# The stochastic power curve analysis of wind turbine.pdf

The stochastic power curve analysis of wind turbinesPatrick MilanÉcole Normale Supérieure de LyonMaster 2 « Sciences de la Matière » – Physics out of Equilibrium2007-2008patrick.milan@ens-lyon.frInternship supervised byJoachim Peinke peinke@uni-oldenburg.deJulia Gottschall julia.gottschall@forwind.deHydrodynamics :::;Pn) at times (t1;:::;tn) reads:W(Pn;tnjPn 1;tn 1;Pn 2;tn 2;:::;P1;t1) = W(Pn;tnjPn 1;tn 1) (1.1)The dynamics of a wind turbine power production will be assumed to be a Markov process [4].1.1.2 GaussianityA stochastic process ~x has a gaussian distribution when its PDF has the form p(x) /e (x x)2= 2x, where x isthe mean value of ~x and x is its standard deviation. In order to greatly simplify the model, we will assumegaussian noise for the stochastic process.1.1.3 StationarityA stochastic process ~x is said to be stationary when its time increment x( ) = x(t+ ) x(t) depends only onthe increment and not on the time. Hence, x( )/j jH where H is called the self-similarity parameter.u(t) gives the wind speed in the direction perpendicular to the rotor blades area. One can show that for a windturbine, the theoretical law that links power outputP(t) and wind speedu(t) isP /u3 [7]. This result is derivedin Appendix A (equation 3.10). This nonlinear (cubic) law introduces problems of stationarity.Let us consider two stochastic processes~a and~b linked by a(t) = b(t)3, and assume that~b is stationary:b( ) = b(t+ ) b(t)/j jH (1.2)a( ) = a(t+ ) a(t) = b(t+ )3 b(t)3 6= b( )3 (1.3)) a( )6=j jH (1.4)The nonlinear law between a(t) and b(t) prevents~a to be stationary even if~b is.1The wind is not stationary. Because the power output P(t) is driven by the wind, we expect P not to bestationary. Furthermore, because of the nonlinear law between u(t) and P(t), the power output P(t) is “evenless“ stationary. In order to overcome the (second) problem, The International Electrotechnical Commission(IEC 61400-12) defined a binning for the wind speed. This means that the wind speed is divided into small binsof size 0:5m=s [1] in which the relation between P and u becomes linear. The binning will be preserved for allfurther analysis. You can observe the effect of the binning on the curve u3 on the following figure:0.0 0.5 1.0 1.5 2.002468u (m/s)u^3 (a.u.)0.0 0.5 1.0 1.5 2.002468u (m/s)Linearization of u^3 (a.u.)Figure 1.1: The graph on the left represents the cubic law u3 and the graph on the right shows the linearizationin bins of size 0:5m=s.1.2 The Langevin model1.2.1 The Langevin equationWe will apply the Langevin equation to the electrical power output P(t) of a wind turbine. As said in theprevious section, the model will be applied using the binning u = 0:5m=s for the wind speed. All quantitieswill be computed in every bin. We also need to discretize the power axis in bins of size P. The (u;P) spacein then discretized in bins of size ( u = 0:5m=s; P). The Langevin equation for the power output (in the Itôdefinition) is a stochastic differential equation:dPdt = D(1)(P;u) +qD(2)(P;u) (t) (1.5)where D(1)(P;u) is called the drift coefficient (or drift field) and D(2)(P;u) is called the diffusion coefficient[6]. These coefficients are given by the following equation:D(n)(P;u) = 1n! lim !0 1 M(n)( ;P;u) = 1n! dM(n)( ;P;u)d j =0 (1.6)M(n)( ;P;u) jP(t)=P (1.7))D(n)(P;u) = 1n! lim !0 1 jP(t)=P (1.8)(t) is a gaussian delta-correlated noise, such that and = 2 (t2 t1).21.2.2 The meaning of the Kramers-Moyal coefficientsThe Langevin equation presented above introduces the first two Kramers-Moyal coefficients D(1)(P;u) andD(2)(P;u). We can notice in equation 1.7 thatM(n) is thenth-order moment of the power increment. Accordingto equation (1.6), D(n) is the slope of the nth-order moment with respect to the increment.! The drift coefficient D(1)(P ;u ) gives the mean (first order) of the time variation of P(t) in the region(P ;u ).!The diffusion coefficientD(2)(P ;u ) gives the deviation (second order) around the mean of the time variationof P(t) in the region (P ;u ).!D(3)(P ;u ) would give the asymmetry (third order) of the time variation of P(t) in the region (P ;u ).! D(4)(P ;u ) would give the intermittency, or flatness (fourth order) of the time variation of P(t) in theregion (P ;u ).The noise term is gaussian. According to Pawula’s theorem, D(n) = 0 (n 4) for such reason [6]. Still forthe same reason (assumed gaussianity), we will setD(3) = 0 by assuming that the evolution is symmetric aroundthe mean value D(1).The noise term (t) (also called Langevin force) is defined as delta-correlated because we assumed that wewere studying a Markov process (section 1.1.1). Under this assumption, the dynamics have no memory and arenot correlated. The gaussian distribution follows the assumption of gaussianity (section 1.1.2).3Chapter 2Wind Turbines And Power Curves2.1 General facts about wind turbinesA wind turbine is a machine that can extract power from the wind to convert it into electrical power output.Wind energy has been exploited for over 3000 years, first with windmills and now with wind turbines. Whilewindmills aimed maximum torque for mechanical purpose, modern wind turbines are designed to extract themaximum power from the wind (more power output = more benefit!!!) [7]. There are many designs of machines,but most of them are horizontal-axis wind turbines (HAWT). We will be studying this kind of machine, yet ourmethod can be applied to all kinds of turbines.It is clear that the primary goal of such a machine is to generate as much power as possible. The power qualityis also taken into account, but we won’t deal with that here. The power production of a wind turbine can berepresented by the so-called power curve. This curve gives the amount of power that can be generated as afunction of the wind speed. As we said in 1.1.3, the theoretical power curve is P /u3. This can be visualizedon the following figure:Figure 2.1: The theoretical power curve that links wind speed (horizontal axis) and power output (vertical axis).The machine extracts more and more power with increasing wind speed up to a rated value. For higher windspeeds, the machine still produces the rated powerPrated, which is one of the main features of any machine. Thetheoretical power curve is of course a generic curve. It gives a typical power extraction, but it cannot be reliablefor every model. For such reason, we have to define a way to characterize the power curve of any model of windturbine.42.2 What to expect from the power curveBecause itgives usthe power production asa functionof thewind speed, the powercurve defines theperformanceof the machine. However, the high level of complexity of the dynamics cannot be analyzed analytically. Wemust use statistics on measurement data. We must define a statistical procedure that is a good balance betweensimplicity (practicality) and complexity (reliability) [5].The goal of the power curve is to characterize the response of the machine. This response should be unique foreach model. It should not depend on the wind condition of the measurement. Of course, a machine won’t havethe same power extraction in the mountains or offshore. However, we must find a tool that can characterizethe machine itself. This model should not depend on the wind fluctuations (the turbulence intensity) of themeasurement and it should give an objective characterization of the turbine.The power curve is a main feature for manufacturers (and maybe more for customers). It should allow us tocompare concurrent machines, make a rough prediction of the power production over long periods of times, aswell as monitor the efficiency of an operating machine over years. For all these reasons, we must create a powercurve that is machine-specific.2.3 How to estimate power curvesIn order to estimate the power curve of a particular machine, we need data measurement. We obviously need thewind speed u(t) and the electrical power output P(t). For a better analysis, we can also use the wind direction~u(t).Figure 2.2: We can measure the wind speed with an anemometer, while the power output is recorded by thegenerator of the turbine.2.4 The standard approach: The IEC methodThe legal procedure to estimate the power curve of a wind turbine is described in IEC 61400-12 [1]. This con-vention was introduced by the International Electrotechnical Commission as a way to fairly estimate the powerextraction of a machine. This procedure collects data for fu(t);P(t)g and transforms it into 10min averagesf10min;10ming. These average quantities are then averaged again in every speed bin of sizeu = 0:5m=s. The result that is given by this method is P(10minbin) [1].The main interest of this method is that it can kill off the fast wind fluctuations through the 10min averaging.Hence, the result is supposed to give the behaviour of the machine without the wind fluctuations. Unfortunately,this is not correct as it introduces systematic averaging errors, as next derived.We can split the wind speed u(t) = +v(t) = V +v(t). Expanding P(u) in a Taylor expansion gives [3]:5P(u) = P(V) + @P(V)@u v + 12 @2P(V)@u2 v2 +o(v3) (2.1)= P(V) + 12 @2P(V)@u2 6= P() = P(V) (2.2)Because P is nonlinear in u, the average of the power is not the power of the average. As we said, the IECprocedure gives P(10minbin), which totally neglects the second-order term in the Taylor expansion.The IEC procedure should be corrected by the second-order term, which is proportional to = 2.Asaconsequenceofthismathematicaloversimplification,theresultappearstodependontheturbulenceintensity[12] because the second-order term was neglected. This means that the method doesn’t extract the response ofthe machine only, as it also depends on the wind condition of the measurement.For such reason, the IEC procedure cannot be fully satisfatory as it doesn’t evaluate the behaviour of themachine objectively. It also suffers systematic averaging errors. Furthermore, the 10min averaging requires verylong measurement times to reach statistical convergence, which is costly and not flexible. A new method mustbe introduced. We can propose the “dynamical“ method.Figure2.3: YoucanobservetheIECpowercurve(squaredots)andthepowercoefficent(rounddots)asevaluatedby the manufacturer Enercon for the commercial machine E70 (Prated = 2:3MW) [13]. The power coefficient isdefined in appendix A (equation 3.11).6Chapter 3The Dynamical Power Curve AnalysisWe will use the Langevin model on the power output P(t) to derive the dynamical power curve. A descriptionof this model is given in 1.2. As we already said, the Langevin equation reads:dPdt = D(1)(P;u) +qD(2)(P;u) (t) (3.1)where D(1)(P;u) is called the drift coefficient (or drift field) and D(2)(P;u) is called the diffusion coefficient.(t) is a gaussian delta-correlated noise, such that and = 2 (t2 t1).As explained in 1.2.2, the drift coefficient D(1)(P ;u ) gives the mean evolution of the power output in theregion (P ;u ). This term desribes the dynamics of the machine and it corresponds exactly to what we expectfrom the power curve. The wind fluctuations are carried by D(2) (second-order term). Hence, the power curvewill be built upon the drift field.3.1 The measurement dataI applied the method to two datasets, both with very different bahaviour:! [DATASET 1] from a MegaWatt-class commercial wind turbine. The model and brand cannot be givenhere as an agreement with the manufacturer, who kindly gave us the dataset. In order to respect their privacy,the value of the power output was normalized (by the maximal power). Very little information is given here,except that the measurement was made at frequency 1Hz for 6 consecutive days (600;000 data points) for thepower output, wind speed and wind direction.! [DATASET 2] from a 2MW wind turbine on the flat terrain of Tjäreborg (Denmark) [8]. The measure-ments were realized according to the IEC requirements. The wind speed was recorded on a met mast at 60mhub height. The timeseries are a collection of 10mindata samples at frequency 25Hz for a total of 24H (2:3 106data points) for the power output and wind speed.5 10 15 20 25 300.20.40.60.81.0[DATASET 1]u (m/s)P (norm.)020004000600080005 10 15 20 25 30 355001000150020002500[DATASET 2]u (m/s)P (kW)050001000015000Figure 3.1: You can see on this figure the density of data points for both datasets.73.2 The drift field and the dynamical power curveThe drift field D(1) and its corresponding error [D(1)] are defined by:D(1)(P;u) = lim !0 1 jP(t)=P (3.2)[D(1)(P;u)] =r2D(2)(P;u)N D(1)(P;u)2N (3.3)where the used for the error estimation corresponds to the we used to compute the limit of D(1), and N isthe number of data points [3].5 10 15 20 25 30 355001000150020002500u (m/s)P (kW)−20−1001020P (kW)D1(P) (kW/s)a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a71a7115001625175018752000212522502375−20−1001020Figure 3.2: [DATASET 2] On the left figure, you can see the entire drift field D(1) for all the regions of (P;u).The vertical lines surround the speed bin u = (20:25 0:25)m=s which is plotted on the right figure. On theright figure, you can observe the drift field D(1)(P;u) for u = (20:25 0:25)m=s with corresponding error bars.The horizontal line represents D(1) = 0. The triangle marks the fixed point, which will be defined next.WhenD(1) 0, that means that the machine tries to increase the power extraction. In the same pattern, thepower extraction decreases when D(1) 11m=s) region so that theestimated error is quite high. A longer measurement time (or higher frequency) would be welcome here.)[DATASET 2]!The faster region (u 12m=s) displays two attractive plateaux for P 1000kW and P 2000kW. This isthe signature of a gearbox. The lower plateau has higher estimated error because the density of data is quitelow there. This would explain that for high wind speed the gearbox can choose between several gears, yet thefastest one is almost always used. The other plateaux (for P 1250kW and P 1500kW) are less attractive,certainly corresponding to transition states.! The slower region (u = 0. y has a memory if R1A0 d = 1. Because it took quite a long time torun, I didn’t finish this computation for P. It looked like this quantity increases quite fast though, so that thedynamics of a wind turbine are not delta-correlated.4.1.2 GaussianityWe assumed that the noise term was gaussian-distributed. We know that the wind u is intermittent, and becausethe power output is driven by the wind, it should be intermittent too.−15 −10 −5 0 5 10 151e−061e−041e−021e+00wind speedN = 2173010 Bandwidth = 0.02672Density−15 −10 −5 0 5 10 151e−061e−031e+001e+03power outputN = 2173010 Bandwidth = 0.02675DensityFigure 4.1: [DATASET 2] Normalised PDFs of signed wind speed and power output increments. From top tobottom, f = 1;5;10;100;500;1000;2000.It is clear that both wind speed and power output are intermittent. The PDFs show fat tails for smallincrements, which is the signature of intermittency. The win