series of P (t) (black) andP ?(t) (red) for 30 minutes. (b) Ten-minute av-erages (full lines) and ± ten-minute standarddeviations (dashed lines) with correspondingcolors.

Table 1: Total ratio of ten-minute averagesand of ten-minute standard deviations.

total〈P ?〉10min

〈P 〉10mintotal

sd(P ?)10min

sd(P )10min

1.061 1.005

In the example presented, the ten-minuteaverage of power is over-estimated by 6.1%,and the ten-minute standard deviation ofpower is over-estimated by 0.5%. These val-ues can vary slightly, due to the randomnessof the model. It should be noted that an opti-mization loop was applied to reach this result.The original ratio of standard deviations wasin the order of 1.25 (over-estimation by 25%),due to natural deviations in the estimation ofD(2), which is typically over-estimated6. D(2)

6Such deviations are expected [15] as equation (2)applies in theory on continuous processes. However,

was reduced in small steps until the total ra-tios presented in table 4.1 reached the de-sired value. The final value of D(2) is retainedas the correct value for the model. As the ten-minute mean value is over-estimated by 6.1%,the optimization technique should be furtherimproved.

4.2 Two-point statistics

All statistical quantities presented above areone-point statistics [16]. Two-point statisticssuch as the power spectral density (morecommonly called spectrum) S(f) are calcu-lated following [17] for the two time series andcompared in figure 5.The overall shape of the spectrum is repro-

Figure 5: Power spectral density S(f) of P(black) and P ? (red). S(f) is normalized bythe variance σ2 of the time series.

duced by the stochastic model. The moststriking deviation is observed at frequencyf ≈ 1Hz , which corresponds to a 1Hz oscilla-tion in the measured time series P (t). Whileit seems possible that the peak observedhere may be caused by the blades passingthe tower of the WEC, no clear evidencewas possible within the present study. Thisoscillation is not reproduced by the model,

the time series used in section 2 are imperfect due tomeasurement noise and to a finite sampling rate. Acontinuous process cannot be measured physically, andremains a purely mathematical object, making the opti-mization procedure necessary in many cases.

but could possibly be added to the model asa succeeding step.

An additional two-point quantity is thepower increment Pτ (t) = P (t + τ) − P (t),where τ is a given time scale. Pτ representsthe change in power over a given time scaleτ . When plotting the probability densityfunction (PDF) p(Pτ ) of the power incrementPτ , one can quantify the occurrence of gustson the given time scale τ [16]. The incrementPDF is given for various scales in figure 6.

Figure 6 illustrates the ability of the model

−100 −50 0 50 1001e−

061e

−04

1e−

021e

+00

1e+

02

Pτ [kW]

p(P

τ) [a

.u.]

P(t)P*(t)

Figure 6: PDF p(Pτ ) of power in-crements Pτ for various scales τ =(0.4, 0.8, 1.6, 6.4, 25.6, 26214.4)s (bottomto top curves shifted upwards for clarity). Thered full lines are estimated from P ? and theblack dashed lines from P . The y-axis isrepresented using a logarithmic scale.

to generate intermittent power gusts. Forexample, changes in power production upto +100kW and −100kW within 0.8s occur,as their probability is more than zero. Thisaspect is especially important on shorter timescales (represented by the lowest curves infigure 6) when fast wind gusts are convertedinto fast power gusts. The model does notcreate as many gusts on the shortest timescale (lowest curve) as deviations betweenthe two curves can be seen. Improvementscould be performed in this direction. Overall,the stochastic model still displays a valuableresult as power gusts are mostly well repro-

duced, but maybe not often enough for thevery fast ones on the shortest time scale. Allresults presented in figures 3 to 6 show agood agreement between the measured timeseries and the result of the stochastic model.For such, it is concluded that the stochasticmodel reproduces the intended statistics.

5 Conclusion

A stochastic model is introduced for the powerproduction of a WEC. This model can gen-erate a high-frequency time series of poweroutput from a high-frequency time series ofwind speed. In a first step, the dynamics ofthe particular WEC are brought into the modelfrom an initial measurement of wind speedand simultaneous power output. From thiscalculation in section 2.2, the drift field D(1)

gives a visual representation of the WEC dy-namics at high frequency. The conversionfrom wind speed to power output appears asthe superposition of an attractive power curveplus some additional fluctuations due to tur-bulence. The WEC would tend towards thepower curve if the wind inflow were laminar,but the turbulent fluctuations always push thedynamics away from the power curve. Whilethe drift field D(1) represents this attractivepower curve, the diffusion field D(2) quanti-fies additional random fluctuations due to tur-bulence.

In a second step, the stochastic Langevinequation is solved using the two fields previ-ously estimated. It is solved here using thewind speed time series used to estimate thefields, such that a direct comparison betweenthe measured and modeled power time se-ries is possible. While a first validation of themodel can be done comparing visually thetime series, a statistical comparison is per-formed in section 4. As a result of this test, thestochastic model reproduces the ten-minuteaverage values of power with deviations of or-der ≈ 6%. This should be further improvedthrough a more advanced optimization tech-nique. The ten-minute standard deviations ofpower deviate from the measurement by lessthan 1%. This result is obtained using an op-timization loop, that could be run further toreach even better correspondence. These re-sults confirm the visual similarity of the two

time series.Additionally, two-point statistics are inves-

tigated. The spectrum of the power outputtime series is estimated. The spectrum of themodeled series displays a qualitative agree-ment with the measurement series, althougha peak at ≈ 1Hz is omitted. This peak is at-tributed to the shadow effect of the tower onthe rotating blades, and could possibly be in-serted manually into the model. The PDF ofpower increments are also investigated. Sat-isfying results are obtained on a large rangeof time scales, indicating that the model repro-duces the power gusts observed on measure-ments. The turbulent structures observed onpower output measurements are reproducedqualitatively well.

Due to its simple structure, such model canbe run fast on any conventional computer.The entire computation for 4 days of data at2.5Hz was performed within 20 minutes on adesktop machine, using the interpreting lan-guage R. An implementation to a compilinglanguage would greatly reduce the computa-tion time, making the model even more flexi-ble. This fast and flexible structure makes apower prediction model on the scale of windfarms (or larger) possible. One would needhowever a high-frequency measurement ofwind speed to feed the model. Current re-search aims towards a similar model thatcould perform without high-frequency time se-ries of wind speed, but simply using ten-minute average wind speed and turbulenceintensity. A fast modeling of power produc-tion, power fluctuations and power "gusts"could then be performed for any WEC in anylocation based on only a wind speed mea-surement, or meteo model. A similar modelfor mechanical loads is also under develop-ment.

Acknowledgements

Parts of this work have been financially sup-ported by the German Environment Ministryunder grant number 0327642A, and by theGerman Ministry for Education and Researchunder grant number 0329372A. We wouldlike to thank Stephan Barth, Julia Gottschall,Edgar Anahua and Michael Hölling for their pi-oneering work on stochastic analysis in wind

energy applications.

References

[1] Böttcher, F., Barth, S., and Peinke, J.,“Small and large scale fluctuations in at-mospheric wind speeds,” Stochastic En-vironmental Research and Risk Assess-ment (SERRA), Vol. 21, 2007, pp. 299–308.

[2] IEC, “Wind Turbine Generator Systems,Part 12: Wind turbine power perfor-mance testing,” International Standard61400-12-1, International Electrotechni-cal Commission, 2005.

[3] Wächter, M., Gottschall, J., Rettenmeier,A., and Peinke, J., “Power curve es-timation using LIDAR measurements,”EWEC 2009 Proceedings, 2009.

[4] Günther, H. and Hennemuth, B., Er-ste Aufbereitung von flächenhaftenWindmessdaten in Höhen bis 150 m,Deutscher Wetter Dienst, BMBF-Projekt0329372A, 1998.

[5] Körber, F., Besel, G., and Rein-hold, H., Meßprogramm an der3MW-Windkraftanlage GROWIAN:Förderkennzeichen 03E-4512-A, GroßeWindenergieanlage Bau- u. Betrieb-sgesellschaft, Hamburg, 1988, Bun-desministerium für Forschung undTechnologie.

[6] Jonkman, J., “NWTC Design CodeFAST,” Tech. Rep. NREL/EL-500-38230,National Renewable Energy Laboratory,Golden, Colorado, USA, 2005.

[7] Milan, P., Mücke, T., Wächter, M., andPeinke, J., “Two numerical modeling ap-proaches for wind energy converters,”Proceedings of Computational Wind En-gineering 2010, 2010.

[8] Risken, H., The Fokker-Planck equation,2nd ed., Springer, Berlin, 1989.

[9] Anahua, E., Barth, S., and Peinke, J.,“Markovian power curves for wind tur-bines,” Wind Energy , Vol. 11, No. 3,2008, pp. 219–232.

[10] Milan, P., Mücke, T., Morales, A.,Wächter, M., and Peinke, J., “Appli-cations of the Langevin power curves,”EWEC 2010 Proceedings, 2010.

[11] Gottschall, J. and Peinke, J., “How to im-prove the estimation of power curves forwind turbines,” Environmental ResearchLetters, Vol. 3, No. 1, 2008, pp. 015005(7pp).

[12] Schneemann, J., Milan, P., Luhur, M. R.,Knebel, P., and Peinke, J., “Lift mea-surements in turbulent flow conditions,”DEWEK 2010 conference proceedings,2010.

[13] Friedrich, R. and Peinke, J., “Descrip-tion of a Turbulent Cascade by a Fokker-Planck Equation,” Physical Review Let-ters, Vol. 78, No. 5, 1997, pp. 863–866.

[14] R Development Core Team, “R: A Lan-guage and Environment for StatisticalComputing,” 2010, ISBN 3-900051-07-0.

[15] Gottschall, J., Modelling the variability ofcomplex systms by means of Langevinprocesses, Ph.D. thesis, Carl von Ossi-etzky Universität Oldenburg, Germany,2009.

[16] Morales, A., Wächter, M., and Peinke,J., “Advanced Characterization of WindTurbulence by Higher Order Statistics,”EWEC Proceedings 2010, 2010.

[17] Press, W. H., Teukolsky, S. A., Vetter-ling, W. T., and Flannery, B. P., Numer-ical recipes in C, Cambridge UniversityPress, 2nd ed., 1992.