Wind Models for Simulation of Power Fluctuationsonwindfarms

7/29/2019 Wind Models for Simulation of Power Fluctuationsonwindfarms

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Journal of Wind Engineeringand Industrial Aerodynamics 90 (2002) 13811402

Wind models for simulation of power fluctuations

from wind farms

Poul Srensena,*, 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 of

the dynamic interaction between large wind farms and the power system to which

they are connected, and to support improvement of the electric design of wind

turbines as well as grid connections.

The work is mainly a result of a Danish project titled Simulation of wind powerplants, but also a Brazilian Ph.D. study titled Power quality and stability issues of

integration of large wind farms has contributed to the model development. This

Ph.D. study is accomplished in Denmark.

*Corresponding author.

0167-6105/02/$ - see front matterr 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


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As part of the Danish project, a dynamic model of a Danish wind farm has been

implemented in the dedicated power system simulation tool DIgSILENT, compris-ing models of grid, wind turbines and wind speeds. General descriptions of the

applied models have been given by Srensen et al. [1], whereas the present paper will

concentrate 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 energy

development is concentrated in areas with good wind resources, where the power

grids are often not so strong, wind energy is becoming a significant source of supply

to the power systems in these areas. As a consequence, wind energy is also playing an

increasingly 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 wind

turbines have been supplemented by a new IEC 61400-21 [2] standard for

measurement 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 of

wind turbines, and to reduce the costs for grid connection to what is required from

the 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 power

fluctuations during continuous operation of the wind farm. The present wind model

combines the stochastic effects caused by the turbulence and deterministic effects

caused 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 known

to move energy to multiples (often denoted ps, e.g. 3p) of the rotor speed from the

lower 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 the

dominating 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 power

output from the wind farm. In IEC 61400-21, it is specified that the maximum 200 ms

average power as well as the maximum 1 min average power must be measured as a

part 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 different

wind 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 200 m s power P0:2S of a wind farm

P. Srensen et al. / J. Wind Eng. Ind. Aerodyn. 90 (2002) 138114021382


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with N wind turbines

P0:2S XNi1

Pn;i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNi1

P0:2;i Pn;i2

r: 1

Assuming that the turbulence at different wind turbines is uncorrelated, a

corresponding relation can be made to predict the standard deviation of the power

from a wind farm, knowing the standard deviation of the power from each wind

turbine. However, analyses of measurements, e.g. Tande et al. [4], have shown that

the assumption of uncorrelation implies an underestimation of the standard

deviation 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 the

distance between the wind turbines is small. The present wind speed model includes

the 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 wind

turbine. The fast fluctuations are particularly important to assess the influence of

wind turbines on the flicker levels in the power system, i.e. the level of voltage

fluctuations causing flicker in the illumination from electric light bulbs.

In many cases, e.g. Srensen [5], measurements have shown that the 3p effect

due to rotational sampling provides the main contribution to flicker emission

from wind turbines to the grid during continuous operation of the wind

turbines.

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 the

wind 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 wind

model is the rotor wind model, which includes the influence of rotational sampling

and integration along the wind turbine blades as the blades rotate. The rotor wind

model 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 simplified

aerodynamic 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 be

independent on the operation of the wind farm, which is reasonable because the

required wind speed vhub;i for each wind turbine i is the wind speed in hub height if

wind 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 to

account for these effects. Jensen [6] suggested a wind farm model to predict the

reduction of the mean wind speed in a wake relative to the ambient mean wind speed.

Frandsen and Thgersen [7] suggested a model for combining the ambient

P. Srensen et al. / J. Wind Eng. Ind. Aerodyn. 90 (2002) 13811402 1383


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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 for

the wind field includes turbulence as well as tower shadow effects. The wind shear is

not included in the model, because it only has a small influence on the power

fluctuations 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 the

azimuth position yWTR;i of the wind turbine rotor. As illustrated in Fig. 1, yWTR;i is

fed 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. v

eq;iis a single

time 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 equivalent

wind speed is that it can be used together with a simple, Cp-based aerodynamic

model, and still include the effect of rotational sampling of the blades over the

rotor disk.

Pa

rkscalewindmodel

vhub,2 veq,2Rotor

wind

Wind

turbine 2

WTR,2

vhub,1 veq,1Rotor

wind

Wind

turbine 1

WTR,1

vhub,n veq,nRotor

wind

Wind

turbine n

WTR,N

Pa

rkscalewindmodel

vhub,2 veq,2Rotor

wind

Wind

turbine 2

WTR,2

vhub,1 veq,1Rotor

wind

Wind

turbine 1

WTR,1

vhub,n veq,nRotor

wind

Wind

turbine n

WTR,N

Fig. 1. Structure of wind model.



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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, Manns [10] simulation method was suggested used in this project, because

the calculation speed of that method is generally faster than cross spectral matrix

method. Some of the results using Manns model were presented in Srensen

et al. [11].

Both the Estanqueiro application of the Shinozuka method and the Mann method

assume Taylors frozen turbulence hypothesis illustrated for a two-dimensional wind

speed field in Fig. 2. The wind speed field is generated in spatial dimensions in the

first place, and then the field is moved forward with the mean wind speed.

Taylors frozen turbulence hypothesis is a reasonable assumption for simulationswhere 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 in

computer programs for simulation of mechanical loads on wind turbines to simulate

the wind speed variations in the rotor plane of a single wind turbine. In that case, the

simulated 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 Taylors frozen turbulence will generate wind speed time series

which are identical for all wind turbines in the line, apart from a delay corresponding

to 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 not

realistic. Another measure, which reveals the error introduced by assuming Taylors

V0V0

Fig. 2. Simulation of park scale wind speeds with the assumption of Taylors frozen turbulence

hypothesis.



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frozen turbulence is the cross correlation function. These aspects are discussed

further 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 Taylors frozen turbulence, we have introduced a new

method for simulation of park scale wind speeds. The new method is based on

Shinozukas cross spectral matrix method, and it uses a complex cross spectral matrix

as Shinozuka originally proposed instead of the real matrices that have been used in

frozen turbulence models for wind turbines. The cross spectral matrix becomes

complex as a result of the time delay between points with longitudinal components of

their 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 turbines

relative to the data grid. The complex cross spectral method directly generates a

single 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 the

fast Mann method.

The cross spectral method is based on the cross power spectrum matrix Sf;which with N points (corresponding to a wind farm with N wind turbines) is an

N N matrix. We have chosen to use the frequency in Hz, f. Each element Srcf in

row r; column c of Sf is determined as the cross power spectrum between theturbulence at wind turbine number r and c: Srcf is defined as the Fourier transformof the cross correlation function Rrct according to

Srcf ZN

N

Rrctej2pft dt: 2

The cross correlation function Rrct between vhub;rt and vhub;ct is defined as

Rrct Efvhub;rt vhub;ct tg; 3

where Efftg denotes the mean value of ft over the time t:The first step in the cross power spectral method is to determine Srcf: 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 yrcfrom north. The mean wind speed V0 and the wind direction yV are also shown.

arc 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 cosarc drc

V0: 4

To represent the time delay trc in the cross power spectrum, we assume that Rrct

as defined in Eq. (3) is symmetric about trc: Then using Eq. (2), it can be shown that



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the argument of the complex number Srcf is 2pftrc; i.e.

Srcf jSrcfj ej2pftrc ; 5

where j is the complex unity number. The size of the cross power spectrum jSrcfj

can be determined using the standard definition of the coherence function

g2f; d; V0;

g2f; drc; V0 jSrcfj

2

SrrfSccf: 6

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

Srcf gf; drc; V0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi

SrrfSccfp

ej2pftrc : 7

The second step is to discretise the frequency to be able to represent the spectra in

a numeric computer code. Simulating time series with the period length TP; thefrequency f is discretised in steps Df 1=TP:; i.e. the ith frequency fi iDf . Thecorresponding discrete value of Srcf is Srci Srcfi Df:

We also discretise the time by the sampled representation of the wind speeds as

time series with time steps Dt 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 TP fs 8

is the number of samples in the simulated time series. Obviously, Ns

must be an

integer, and preferably an exponent of 2 which enables the use of an FFT to speed up

the Fourier transformation used in the end of the method. This can be obtained by

adjusting 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

rc

V0

drc

V0rc

c

r

V

rcrc

V0

drc

V0rc

c

r

V

rc

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 y

rcfrom north. V

0is the mean wind speed, and y

Vis the wind direction. a

rcis

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



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preserved according to

s2r

ZN

N

Srrf df

E

Zfs=2fs=2

Srrf df

E

XNs=2iNs=2

Srri: 9

The discretisation is only done for frequency indices iX0; because the values forio0 are given by Srci Src i; where denotes complex conjugation.

The third step is for each frequency index iX0 to resolve the discrete matrix Si

with the elements Srci into a product of the transformation matrix Hi and the

transpose of its conjugateH

T

i; i.e.Si HiHTi: 10

Choosing the solution where Hi is a lower triangular matrix, i.e. the element

Hrci 0 if c > r; the elements can be found one by one. The diagonal elements aredetermined according to

Hrri

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSrri

Xr1k1

HrkiHrki

r11

and the elements below the diagonal are determined according to

Hrci Srci

Pc1k1 HrkiH

cki

Hcci: 12

It can be seen from Eqs. (11) and (12) that Hi gets the same phases as Si; i.e.zero phase shift in the diagonal and a phase shift 2pftrc 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 Ei of

unity complex numbers with a random phase. This is done by simulating N random

phase angles jri using a random generator with uniform distribution in the interval

0; 2p; and calculate each element Er in row r ofE according to

Er ejjri: 13

The fifth step is for each frequency index iX0 to calculate a vector Vhubi

containing the ith Fourier coefficients of all N wind speed time series according to

Vhubi HhubiEi: 14

The imaginary part ofVhub0 should be set to zero, because an imaginary part of

this frequency component does not make sense.

Finally, for each wind turbine r; the Fourier coefficients are joined in an arrayVhub;r; and an inverse Fourier transform is performed to obtain the time seriesvhub;rt:



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3.3. Spectral distributions

At the present state, the model is capable of simulating wind speeds with power

spectra of either Kaimal or Hjstrup type, but implementation with another spectrum

is 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 Hjstrup spectrum was selected because it includes more energy than the

Kaimal spectrum at the low frequencies, and this has shown to agree better in a

number of cases.

The two-sided Kaimal [12] spectrum SKaif is determined according to

SKaif u2

52:5z=V0

1 33z=V0f5=3

15

and the Hjstrup [13] spectrum SHojf according to

SHojf u2

2:5AHoj=V0

1 2:2AHoj=V0f5=3

52:5z=V0

1 33z=V0f 5=3

!

1

1 7:4z=AHoj:

16

In Eqs. (15) and (16), we have used the (hub)height z and the friction velocity uwhich can be determined by the roughness length z0 according to

lnz

z0

kV0

u; 17

where k 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 are

far above the lower boundary layer today. For large wind turbines, it is often

assumed that the length scale LE20z only increases with height up to z 30 m, 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 the

PARKWIND method. We have chosen to implement a Davenport [15] type

coherence, and use the decay factors recommended by Schlez and Infield [16] as

default 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 on

estimates 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

gf; drc; V0 earc drc=V0f; 18

where arc is the decay factor. Schlez and Infield uses a decay factor which depends on

the inflow angle arc shown in Fig. 3. The figure shows that arc 0 corresponds to



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points separated in the longitudinal direction, and arc 901 corresponds to points

separated in the lateral direction.With a given arc; the decay factor can be expressed according to

arc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffialong cos arc

2 alat sin arc2

q; 19

where along and alat are the decay factors for separations in the longitudinal and the

lateral directions, respectively. Using our definition of coherence decay factors in

Eqs. (18) and (19), the recommendation of Schlez and Infield can be rewritten as to

use the decay factors

along 1575 s

V0; 20

alat 17:575m=s1 s; 21

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 speed

veq; 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 equivalent

wind speed is derived from the rotor wind speed field.

Fig. 4 shows the three bladed wind turbine with the wind field vt; r; y: It is seen

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

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

Taet X3b1

Mbt: 22

Fig. 4 indicates that the blades are profiled from the inner radius r0 to the outer

radius R of the rotor disk. Linearising the blade root moment dependence on the

wind speed we obtain

Mbt MV0

ZRr0

crvt; r; yb V0 dr; 23

where MV0 is the steady state blade root moment corresponding to the mean wind

speed V0; and cr 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 cr to be proportional to r and r0 0:1R; which has also been assumed in



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the implemented model. However, for the sake of completeness, we will formally

keep cr and r0 here.

Inserting Eq. (23) in Eq. (22) we obtain

Taet 3MV0 X3b1

ZRr0

crvt; r; yb V0 dr: 24

Now we define the equivalent wind speed veqt as the single wind speed time series

which would give the same aerodynamic torque as the actual wind speed field, i.e.veqt must fulfil

Taet 3MV0 X3b1

ZRr0

crveqt V0 dr: 25

1

2

3

v(t,r,)

(R,1)

(r0,1)

(r,)

1

2

3

v(t,r,)

(R,1)

(r0,1)

(r,)

Fig. 4. The wind speed field in the rotor plane is given as vt; r; y; 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.



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Combining Eq. (25) with Eq. (24) gives the equivalent wind speed a the mean

value of contributions from all three blades:

veqt 1

3

X3b1

vct; yb; 26

where we have used the weighted wind speed vct; yb defined as

vct; yb

RRr0crvt; r; yb drRR

r0cr dr

: 27

Eqs. (26) and (27) express the equivalent wind speed as a weighting of all the wind

speeds 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 the

method. In this paper, it is combined with the park scale model.

The equivalent wind speed simulation method is based on Riss frequency domain

models [1921]. These models are based on expansion of the wind speed field in the

rotor 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.

vct; yb XN

kN

*vc;ktejkyb ; 28

where *vc;kt is the kth azimuth expansion coefficient ofvt; yb determined accordingto

*vc;kt 1

2p

Z2p0

vct; ybejny dy: 29

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

veqt XN

kN

*vc;3ktej3kyWTR ; 30

where yWTR y1 is the wind turbine rotor position obtained from the mechanical

model.

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 symmetric

structure of rotor, which causes the contributions from the other orders to even out

when they are summed up for all three blades. If a 1p and/or 2p are still significant in

a measurement of torque or power, this is often an indication that the blades are not

pitched with exactly the same angle.



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In the present implementation of the rotor wind model, we have only included the

0th and 3rd harmonics, i.e. Eq. (30) has been approximated toveqtE *vc;0t

2Ref*vc;3tgcos3yWTR

2Imf*vc;3tgsin3yWTR: 31

The idea of the rotor wind model is to simulate the azimuth expansion coefficients

*vc;kt in the first place as independent on the azimuth position of the rotor, and then

use Eq. (31) to generate the equivalent wind speed which includes the azimuth

dependence.

The azimuth expansion coefficients *vc;kt are sums of contributions from the

turbulence model, tower model and the mean wind speed. The mean wind speed

contributes with V0 to *vc;0t: 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 the

azimuth expansion coefficients *vc;k;turbt of the turbulence field. It has been shown

that [19,21] the power spectral density (PSD) S*vc;kf of *vc;k;turbt can be determined

according to

S*vc;kf F*vc;kf Svf; 32

where Svf is the PSD of the wind speed in a fixed point, and F*

vc;kf is denoted theadmittance function. F*vc;kf can be determined by a triple integral, which can be

resolved into the double integral

F*vc;kf

RRr0

RRr0cr1cr2Fkf; r1; r2 dr1 dr2

RR

r0cr dr2

33

and the single integral

Fkf; r1; r2 1

2p

Z2p0

glvf;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi

r21 r22 2r1r2 cosy

q cosny dy: 34

Here, glvf; d is the square root coherence function between two points with adistance d in the rotor plane. glvf; d is assumed to be the same horizontally (i.e.laterally) and vertically in the plane.

Eqs. (33) and (34) have been solved numerically by Srensen [21]. Using the

Laplace operator s jo j2pf; Langreder [17] defined the transfer functionsH*vc;kj2pf with the size defined as

jHc;kj2pfj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

F*vc;kfq

35



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and used the numerical results to fit H*vc;0 j2pf and H*vc;3 j2pf to linear filters.

Defining the constant d R=V0; the results of these fittings are

Hc;0sE0:99 4:79ds

1 7:35ds 7:68ds2; 36

Hc;3sE0:0307 0:277ds

1 1:77ds 0:369ds2: 37

Using vhubt from the park scale model as input and a linear filter with the transfer

function Hc;0j2pf; *vc;0;turbt is now simulated according to

*Vc;0;turbf Hc;0j2pf Vhubf; 38

where *Vc;0f and Vhubf are the Fourier transforms of *vc;0;turbt and vhubt;respectively.

*vc;3;turbt is a complex variable, and it was shown [19,21] that the real and

imaginary parts are uncorrelated with each other and with azimuth expansion

coefficients of other orders k: Distributing the variance between the real andimaginary parts of *vc;3;turbt evenly, they are determined by the relations between

Fourier transforms

Ref *Vc;3;turbfg 1ffiffiffi

2pHc;3j2pf V3;Ref; 39

Imf *Vc;3;turbfg 1ffiffiffi

2pHc;3j2pf V3;Imf; 40

where V3;Ref and V3;Imf are Fourier transforms of uncorrelated stochastic signals

with the same PSD as the wind speed in a fixed point.

To support the simulation of V3;Ref and V3;Imf; Langreder also fitted a filterwhich converts uniformly distributed white noise to a signal with the PSD as the

Kaimal spectrum.

4.3. Tower shadow model

Today most wind turbines are constructed with a rotor upwind of the tower to

reduce the tower interference of the wind flow. Early wind turbines often had lattice

towers, 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 V0a2 x2

y2

x2 y22; 41

where a is the tower radius, and x and y are the components of the distance from

each blade to the tower centre in the lateral and the longitudinal directions,

respectively.



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Rosas [18] used Eqs. (41), (27) and (29) to calculate the azimuth expansion

coefficients 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 sea

masts SMW and SMS on the Vindeby offshore site. The offshore site is shown in

Fig. 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 offshore

wind farm. The distance between the two sea masts SMS and SMW was

dSW 807 m, and the angle ySW 2971: The measurements used in this paper wereacquired in 1994.

Fig. 6 shows 1 h of 1 min block average values of wind speed measurements with

the mean wind speed V0 11 m/s and the mean wind direction yV 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.



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Both measurements and PARKWIND indicate a delay of SMS with a little more

than 1 min relative to SMW. This corresponds very well to the expected delay for

11 m/s with 807 m distance.

A more consistent verification is to compare the coherence functions. However,

this requires a substantial data set, because the coherence in 807 m 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 standard

deviation of the 234 10-min mean values of wind directions is 4.3 1, with maximum

wind 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 factor

for the coherence according to Eq. (19) for each of the six simulations. The

corresponding coherence (Davenport) is also indicated in the figure as a straight

line.

The coherence comparisons show very good agreement between measurements

and simulations, indicating that the Davenport type coherence and Schlez and

Infields decay factors are reasonable in this case.

Finally, the simulated and measured cross correlation functions are compared in

Fig. 8. 8 10 min with average inflow angles a 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]

windspeed[m/s]

Measured

0 10 20 30 40 50 608

10

12

14

time [min]

windspe

ed[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.



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band, but as it can be seen from Fig. 8, the estimated cross correlation functions are

reasonably smooth.

The exactly measured inflow angle has been used in the simulations, ensuring that

also distributed inflow angles in the interval a 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(coherence)

MeasuredSimulatedDavenport

Fig. 7. Square root coherence function of measured and simulated 6 6.5h 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

Normalisedcrosscorrelationfunction

MeasuredSimulatedSim-frozen

MeasuredSimulatedSim-frozen

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

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

Taylors frozen turbulence.



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simulations have been done for each of the time series, one with the present

PARKWIND model using the Schlez and Infield decay factors along and alat; and oneassuming Taylors frozen turbulence. The frozen turbulence is also simulated with

PARKWIND, using the longitudinal decay factor along 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 mean

wind speed in each of the 10 min time series. The cross correlation functions are

normalised with the standard deviations of wind speeds, so that maximum=1

corresponds to identical, only time delayed time series at the two masts. The shown

curves 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, and

also reveals the problem with the assumption of frozen turbulence. It has a very high

spike, 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] compared

simulations of the equivalent wind speed to measurements of the wind speed

performed by a pitot tube mounted on a rotating wind turbine blade. These

measurements 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 turbulence

intensity as input parameters to the simulations. The turbulence intensity was 0.16,

whereas the mean wind speed was 10.5 m/s.

The pitot tube was mounted in 15 m radius. The main parameters of the wind

turbine are a rotor diameter of 41 m, a rotor speed of 30 rpm, a distance from the

rotor disc to the tower of 2.9 m and a tower diameter of 1.7 m. These values were also

used 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 all

three blades. Therefore, the measured wind speed was processed to reflect the

average wind speed from 3 blades before it was compared to the simulation. This was

done by averaging the measured wind speed at times tDt, t and t+Dt, where Dt is

the 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 shows

significantly higher measurement values than simulation values around the 3p

frequency (1.5 Hz). The main reason for that difference is that the measurements are

done in a single radius, whereas the model intends to include the averaging along the

blades.

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 be

included in the simulations, but the reason why we have not included is that it has



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only little influence on the power output from the wind turbines, because the wind

turbine 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 simulated

power on a 2 MW NM2000/72 NEG-Micon wind turbine.

10-2 10-1 100 101

100

102

104

106

Frequency [Hz]

PowerPSD[(kW)2/Hz]

12 March 2001 18:17

MeasuredSimulatedMeasuredSimulated

Fig. 10. PSDs of measured and simulated power of a 2 MW 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]

PSD[(m/s)2/Hz]

MeasuredSimulated

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

a rotating blade.



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Here we can also see the concentration of energy around the 3p frequency, which

is 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 also

reflects the general aspect that the mechanical structure of the wind turbine low-pass

filters the wind speed. Consequently, the higher ps are generally more significant in

the wind speed than in the power.

6. Conclusion

Models for the wind speeds in wind farms have been developed. The models

include 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 power

quality 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 rotational

sampling in the rotor. Another way to classify the effects is as turbulence effects and

tower 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 is

because 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 of

the wind speed between two points with longitudinal separation, and the coherence

between the simulated wind speeds shows good agreement with the correspondingcoherence between measured wind speeds. Comparisons of the cross correlation

functions have shown that the present model is more reliable than models assuming

frozen turbulence, because we include the decay in coherence for longitudinal

separation.

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 cannot

be measured directly. However, the equivalent wind speed has been compared

to wind speeds measured with a pitot tube mounted on a rotating wind turbine

blade. The result of this comparison is reasonable up to approximately four

times the rotor speed, taking into account the difference between the (measured)

single radius wind speed and the (simulated) radius weighted equivalent wind

speed.

The rotor wind model has also been verified indirectly by a comparison of the

PSDs of measured and simulated power. This comparison also shows good

agreement for frequencies up to four times the rotor speed, and it demonstrates that

the structure of the wind turbine acts as a filter, which reduces the fluctuations at

higher frequencies.



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Acknowledgements

The Danish Energy Agency is acknowledged for the funding of this work in

contract 1363/00-0003. Besides, a special thanks is given to our partners in the

Danish project, Aalborg University and Dancontrol Engineering and North-West

Sealand Energy Supply Company, NVE. Also thanks to the Danish transmission

system operators Eltra and Elkraft System, who have participated as members of an

advisory committee for the project. Finally thanks to SEAS Wind Energy Centre for

funding of maintenance of Vindeby measurements. Finally, Capes is acknowledged

for funding the Brazilian Ph.D. project.

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