Wind models for simulation of power fluctuations from wind farms

Journal of Wind Engineering

and Industrial Aerodynamics 90 (2002) 1381–1402

Wind models for simulation of power fluctuationsfrom wind farms

Poul S�rensena,*, Anca D. Hansena, Pedro Andr!eCarvalho Rosasa,b

aWind Energy Department, Ris� National Laboratory, P.O. Box 49, DK-4000 Roskilde, DenmarkbDepartment of Electrical Power Engineering, Technical University of Denmark, DK-2800 Lyngby,

Denmark

Abstract

This paper presents a wind model, which has been developed for studies of the dynamic

interaction between wind farms and the power system to which they are connected. The wind

model is based on a power spectral description of the turbulence, which includes the coherence

between wind speeds at different wind turbines in a wind farm, together with the effect of

rotational sampling of the wind turbine blades in the rotors of the individual wind turbines.

Both the spatial variations of the turbulence and the shadows behind the wind turbine towers

are included in the model for rotational sampling. The model is verified using measured wind

speeds and power fluctuations from wind turbines.

r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Wind models; Turbulence; Wind turbines; Power fluctuations; Tower shadow effect; 3p;

Coherence

1. Introduction

This paper presents a wind model, which has been developed to support studies ofthe dynamic interaction between large wind farms and the power system to whichthey are connected, and to support improvement of the electric design of windturbines as well as grid connections.The work is mainly a result of a Danish project titled ‘‘Simulation of wind power

plants’’, but also a Brazilian Ph.D. study titled ‘‘Power quality and stability issues ofintegration of large wind farms’’ has contributed to the model development. ThisPh.D. study is accomplished in Denmark.

*Corresponding author.

0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 2 6 0 - X

As part of the Danish project, a dynamic model of a Danish wind farm has beenimplemented in the dedicated power system simulation tool DIgSILENT, compris-ing models of grid, wind turbines and wind speeds. General descriptions of theapplied models have been given by S�rensen et al. [1], whereas the present paper willconcentrate on the wind models.The background for the Danish project and the Brazilian Ph.D. study is the fast

development and ambitious targets for wind energy. As the wind energydevelopment is concentrated in areas with good wind resources, where the powergrids are often not so strong, wind energy is becoming a significant source of supplyto the power systems in these areas. As a consequence, wind energy is also playing anincreasingly important role in the operation of these power systems.The increased wind energy development is also reflected in the requirements for

grid connection of wind turbines. National standards for power quality of windturbines have been supplemented by a new IEC 61400-21 [2] standard formeasurement and assessment of power quality of grid connected wind turbines.The wind models presented in this paper have been developed with the intention to

support simulation of the power fluctuations, which are quantified in IEC 61400-21.Such simulations can help the wind turbine industry to improve the electric design ofwind turbines, and to reduce the costs for grid connection to what is required fromthe point of view of keeping a high power quality in the systems.

2. Wind model structure

The wind model is essential to obtain realistic simulations of the powerfluctuations during continuous operation of the wind farm. The present wind modelcombines the stochastic effects caused by the turbulence and deterministic effectscaused by the tower shadow.The stochastic part includes the (park scale) coherence between the turbulence at

different wind turbines as well as the effects of rotational sampling, which is knownto move energy to multiples (often denoted p’s, e.g. 3p) of the rotor speed from thelower frequencies [3].Only the longitudinal component of the wind speed is included, which is normally

a reasonable assumption for wind turbines, because this component has thedominating influence on the aero loads on wind turbines.The park scale coherence is included, because it ensures realistic fluctuations in the

sum of the power from all wind turbines, which is important for the maximum poweroutput from the wind farm. In IEC 61400-21, it is specified that the maximum 200msaverage power as well as the maximum 1min average power must be measured as apart of the power quality test of wind turbines.To be able to extend the results from measurement on a single wind turbine to a

wind farm, IEC 61400-21 assumes that the turbulence of the wind at the differentwind turbines is uncorrelated. Moreover, it is assumed that for each wind turbine i;the maximum 200ms power P0:2;i appears at rated power Pn;i: These assumptionsleads to the following estimate of the maximum 200m s power P0:2S of a wind farm

P. S�rensen et al. / J. Wind Eng. Ind. Aerodyn. 90 (2002) 1381–14021382

with N wind turbines

P0:2S ¼XN

i¼1

Pn;i þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i¼1ðP0:2;i � Pn;iÞ

2

r: ð1Þ

Assuming that the turbulence at different wind turbines is uncorrelated, acorresponding relation can be made to predict the standard deviation of the powerfrom a wind farm, knowing the standard deviation of the power from each windturbine. However, analyses of measurements, e.g. Tande et al. [4], have shown thatthe assumption of uncorrelation implies an underestimation of the standarddeviation of the power from a wind farm.The main reason for this underestimation is that the actual correlation between the

turbulence at different wind turbines has some influence, particularly when thedistance between the wind turbines is small. The present wind speed model includesthe coherence of the turbulence to be able to obtain better estimates of maximumpower and power standard deviation.The effect of the rotational sampling is included because it is a very important

source to the fast power fluctuations during continuous operation of the windturbine. The fast fluctuations are particularly important to assess the influence ofwind turbines on the flicker levels in the power system, i.e. the level of voltagefluctuations causing flicker in the illumination from electric light bulbs.In many cases, e.g. S�rensen [5], measurements have shown that the 3p effect

due to rotational sampling provides the main contribution to flicker emissionfrom wind turbines to the grid during continuous operation of the windturbines.The structure of the wind model is shown in Fig. 1. It is built as a two-step model.

The first step of the wind model is the park scale wind model, which simulates thewind speeds vhub;1;y; vhub;N in a fixed point (hub heigh) at each of the N windturbines, taking into account the park scale coherence. The second step of the windmodel is the rotor wind model, which includes the influence of rotational samplingand integration along the wind turbine blades as the blades rotate. The rotor windmodel provides an equivalent wind speed veq;i for each wind turbine i; i.e. a singletime series for each wind turbine, which is conveniently used as input to a simplifiedaerodynamic model of the wind turbine.The park scale wind model is implemented in an external program PARKWIND

that generates a file with wind speed time series, which are then read by DIgSILENT.This model interface is possible because the wind speeds are assumed to beindependent on the operation of the wind farm, which is reasonable because therequired wind speed vhub;i for each wind turbine i is the wind speed in hub height ifwind turbine i was not erected.The present version of PARKWIND does not include the effects of wakes in the

wind farm, but the mean wind speed and turbulence intensity could be modified toaccount for these effects. Jensen [6] suggested a wind farm model to predict thereduction of the mean wind speed in a wake relative to the ambient mean wind speed.Frandsen and Th�gersen [7] suggested a model for combining the ambient

P. S�rensen et al. / J. Wind Eng. Ind. Aerodyn. 90 (2002) 1381–1402 1383

turbulence and the wake-induced turbulence to predict the influence of the windfarm wakes on fatigue loadings on the wind turbines.The rotor wind models describes the influence of rotational sampling and

integration along the wind turbine blades as the blades rotate. The present model forthe wind field includes turbulence as well as tower shadow effects. The wind shear isnot included in the model, because it only has a small influence on the powerfluctuations as discussed in Section 4.1.The rotor effects are included for each of the n wind turbines individually. The

wind speed seen by the rotating blades of the ith wind turbine depends on theazimuth position yWTR;i of the wind turbine rotor. As illustrated in Fig. 1, yWTR;i isfed back from the mechanical part of the wind turbine model.The wind model provides an equivalent wind speed veq;i for each wind turbine i;

which is used as input to the aerodynamic model of that wind turbine. veq;i is a singletime series for the wind turbine i; which takes into account the variations inthe whole wind speed field over the rotor disk. The advantage of using the equivalentwind speed is that it can be used together with a simple, Cp-based aerodynamicmodel, and still include the effect of rotational sampling of the blades over therotor disk.

Park

sca

le w

ind

mod

el

vhub,2 veq,2Rotorwind

Windturbine 2

θWTR,2


Windturbine 1

θWTR,1

vhub,n veq,nRotorwind

Windturbine n

θWTR,N

Park

sca

le w

ind

mod

el


Windturbine 2

θWTR,2


Windturbine 1

θWTR,1

vhub,n veq,nRotorwind

Windturbine n

θWTR,N

Fig. 1. Structure of wind model.


3. Park scale model

3.1. Review of methods

Different methods can be applied to simulate the wind speeds in a wind farm.Estanqueiro [8] used the Shinozuka [9] method based on a cross spectral matrix.Initially, Mann’s [10] simulation method was suggested used in this project, becausethe calculation speed of that method is generally faster than cross spectral matrixmethod. Some of the results using Mann’s model were presented in S�rensenet al. [11].Both the Estanqueiro application of the Shinozuka method and the Mann method

assume Taylor’s frozen turbulence hypothesis illustrated for a two-dimensional windspeed field in Fig. 2. The wind speed field is generated in spatial dimensions in thefirst place, and then the field is moved forward with the mean wind speed.Taylor’s frozen turbulence hypothesis is a reasonable assumption for simulations

where the simulated time series only pass the object once, like a wind turbine rotor.Both the Shinozuka method (with frozen wake) and the Mann method are used incomputer programs for simulation of mechanical loads on wind turbines to simulatethe wind speed variations in the rotor plane of a single wind turbine. In that case, thesimulated wind speed time series only pass the object once.But for park scale wind simulations, the Taylor hypothesis is not so realistic,

especially when the wind direction is along a line of wind turbines. In that case,simulations assuming Taylor’s frozen turbulence will generate wind speed time serieswhich are identical for all wind turbines in the line, apart from a delay correspondingto the travel time for the wind from one wind turbine to the other.One consequence of assuming frozen turbulence is that the coherence between the

turbulence in two points on a line in the wind direction will be one, which is notrealistic. Another measure, which reveals the error introduced by assuming Taylor’s

V0V0

Fig. 2. Simulation of park scale wind speeds with the assumption of Taylor’s frozen turbulence

hypothesis.


frozen turbulence is the cross correlation function. These aspects are discussedfurther in Section 5. The assumption of frozen turbulence will significantly affect thesummation of power fluctuations from the wind turbines in a line.

3.2. The complex cross spectral method

To avoid the assumption of Taylor’s frozen turbulence, we have introduced a newmethod for simulation of park scale wind speeds. The new method is based onShinozukas cross spectral matrix method, and it uses a complex cross spectral matrixas Shinozuka originally proposed instead of the real matrices that have been used infrozen turbulence models for wind turbines. The cross spectral matrix becomescomplex as a result of the time delay between points with longitudinal components oftheir separation.The new method also has the advantage that it does not produce more data than

what is needed. The Mann method on the other hand produces a grid of data, whichthen in turn has to be interpolated in at the initial positions of the wind turbinesrelative to the data grid. The complex cross spectral method directly generates asingle time series at the position of each wind turbine. Because of this data reduction,the new method also reduces the computation time considerably compared to thefast Mann method.The cross spectral method is based on the cross power spectrum matrix Sðf Þ;

which with N points (corresponding to a wind farm with N wind turbines) is anN � N matrix. We have chosen to use the frequency in Hz, f. Each element Srcðf Þ inrow r; column c of Sðf Þ is determined as the cross power spectrum between theturbulence at wind turbine number r and c: Srcðf Þ is defined as the Fourier transformof the cross correlation function RrcðtÞ according to

Srcðf Þ ¼Z

N

�N

RrcðtÞe�j2pf t dt: ð2Þ

The cross correlation function RrcðtÞ between vhub;rðtÞ and vhub;cðtÞ is defined as

RrcðtÞ ¼ Efvhub;rðtÞ � vhub;cðt � tÞg; ð3Þ

where Eff ðtÞg denotes the mean value of f ðtÞ over the time t:The first step in the cross power spectral method is to determine Srcðf Þ: Fig. 3

shows the two points r and c each corresponding to a wind turbine.The distance between the two wind turbines is drc; with the angular direction yrc

from north. The mean wind speed V0 and the wind direction yV are also shown.arc ¼ yV � yrc is the inflow angle.Fig. 3 also indicates the delay time trc for the wind field to travel from wind

turbine c to wind turbine r: Simple geometry yields trc determined as

trc ¼cosðarcÞ � drc

V0: ð4Þ

To represent the time delay trc in the cross power spectrum, we assume that RrcðtÞas defined in Eq. (3) is symmetric about trc: Then using Eq. (2), it can be shown that


the argument of the complex number Srcðf Þ is �2pf trc; i.e.

Srcðf Þ ¼ jSrcðf Þj � e�j2pf trc ; ð5Þ

where j is the complex unity number. The size of the cross power spectrum jSrcðf Þjcan be determined using the standard definition of the coherence functiong2ðf ; d;V0Þ;

g2ðf ; drc;V0Þ ¼jSrcðf Þj

2

Srrðf ÞSccðf Þ: ð6Þ

Combining Eqs. (6) and (5), we can express the complex cross power spectrum as

Srcðf Þ ¼ gðf ; drc;V0ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSrrðf ÞSccðf Þ

p� e�j2pf trc : ð7Þ

The second step is to discretise the frequency to be able to represent the spectra ina numeric computer code. Simulating time series with the period length TP; thefrequency f is discretised in steps Df ¼ 1=TP:; i.e. the ith frequency f ½i� ¼ i�Df . Thecorresponding discrete value of Srcðf Þ is Src½i� ¼ SrcðfiÞ � Df :We also discretise the time by the sampled representation of the wind speeds as

time series with time steps Dt ¼ 1=fs; where fs is the sampling frequency in Hz. Thissampling limits the frequency to 7fs=2 and consequently the frequency index i to7Ns=2; where

Ns ¼ TP � fs ð8Þ

is the number of samples in the simulated time series. Obviously, Ns must be aninteger, and preferably an exponent of 2 which enables the use of an FFT to speed upthe Fourier transformation used in the end of the method. This can be obtained byadjusting either TP or fs according to Eq. (8).Selecting an appropriate sampling frequency and assuming two-sided spectra, this

discretisation ensures that the variance s2r of the wind speed at wind turbine r is

αrc

V0

drc

V0·τrc

c

r

θV

θrcαrc

V0

drc

V0·τrc

c

r

θV

θrc

Fig. 3. Two points r and c each corresponding to a wind turbine. The distance between the two points r

and c is drc; with a direction yrc from north. V0 is the mean wind speed, and yV is the wind direction. arc is

the resulting inflow angle, and trc is the delay time.


preserved according to

s2r ¼Z

N

�N

Srrðf Þ df

EZ fs=2

�fs=2Srrðf Þ df

EXNs=2

i¼�Ns=2

Srr½i�: ð9Þ

The discretisation is only done for frequency indices iX0; because the values forio0 are given by Src½�i� ¼ Src ½i�; where denotes complex conjugation.The third step is for each frequency index iX0 to resolve the discrete matrix S½i�

with the elements Src½i� into a product of the transformation matrix H ½i� and thetranspose of its conjugate H T ½i�; i.e.

S½i� ¼ H ½i�H T½i�: ð10Þ

Choosing the solution where H ½i� is a lower triangular matrix, i.e. the elementHrc½i� ¼ 0 if c > r; the elements can be found one by one. The diagonal elements aredetermined according to

Hrr½i� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSrr½i� �

Xr�1

k¼1Hrk½i�H

rk½i�

rð11Þ

and the elements below the diagonal are determined according to

Hrc½i� ¼Src½i� �

Pc�1k¼1 Hrk½i�H

ck½i�Hcc½i�

: ð12Þ

It can be seen from Eqs. (11) and (12) that H ½i� gets the same phases as S½i�; i.e.zero phase shift in the diagonal and a phase shift �2pf trc below the diagonal,corresponding to the delay of wind speed between two wind turbines r and c.

The fourth step is for each frequency index iX0 to generate an N � 1 vector E½i� ofunity complex numbers with a random phase. This is done by simulating N randomphase angles jr½i� using a random generator with uniform distribution in the interval½0; 2p�; and calculate each element Er in row r of E according to

Er ¼ ejjr½i�: ð13Þ

The fifth step is for each frequency index iX0 to calculate a vector Vhub½i�containing the ith Fourier coefficients of all N wind speed time series according to

Vhub½i� ¼ Hhub½i�E½i�: ð14Þ

The imaginary part of Vhub½0� should be set to zero, because an imaginary part ofthis frequency component does not make sense.Finally, for each wind turbine r; the Fourier coefficients are joined in an array

Vhub;r; and an inverse Fourier transform is performed to obtain the time seriesvhub;rðtÞ:


3.3. Spectral distributions

At the present state, the model is capable of simulating wind speeds with powerspectra of either Kaimal or H�jstrup type, but implementation with another spectrumis straightforward, as the spectra are used explicitly in the model according to Eq. (7).The Kaimal spectrum has been selected in the first place because it is used widely,

while the H�jstrup spectrum was selected because it includes more energy than theKaimal spectrum at the low frequencies, and this has shown to agree better in anumber of cases.The two-sided Kaimal [12] spectrum SKaiðf Þ is determined according to

SKaiðf Þ ¼ u2 52:5ðz=V0Þ

ð1þ 33ðz=V0Þf Þ5=3

ð15Þ

and the H�jstrup [13] spectrum SHojðf Þ according to

SHojðf Þ ¼ u2 �2:5ðAHoj=V0Þ

1þ 2:2ððAHoj=V0Þf Þ5=3

þ52:5ðz=V0Þ

1þ 33ðz=V0Þf� �5=3

! �

1

1þ 7:4ðz=AHojÞ:

ð16Þ

In Eqs. (15) and (16), we have used the (hub)height z and the friction velocity u

which can be determined by the roughness length z0 according to

lnz

z0

� �¼

kV0

u ; ð17Þ

where k ¼ 0:4 is the von Karman constant. The spectra in Eqs. (15) and (16) aregiven as they are developed in the lower boundary layer. However, wind turbines arefar above the lower boundary layer today. For large wind turbines, it is oftenassumed that the length scale LE20z only increases with height up to z ¼ 30m, e.g.Danish code of practice for loads and safety of wind turbine structures [14].

3.4. Coherence

As it is seen from Section 3.2, it is simple to use any coherence function with thePARKWIND method. We have chosen to implement a Davenport [15] typecoherence, and use the decay factors recommended by Schlez and Infield [16] asdefault values in the program.Schlez and Infield studied the horizontal two-point coherence for separations

greater than the measurement height, and their recommendations are based onestimates on own measurements and several other measurements.The Davenport type coherence function between the two points r and c (see Fig. 3)

can be defined in the square root form

gðf ; drc;V0Þ ¼ e�arcðdrc=V0Þf ; ð18Þ

where arc is the decay factor. Schlez and Infield uses a decay factor which depends onthe inflow angle arc shown in Fig. 3. The figure shows that arc ¼ 0 corresponds to


points separated in the longitudinal direction, and arc ¼ 901 corresponds to pointsseparated in the lateral direction.With a given arc; the decay factor can be expressed according to

arc ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðalong cos arcÞ

2 þ ðalat sin arcÞ2

q; ð19Þ

where along and alat are the decay factors for separations in the longitudinal and thelateral directions, respectively. Using our definition of coherence decay factors inEqs. (18) and (19), the recommendation of Schlez and Infield can be rewritten as touse the decay factors

along ¼ ð1575Þ �s

V0; ð20Þ

alat ¼ ð17:575Þðm=sÞ�1 � s; ð21Þ

where s is the standard deviation of the wind speed in m/s.

4. Rotor wind model

4.1. Equivalent wind speed

As described in Section 2, the rotor wind model provides an equivalent wind speedveq; which takes into account the variations due to turbulence and tower shadow inthe wind speed field over the rotor disk. This section describes how the equivalentwind speed is derived from the rotor wind speed field.Fig. 4 shows the three bladed wind turbine with the wind field vðt; r; yÞ: It is seen

that the positions are given in the polar coordinates ðr; yÞ; where y is denoted theazimuth angle.The aerodynamic torque TaeðtÞ is given as the sum of the blade root moments

MbðtÞ in the drive direction of each blade b; i.e.

TaeðtÞ ¼X3b¼1

MbðtÞ: ð22Þ

Fig. 4 indicates that the blades are profiled from the inner radius r0 to the outerradius R of the rotor disk. Linearising the blade root moment dependence on thewind speed we obtain

MbðtÞ ¼ MðV0Þ þZ R

r0

cðrÞðvðt; r; ybÞ � V0Þ dr; ð23Þ

where MðV0Þ is the steady state blade root moment corresponding to the mean windspeed V0; and cðrÞ is the influence coefficient of the aero load on the blade rootmoment in radius r:For aero torque, a typical load distribution along the blades can be obtained by

assuming cðrÞ to be proportional to r and r0 ¼ 0:1R; which has also been assumed in


the implemented model. However, for the sake of completeness, we will formallykeep cðrÞ and r0 here.Inserting Eq. (23) in Eq. (22) we obtain

TaeðtÞ ¼ 3MðV0Þ þX3b¼1

Z R

r0

cðrÞðvðt; r; ybÞ � V0Þ dr: ð24Þ

Now we define the equivalent wind speed veqðtÞ as the single wind speed time serieswhich would give the same aerodynamic torque as the actual wind speed field, i.e.veqðtÞ must fulfil

TaeðtÞ ¼ 3MðV0Þ þX3b¼1

Z R

r0

cðrÞðveqðtÞ � V0Þ dr: ð25Þ

1

2

3

v(t,r,θ)

(R,θ1)

(r0,θ1)

(r,θ)

1

2

3

v(t,r,θ)

(R,θ1)

(r0,θ1)

(r,θ)

Fig. 4. The wind speed field in the rotor plane is given as vðt; r; yÞ; the blades are profiled from the inner

radius r0 to the outer radius R: The figure also indicates azimuth position y1 of blade number 1.


Combining Eq. (25) with Eq. (24) gives the equivalent wind speed a the meanvalue of contributions from all three blades:

veqðtÞ ¼1

3

X3b¼1

vcðt; ybÞ; ð26Þ

where we have used the weighted wind speed vcðt; ybÞ defined as

vcðt; ybÞ ¼

RR

r0cðrÞvðt; r; ybÞ drRR

r0cðrÞ dr

: ð27Þ

Eqs. (26) and (27) express the equivalent wind speed as a weighting of all the windspeeds which are instantaneously seen by the wind turbines along the blades.We could now simulate the wind speeds in a number of points in the rotor plane

like it is typically done in codes for simulation of mechanical loads on wind turbines.However, a much more computer time saving simulation method for simulation ofthe equivalent wind speed has been developed in Ris� National Laboratory.This simulation method was first presented by Langreder [17] for the contribution

from turbulence. Later, Rosas et al. [18] included tower shadow effects in themethod. In this paper, it is combined with the park scale model.The equivalent wind speed simulation method is based on Ris�s frequency domain

models [19–21]. These models are based on expansion of the wind speed field in therotor plane in the azimuth angle.To understand the simulation model, we first expand the weighted wind speed for

a single blade in the azimuth angle, i.e.

vcðt; ybÞ ¼XN

k¼�N

*vc;kðtÞejkyb ; ð28Þ

where *vc;kðtÞ is the kth azimuth expansion coefficient of vðt; ybÞ determined accordingto

*vc;kðtÞ ¼1

2p

Z 2p

0

vcðt; ybÞe�jny dy: ð29Þ

Inserting Eq. (28) in Eq. (26) yields

veqðtÞ ¼XN

k¼�N

*vc;3kðtÞej3kyWTR ; ð30Þ

where yWTR ¼ y1 is the wind turbine rotor position obtained from the mechanicalmodel.It is seen from Eq. (30) that only the azimuth expansion coefficients with orders

which are multiple of 3 contribute to the sum. This is because of the symmetricstructure of rotor, which causes the contributions from the other orders to even outwhen they are summed up for all three blades. If a 1p and/or 2p are still significant ina measurement of torque or power, this is often an indication that the blades are notpitched with exactly the same angle.


In the present implementation of the rotor wind model, we have only included the0th and 3rd harmonics, i.e. Eq. (30) has been approximated to

veqðtÞE *vc;0ðtÞ

þ 2Ref*vc;3ðtÞgcosð3yWTRÞ

þ 2Imf*vc;3ðtÞgsinð3yWTRÞ: ð31Þ

The idea of the rotor wind model is to simulate the azimuth expansion coefficients*vc;kðtÞ in the first place as independent on the azimuth position of the rotor, and thenuse Eq. (31) to generate the equivalent wind speed which includes the azimuthdependence.The azimuth expansion coefficients *vc;kðtÞ are sums of contributions from the

turbulence model, tower model and the mean wind speed. The mean wind speedcontributes with V0 to *vc;0ðtÞ: The contributions from the turbulence model and thetower model are determined in the next subsections.Other effects like wind shear and yaw error could be included, but these effects

mainly contribute to the 1p; which is filtered away by the summation of the 3symmetric blades as mentioned above.

4.2. Turbulence model

As a consequence of the description above, the turbulence model generates theazimuth expansion coefficients *vc;k;turbðtÞ of the turbulence field. It has been shownthat [19,21] the power spectral density (PSD) S*vc;k ðf Þ of *vc;k;turbðtÞ can be determinedaccording to

S*vc;k ðf Þ ¼ F*vc;k ðf Þ � Svðf Þ; ð32Þ

where Svðf Þ is the PSD of the wind speed in a fixed point, and F*vc;k ðf Þ is denoted theadmittance function. F*vc;k ðf Þ can be determined by a triple integral, which can beresolved into the double integral

F*vc;k ðf Þ ¼

RR

r0

RR

r0cðr1Þcðr2ÞFkðf ; r1; r2Þ dr1 dr2

ðRR

r0cðrÞ drÞ2

ð33Þ

and the single integral

Fkðf ; r1; r2Þ ¼1

2p

Z 2p

0

glvðf ;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 þ r22 � 2r1r2 cosðyÞ

qÞ cosðnyÞ dy: ð34Þ

Here, glvðf ; dÞ is the square root coherence function between two points with adistance d in the rotor plane. glvðf ; dÞ is assumed to be the same horizontally (i.e.laterally) and vertically in the plane.Eqs. (33) and (34) have been solved numerically by S�rensen [21]. Using the

Laplace operator s ¼ jo ¼ j2pf ; Langreder [17] defined the transfer functionsH *vc;k ðj2pf Þ with the size defined as

jHc;kðj2pf Þj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF*vc;k ðf Þ

qð35Þ


and used the numerical results to fit H *vc;0 ðj2pf Þ and H *vc;3 ðj2pf Þ to linear filters.Defining the constant d ¼ R=V0; the results of these fittings are

Hc;0ðsÞE0:99þ 4:79ds

1þ 7:35ds þ 7:68ðdsÞ2; ð36Þ

Hc;3ðsÞE0:0307þ 0:277ds

1þ 1:77ds þ 0:369ðdsÞ2: ð37Þ

Using vhubðtÞ from the park scale model as input and a linear filter with the transferfunction Hc;0ðj2pf Þ; *vc;0;turbðtÞ is now simulated according to

*Vc;0;turbðf Þ ¼ Hc;0ðj2pf Þ � Vhubðf Þ; ð38Þ

where *Vc;0ðf Þ and Vhubðf Þ are the Fourier transforms of *vc;0;turbðtÞ and vhubðtÞ;respectively.

*vc;3;turbðtÞ is a complex variable, and it was shown [19,21] that the real andimaginary parts are uncorrelated with each other and with azimuth expansioncoefficients of other orders k: Distributing the variance between the real andimaginary parts of *vc;3;turbðtÞ evenly, they are determined by the relations betweenFourier transforms

Ref *Vc;3;turbðf Þg ¼1ffiffiffi2

p Hc;3ðj2pf Þ � V3;Reðf Þ; ð39Þ

Imf *Vc;3;turbðf Þg ¼1ffiffiffi2

p Hc;3ðj2pf Þ � V3;Imðf Þ; ð40Þ

where V3;Reðf Þ and V3;Imðf Þ are Fourier transforms of uncorrelated stochastic signalswith the same PSD as the wind speed in a fixed point.To support the simulation of V3;Reðf Þ and V3;Imðf Þ; Langreder also fitted a filter

which converts uniformly distributed white noise to a signal with the PSD as theKaimal spectrum.

4.3. Tower shadow model

Today most wind turbines are constructed with a rotor upwind of the tower toreduce the tower interference of the wind flow. Early wind turbines often had latticetowers, but because of the visual impact, tubular towers are the most common today.The tubular towers have more effect on the flow than lattice towers. In the upwind

rotor case, the tower disturbance vtow can be modelled using potential flow theory.Ekelund [22] found

vtow ¼ V0a2 x2 � y2

ðx2 þ y2Þ2; ð41Þ

where a is the tower radius, and x and y are the components of the distance fromeach blade to the tower centre in the lateral and the longitudinal directions,respectively.


Rosas [18] used Eqs. (41), (27) and (29) to calculate the azimuth expansioncoefficients caused by the tower shadow. Neglecting the effect of the blade bending,these coefficients become constants *vc;k;tow; which can be added to contributionsfrom the turbulence.

5. Verification

The parkscale model has been verified using wind speed measurements on two seamasts SMW and SMS on the Vindeby offshore site. The offshore site is shown inFig. 5.Eleven wind turbines are installed on this site, and three meteorological masts

were installed as part of a major data collection on the worlds first offshorewind farm. The distance between the two sea masts SMS and SMW wasdSW ¼ 807m, and the angle ySW ¼ 2971: The measurements used in this paper wereacquired in 1994.Fig. 6 shows 1 h of 1min block average values of wind speed measurements with

the mean wind speed V0 ¼ 11m/s and the mean wind direction yV ¼ 2931; togetherwith PARKWIND simulations with the same mean wind speed and wind direction.

-2000

-1500

-1000

-500

0

500

1000

1500

-500 0 500 1000

Wind turbines

Met. mastsSMS

SMW

LM

Wind turbines

Met. masts

Fig. 5. Vindeby offshore site with 11 wind turbines, two offshore masts SMW and SMS and the land mast

LM.


Both measurements and PARKWIND indicate a delay of SMS with a little morethan 1min relative to SMW. This corresponds very well to the expected delay for11m/s with 807m distance.A more consistent verification is to compare the coherence functions. However,

this requires a substantial data set, because the coherence in 807m distance is verysmall.Fig. 7 shows the square root coherence of 6� 6.5 h wind speed with average wind

direction 2911. To get an idea of the variation of the wind direction, the standarddeviation of the 234 10-min mean values of wind directions is 4.31, with maximumwind direction 3001 and minimum 2771.The inflow angle of the measurements has been calculated from the mean value of

the wind direction in each (6.5 h) time series, and used to determine the decay factorfor the coherence according to Eq. (19) for each of the six simulations. Thecorresponding coherence (‘‘Davenport’’) is also indicated in the figure as a straightline.The coherence comparisons show very good agreement between measurements

and simulations, indicating that the Davenport type coherence and Schlez andInfield’s decay factors are reasonable in this case.Finally, the simulated and measured cross correlation functions are compared in

Fig. 8. 8� 10min with average inflow angles a ¼ 70:51 have been found in the6� 6.5 h wind speed time series and used to estimate the cross correlation functions.This is not enough data to estimate the coherence in this narrow wind direction

0 10 20 30 40 50 608

10

12

14

time [min]

win

d sp

eed

[m/s

]Measured

0 10 20 30 40 50 608

10

12

14

time [min]

win

d sp

eed

[m/s

]

Simulated

SMW

SMSSMS

SMW

SMS

SMW

SMS

Fig. 6. Simulation of wind speeds compared to measurements on two sea masts in Vindeby offshore wind

farm.


band, but as it can be seen from Fig. 8, the estimated cross correlation functions arereasonably smooth.The exactly measured inflow angle has been used in the simulations, ensuring that

also distributed inflow angles in the interval a ¼ 70:51 for the simulations. Two

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0110-1

100

frequency [Hz]

sqrt

(coh

eren

ce)

MeasuredSimulatedDavenport

Fig. 7. Square root coherence function of measured and simulated 6� 6.5 h wind speed time series on

Vindeby SMW and SMS.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalised delay

Nor

mal

ised

cro

ss c

orre

lati

on f

unct

ion

Measured Simulated Sim-frozen

Measured Simulated Sim-frozen

Fig. 8. Measured and simulated cross correlation functions of the eight 10min time series with average

inflow angles a ¼ 70:51: Simulation is done with the present model, whereas Sim frozen is done assuming

Taylor’s frozen turbulence.


simulations have been done for each of the time series, one with the presentPARKWIND model using the Schlez and Infield decay factors along and alat; and oneassuming Taylor’s frozen turbulence. The frozen turbulence is also simulated withPARKWIND, using the longitudinal decay factor along ¼ 0:The values shown in Fig. 8 have been normalised, so that the delay=1

corresponds to the travel time of the wind form SMW to SMS, using the meanwind speed in each of the 10min time series. The cross correlation functions arenormalised with the standard deviations of wind speeds, so that maximum=1corresponds to identical, only time delayed time series at the two masts. The showncurves are the averages of the eight cross correlation functions.The comparison of the measured and simulated cross correlation functions in

Fig. 8 shows a good agreement between our simulations and the measurements, andalso reveals the problem with the assumption of frozen turbulence. It has a very highspike, because the turbulence at the two masts are assumed to be (almost) identical,only with a delay corresponding to the travel time from SMW to SMS.The rotor wind model is difficult to verify, because the equivalent wind speed is a

fictitious concept, which cannot be measured directly. Still, Rosas [18] comparedsimulations of the equivalent wind speed to measurements of the wind speedperformed by a pitot tube mounted on a rotating wind turbine blade. Thesemeasurements are described in Petersen and Madsen [23].The undisturbed wind speed measured at hub height on a meteorological mast in

front of the wind turbine was used to determine the mean wind speed and turbulenceintensity as input parameters to the simulations. The turbulence intensity was 0.16,whereas the mean wind speed was 10.5m/s.The pitot tube was mounted in 15m radius. The main parameters of the wind

turbine are a rotor diameter of 41m, a rotor speed of 30 rpm, a distance from therotor disc to the tower of 2.9m and a tower diameter of 1.7m. These values were alsoused as input to the simulations.The measured wind speed is seen from a single point on a rotating blade, whereas

the rotor wind model describes the summed effect of the wind speed on allthree blades. Therefore, the measured wind speed was processed to reflect theaverage wind speed from 3 blades before it was compared to the simulation. This wasdone by averaging the measured wind speed at times t�Dt, t and t+Dt, where Dt isthe time corresponding to 1201 rotation with 30 rpm.The comparison of the simulation with the processed measurements is shown in

Fig. 9. The comparison between the measured and simulated wind speeds showssignificantly higher measurement values than simulation values around the 3p

frequency (1.5Hz). The main reason for that difference is that the measurements aredone in a single radius, whereas the model intends to include the averaging along theblades.One exception from that is the spike on the simulated wind speed exactly at the 3p

frequency, which is probably due to overestimation of the tower shadow effects.Another significant difference between measurements and simulations in Fig. 9 is

the 6p on the measurements, which we do not simulate. The 6p could easily beincluded in the simulations, but the reason why we have not included is that it has


only little influence on the power output from the wind turbines, because the windturbine structures act as a low-pass filter.This statement can be confirmed by comparisons of measured and simulated

power from a wind turbine. Fig. 10 shows the PSDs of measured and simulatedpower on a 2 MW NM2000/72 NEG-Micon wind turbine.

10-2 10-1 100 101

100

102

104

106

Frequency [Hz]

Pow

er P

SD [

(kW

)2 /Hz]

12 March 2001 18:17

MeasuredSimulatedMeasuredSimulated

Fig. 10. PSDs of measured and simulated power of a 2MW NM2000/72 wind turbine.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Frequency [Hz]

PS

D [

(m/s

)2 /H

z]MeasuredSimulated

Fig. 9. PSDs of simulated equivalent wind speed and measured wind speed measured with a pitot tube on

a rotating blade.


Here we can also see the concentration of energy around the 3p frequency, whichis 0.9Hz in this case. The 6p; 9p and 12p frequencies can also be seen in themeasurements, but they are much weaker than the 3p:In Fig. 10, the reduction of the measured 6p compared to the 3p is much more

significant. Of course, it is different wind turbines and Fig. 10, but the difference alsoreflects the general aspect that the mechanical structure of the wind turbine low-passfilters the wind speed. Consequently, the higher p’s are generally more significant inthe wind speed than in the power.

6. Conclusion

Models for the wind speeds in wind farms have been developed. The modelsinclude the effects which are assumed to be important to predict the influence of thewind farm on the power quality as characterised in a new IEC standard for powerquality of wind turbines.The selected effects can be summarised as the park scale coherence between the

wind speeds at different wind turbines in the farm, and the effects of rotationalsampling in the rotor. Another way to classify the effects is as turbulence effects andtower shadow effects.Other effects like wind shear and yaw errors, which are important for the

structural loading of wind turbine blades, have not been included here. This isbecause the summation of the torque from the blades removes most of these effects.The models have been verified using a few sets of data, and the results are very

promising. The park scale model is able to simulate the delay of the coherent part ofthe wind speed between two points with longitudinal separation, and the coherencebetween the simulated wind speeds shows good agreement with the correspondingcoherence between measured wind speeds. Comparisons of the cross correlationfunctions have shown that the present model is more reliable than models assumingfrozen turbulence, because we include the decay in coherence for longitudinalseparation.The rotor wind model is more difficult to verify directly, because the eq-

uivalent wind speed generated by this model is a fictitious concept, which cannotbe measured directly. However, the equivalent wind speed has been comparedto wind speeds measured with a pitot tube mounted on a rotating wind turbineblade. The result of this comparison is reasonable up to approximately fourtimes the rotor speed, taking into account the difference between the (measured)single radius wind speed and the (simulated) radius weighted equivalent windspeed.The rotor wind model has also been verified indirectly by a comparison of the

PSDs of measured and simulated power. This comparison also shows goodagreement for frequencies up to four times the rotor speed, and it demonstrates thatthe structure of the wind turbine acts as a filter, which reduces the fluctuations athigher frequencies.


Acknowledgements

The Danish Energy Agency is acknowledged for the funding of this work incontract 1363/00-0003. Besides, a special thanks is given to our partners in theDanish project, Aalborg University and Dancontrol Engineering and North-WestSealand Energy Supply Company, NVE. Also thanks to the Danish transmissionsystem operators Eltra and Elkraft System, who have participated as members of anadvisory committee for the project. Finally thanks to SEAS Wind Energy Centre forfunding of maintenance of Vindeby measurements. Finally, Capes is acknowledgedfor funding the Brazilian Ph.D. project.

References

[1] P. S�rensen, A.D. Hansen, L. Janosi, B. Bak-Jensen, F. Blaabjerg, J. Bech, Simulation of wind farm

interaction with grid, 2001 European Wind Energy Conference and Exhibition, 2–6 July 2001,

Copenhagen, DK.

[2] IEC 61400-21, Ed.1: Wind turbine generator systems—Part 21: measurement and assessment of

power quality characteristics of grid connected wind turbines, Final Draft International Standard

88/144/FDIS International Electrotechnical Commission, IEC 2001-07-01.

[3] L. Kristensen, S. Frandsen, Model for power spectra of the blade of a wind turbine measured from

the moving frame of reference, J. Wind Eng. Ind. Aerodyn. 10 (1982).

[4] J. Tande, P. S�rensen, M. Galal, U. Naoman, E. Mansour, A. Hewehy, A. Maghawry, I. Darweesh,

Hurghada Wind Energy Technology Centre Demostration Wind Farm Studies, Power Quality

Assessment, Ris�-I-1141(EN), September 1997.[5] P. S�rensen, Methods for calculation of the flicker contribution from wind turbines Ris�-I-939, 1995,

Ris� National Laboratory, Roskilde, Denmark.

[6] N.O. Jensen, Anote on wind generator interaction, Ris� M 2411, November 1983.

[7] S. Frandsen, M.L. Th�gersen, Integrated fatigue loading for wind turbines in wind farms by

combining ambient turbulence and wakes, Wind Eng. 23 (6) (1999) 327–340.

[8] A.I. Estanqueiro, R.F. Aguiar, J.A.G. Saraiva, R.M.G. Castro, J.M.F.D. Jesus, On the effect of

utility grid characteristics on wind park power output fluctuations, Proceedings of British Wind

Energy Conference BWEA’15, York, 1993.

[9] M. Shinozuka, C.M. Jan, Digital simulations of random processes and its applications, J. Sound Vib.

25 (1) (1972) 111–128.

[10] J. Mann, Wind field simulation, Prob. Eng. Mech. 13 (4) (1998) 269–282.

[11] P. S�rensen, B. Bak-Jensen, J. Kristiansen, A.D. Hansen, L. Janosi, J. Bech, Power plant

characteristics of wind farms, in: Proceedings (on DC-ROM), Wind Power for the 21 Century:

EUWEC Special Topic Conference and Exhibition, 25–27 September 2000, Kassel, DE (WIP—

Renewable Energies, M .unchen, 2001) pp. 176–179.

[12] J.C. Kaimal, J.C. Wyngaard, Y. Izumi, O.R. Cot!e, Spectral characteristics of surface layer

turbulence, Q. J. R. Meteorol. Soc. 98 (1972) 563–598.

[13] J. H�jstrup, S.E. Larsen, P.H. Madsen, Power spectra of horizontal wind components in the neutral

atmospheric boundary layer, in: N.O. Jensen, L. Kristiansen, S.E. Larsen (Eds.), Ninth Symposium

of Turbulence and Diffusion, American Meteorology Society, 1990, pp. 305–308.

[14] Dansk Ingeni�rforenings og Ingeni�r-sammenslutningens norm for last og sikkerhed for

vindm�llekonstruktioner. 1. udgave maj 1992. Dansk Standard DS 472.

[15] A.G. Davenport, The spectrum of horizontal gustiness near the ground in high winds, Q. J. R.

Meteorol. Soc. 87 (1961) 194–211.

[16] W. Schlez, D. Infield, Horizontal, two point coherence for separations greater than the measurement

height, Boundary-Layer Meteor. 87 (1998) 459–480.


[17] W. Langreder, Models for Variable Speed Wind Turbines, M.Sc. Thesis, CREST Loughborough

University and Ris� National Laboratory, 1996.

[18] P.A.C. Rosas, P. S�rensen, H. Bindner, Fast wind modelling for wind turbines, in: Proceedings

(on CD-ROM), Wind power for the 21 Century: EUWEC Special Topic Conference and Exhibition,

25–27 September 2000, Kassel, DE, (WIP—Renewable Energies, M .unchen, 2001) pp. 184–187.

[19] P.H. Madsen, F. Rasmussen, Rotor loading on a three-bladed wind turbine, European Wind Energy

Conference EWEC’89, Glasgow, 1989.

[20] P. S�rensen, G.C. Larsen, C.J. Christensen, A complex frequency domain model of wind turbine

structures, J. Sol. Energy Eng. 117 (1995) 311–317.

[21] P. S�rensen, Frequency domain modelling of wind turbine structures. Ris�-R749(EN), 1994.[22] F.A. Ekelund, Hydrodynamik, Newtonske v!cskers mekanik. (Den private Ingeni�rfond). 322pp.

1968.

[23] J.T. Petersen, H.Aa. Madsen, Local inflow and dynamics—measured and simulated on a rotating

wind turbine blade, DRAFT, Ris�-R-993(EN), September 1997.


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