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132 D. Kleinhans and R. Friedrich D (1) E(x=0.5) 0.5 0 −0.5 −1 −1.5 10 100 1000 N 10000 100000 Fig. 23.1. Estimated drift function D (1) E (x =0.5) of synthetic Ornstein-Uhlenbeck process as function of N. Dashed region marks the symmetric error-bars corresponding to (23.7) E[D (1) (x=0.5)] 40 35 30 25 20 15 10 5 0 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 τ Fig. 23.2. Error of drift estimate as function of time increment τ (triangles). A divergent behaviour for τ → 0 is evident. The solid line represents the best fit f(τ) =0.0045/ √ τ � EN D (1) E (x0,τ) � � � � � 2 D = τ (2) E (x0,τ) � D − N (1) E (x0,τ) �2 . (23.7) N In sum the statistical error in the estimated drift function is proportional to (Nτ) −1/2 . Figures 23.1 and 23.2 illustrate the divergent behaviour of the estimated drift coefficient D (1) E in the cases of few data-points and small time increments considering as example data from an Ornstein-Uhlenbeck process. 23.5 Conclusion In conclusion there is simple method to estimate the dynamical drift and diffusion functions from measured data. This method can be used to describe
23 Quantitative Estimation of Drift and Diffusion Functions 133 the dynamical behaviour of complex systems such as wind energy converters. Quantitative results from this method have to be considered carefully for the reasons discussed in Sects. 23.3 and 23.4. We would like to stress that there is a more recent extension that avoids the evaluation of the conditional moments in the limit of small time increments and improves the accuracy of the results substantially [4]. References 1. E. Anahua, F. Böttcher, S. Barth, and J. Peinke, Proceedings of the European <strong>Wind</strong> <strong>Energy</strong> Conference (EWEC), London, UK (2004) 2. S. Siegert, R. Friedrich, and J. Peinke, Phys Letts A 243, 275 (1998) 3. H. Risken, The Fokker-Planck equation, 2nd ed. (Springer, Berlin Heidelberg New York (1989) 4. D. Kleinhans, R. Friedrich, A. Nawroth, and J. Peinke, Phys Lett A 346, 42 (2005).
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Wind Energy Proceedings of the Euro
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Prof. Dr. Joachim Peinke Dr. Stepha
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VI Preface been presented, spanning
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VIII Contents 3.3 Stochastic Wind F
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X Contents 12.6 Subgrid Scale Turbu
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XII Contents 22 Stochastic Small-Sc
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XIV Contents 32 Handling Systems Dr
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XVI Contents 41 3D Numerical Simula
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XVIII Contents 51 Comparing WAsP an
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XX Contents 58.3 Ultra-High Perform
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XXII List of Contributors Lars Berg
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XXIV List of Contributors Renata Gn
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XXVI List of Contributors A. Kiss D
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XXVIII List of Contributors El˙zbi
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XXX List of Contributors Ivo Sláde
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1 Offshore Wind Power Meteorology B
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1 Offshore Wind Power Meteorology 3
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1 Offshore Wind Power Meteorology 5
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2 Wave Loads on Wind-Power Plants i
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Trubaduren 2 Wave Loads on Wind-Pow
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Base moment (10 6 Nm) 0.4 0.2 −0.
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2 Wave Loads on Wind-Power Plants i
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3 Time Domain Comparison of Simulat
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3 Time Domain Comparison Using Cons
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7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 3 T
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4 Mean Wind and Turbulence in the A
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5 Wind Speed Profiles above the Nor
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5 Wind Speed Profiles above the Nor
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Height [m] 100 90 80 70 60 50 40 30
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6 Fundamental Aspects of Fluid Flow
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f Suu(f, z) / σ 2 u(z) 10 0 10 −
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a) c) −0,375 6 Fluid Flow over Co
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7 Models for Computer Simulation of
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y/h 2.5 2 1.5 1 0.5 0 7 Models for
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8 Power Performance via Nacelle Ane
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9 Pollutant Dispersion in Flow Arou
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x1/B 0.3 0.5 0.8 1.0 9 Pollutant Di
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9 Pollutant Dispersion in Flow Arou
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10 On the Atmospheric Flow Modellin
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11 Comparison of Logarithmic Wind P
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Height above ground in m 100 90 80
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12 Turbulence Modelling and Numeric
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13 Gusts in Intermittent Wind Turbu
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Durations (s) 25 20 15 10 5 13 Gust
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Page 117 and 118: 15 Simulation of Turbulence, Gusts
Page 119 and 120: S(ƒ)(m/s) 2 /Hz 15 Simulation of T
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Page 131 and 132: Log10 E(k)*k**(5/3) 6 4 2 0 5/3 cor
Page 133 and 134: 17.3 Discussion and Conclusion 17 W
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Page 147 and 148: h(u) 0.5 0.3 0.1 20 Superposition M
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Page 153 and 154: 22 Stochastic Small-Scale Modelling
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Page 159 and 160: 23 Quantitative Estimation of Drift
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Page 165 and 166: 24 Scaling Turbulent Atmospheric St
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Page 169 and 170: 25 Wind Farm Power Fluctuations P.
Page 171 and 172: 25 Wind Farm Power Fluctuations 141
Page 173 and 174: d rc q rc V 0 q V 25 Wind Farm Powe
Page 175 and 176: References 25 Wind Farm Power Fluct
Page 177 and 178: 26 Network Perspective of Wind-Powe
Page 183 and 184: 27 Phenomenological Response Theory
Page 189 and 190: 28 Turbulence Correction for Power
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Page 197 and 198: 30 Uncertainty of Wind Energy Estim
Page 203 and 204: 31 Characterisation of the Power Cu
Page 209 and 210: 32 Handling Systems Driven by Diffe
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33 Experimental Researches of Chara
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33 Experimental Researches of Chara
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34 Methodical Failure Detection in
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34 Methodical Failure Detection in
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35 Modelling of the Transition Loca
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35 Modelling an Airfoil with Surfac
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(Cl/Cd)max 120 110 100 90 80 70 60
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35 Modelling an Airfoil with Surfac
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36 Helicopter Aerodynamics with Emp
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37 Determination of Angle of Attack
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37 Determination of Angle of Attack
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Drag coefficient 0.5 0.4 0.3 0.2 0.
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38 Unsteady Characteristics of Flow
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PSD PSD 38 Unsteady Characteristics
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39 Aerodynamic Multi-Criteria Shape
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39 Multi-Criteria Shape Optimizatio
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39 Multi-Criteria Shape Optimizatio
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40 Rotation and Turbulence Effects
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Cp Xsep/c 1 0.6 0.2 −0.2 −0.6
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Standard deviation of Cp 0.7 0.6 0.
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41 3D Numerical Simulation and Eval
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41 3D-CFD-Simulation of a Wind Turb
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42 Performance of the Risø-B1 Airf
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42 Performance of the Risø-B1 Airf
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43 Aerodynamic Behaviour of a New T
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44 Numerical Simulation of Dynamic
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44 Numerical Simulation of Dynamic
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45 Modeling of the Far Wake behind
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8 6 uz 4 2 45 Modeling of the Far W
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46 Stability of the Tip Vortices in
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46 Stability of the Tip Vortices in
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47 Modelling Turbulence Intensities
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48 Numerical Computations of Wind T
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Z X Y 48 Numerical Computations of
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References 48 Numerical Computation
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49 Modelling Wind Turbine Wakes wit
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U mean / U ref 1 0.9 0.8 0.7 49 Mod
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49.5 Conclusion 49 Modelling Wind T
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50 Prediction of Wind Turbine Noise
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0.8 0.6 0.4 0.2 −0.2 −0.4 −0.
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51 Comparing WAsP and Fluent for Hi
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Table 51.1. Measurements and simula
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51 Comparing WAsP and Fluent for Hi
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52 Fatigue Assessment of Truss Join
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53 Advances in Offshore Wind Techno
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Simulation Measurement pitch angle
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53 Advances in Offshore Wind Techno
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54 Benefits of Fatigue Assessment w
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∆σN [N/mm²] 100 10 54 Benefits
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55 Extension of Life Time of Welded
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Transverse residual stresses [N/mm
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56 Damage Detection on Structures o
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56 Damage Detection on Structures o
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57 Influence of the Type and Size o
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[W/m 2 ] q t [W/m 5000 4000 3000 20
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58 High-cycle Fatigue of “Ultra-H
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(a) (b) 450 Load [kN] 400 350 300 2
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59 A Modular Concept for Integrated
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60 Solutions of Details Regarding F
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SR 1000 800 600 400 200 100 80 60 4
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60 Solutions of Details Regarding F
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61 On the Influence of Low-Level Je
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Relative Power and Loading [%] 61 O
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62 Reliability of Wind Turbines Exp
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62 Reliability of Wind Turbines 331
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