Gasch R., Twele J. (Eds.) Wind Power Plants: Fundamentals, Design, Construction and Operation

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4.5 Prediction of the wind regime

162

Fig. 4-43 Principle of conversion of the wind statistics between different locations according to

WAsP [5]

One of the biggest problems of WAsP is that the large-scale dynamic effects in

mountainous terrain are not considered and implemented up to now. These effects

are among others the dependency of the vertical wind speed profile on the free

convection. The heating and cooling of the ground leads to different lift forces

which influence the dynamics of turbulence, cf. section 4.1.3. If on one hand the

ground is cooling down at night (stable stratification) turbulence decreases, and

therefore the wind gradient increases. On the other, the heating of the ground dur-

ing the day leads to an increased turbulence and mixing of the air masses (unstable

stratification), so the wind speed gradient decreases. In order to take account for

this effect the model in WAsP is based on a simplified input of the climatologic

mean and the standard deviation. If the annual cycle at the site differs from the

basic assumptions of WAsP there may be a false estimation of the wind profile.

4 The wind

163

One way to quantify the complexity of a site is the ruggedness index (RIX) which

describes the percentage of the area around a site which exceeds a critical inclina-

tion of e.g. 17°. Thus, the RIX serves for a rough estimation of the degree of flow

separation and therefore to which extent the assumptions of the linearised model

are violated.

-10

-20

-30

-40

Wind speed error in %

-30 -20 -10 0 10 20 30

'RIX in %

-10

-20

-30

-40

Wind speed error in %

-30 -20 -10 0 10 20 30

'RIX in %

Fig. 4-44 Prediction errors of WAsP versus difference between 'RIX of the investigated site

and the reference site [6]

Comparing the RIX at the planning site (chosen wind turbine) with the RIX of the

reference site (measuring mast), the relative difference ǻRIX may serve as an

indicator for the prediction error of WAsP [6]. Fig. 4-44 illustrates the strong

influence of flow separation on the prediction error: If the complexity of the

terrain at the planning site and the reference site is quite similar (ǻRIX = 0), the

prediction error is small. If the planning site is located in a more complex terrain

than the reference site (ǻRIX > 0), WAsP tends to over-estimate the wind speed.

On the contrary, if the planning site is less complex than the reference site

(ǻRIX < 0), WAsP tends to under-estimate the wind speed.

The advantages and disadvantages of WAsP are briefly summarized as follows:

Advantages Disadvantages

x Simple operation x Valid only for a defined atmospheric

stratification close to neutrally stable

atmosphere

x Cost effective and fast

x Problems with turbulent flow separa-

tion in complex terrain

x Validated method, ap-

plication limits are

known

x Problems with the wind speed calcu-

lation outside the Prandtl layer (sur-

face boundary layer)

4.5 Prediction of the wind regime

164

4.5.2 Meso-Scale models

Meso-scale models describe the air flow with a spatial scale of up to 300 km and a

time scale of the fluctuations ranging from hours (storms) to several days (passage

of cyclones), cf. Fig. 4-13. Meso-scale models are applied to generate maps of the

wind potential for large areas. The used systems of equations describe the fluid

mechanics (momentum, mass flow, energy flow) among other based on the

Navier-Stokes equations of motion. Different models vary strongly in their

complexity.

Input data of the systems of equations are long-time data series for the wind

speed and direction, the geostrophic wind as well as geometric data of the terrain

and its roughness. Further parameters influencing the calculations are the tempera-

ture profiles, the description of turbulence and also the structure and density of the

generated simulation grid.

While in the Northern latitudes the surface wind is generally dominated by the

geostrophic wind, the influence of local thermal wind systems increases when

approaching the equator. The treatment of thermal effects in meso-scale models is

generally possible but requires far more complex input data, e.g. the albedo rate of

the surface, which increases rapidly and heavily the computational efforts.

Since an adequately high spatial and temporal resolution requires high compu-

tational efforts, it is useful to combine meso-scale models with WAsP. A wind

atlas is generated from the meso-scale calculations which are then the basis for the

very detailed calculations with WAsP. But it is by all means recommendable to

validate the simulation results with wind measurements at the site.

4.5.3 Measure-Correlate-Predict-Methode

For most planning sites wind measurements seldom exist for a sufficiently long

period (minimum 20 years), on one hand. On the other, the wind regime at the site

should be predicted as exact as possible for the expected operating period of 20

years because of the strong variations of the mean annual wind speed from year to

year (cf. Fig. 4-17).

If a long-term time series of wind speed and direction is available for at least

one reference site in the region of the planning site (minimum measuring period

20 years) then there is the possibility to generate a long-term wind prediction by

correlation between the reference site and the planning site.

The purely statistical method Measure-Correlate-Predict (MCP) is based on

the assumption that there is a linear correlation between simultaneous measure-

ments at the planning site and the reference site. Using, e.g., the hourly mean

values of the wind speed v

at the reference site as x-coordinate and the simulta-

neous ones v

from the planning site as y-coordinate, these pairs of values may be

4 The wind

165

drawn in a Cartesian coordinate system. Under the assumption of a linear relation

it is then allowed to draw a regression line through the points. The gradient of the

line is a measure for the relation of wind speed v

at the planning site to the wind

speed of the reference site v

. Calculating the correlation between the measuring

data from reference site and planning site allows estimating statistically the rela-

tion between the wind regimes at the considered sites using the correlation coeffi-

cient R². It gives the linear correlation between the simultaneously recorded data

as well as the scattering (variance). From the obtained averages and standard

deviations of the sample it is possible to determine e.g. the Weibull distribution

function, cf. section 4.2.2. If the correlation is sufficiently high, i.e. R

! 0.70, it is

allowed to transfer the factors of the wind speed distribution at the reference site

to the planning site.

Wind farm planning is in general based on the sectorised wind data, with usu-

ally 12 sectors, cf. Fig. 4-22. It follows that also the correlation between reference

site and planning site has to be performed sector by sector: The distributions

factors obtained for one individual wind direction sector of the wind rose at the

reference site are transferred by the correlation to the corresponding sector of the

planning site to achieve its individual distribution factors. In course of this proce-

dure, many problems arise: If in the considered region a certain wind direction

dominates the wind regime of the year or season there will be some sectors of the

wind rose where there are less individual events than required for a correlation,

meaning that the transformation from the reference site to the planning site is not

possible. Moreover there may be a time delay between the reference and the plan-

ning site for the occurrences of distinct meteorological events, e.g. weather fronts.

This leads to simultaneously recorded pairs of data which are not correlated. Due

to the time delay it is moreover possible that in course of the passage of weather

fronts there is a shift of the prevailing wind direction between the sites, and con-

sequently the wind speeds of the meteorological event are located at the reference

and planning site in different wind rose sectors.

All these problematic issues which reduce the quality of the correlation are

reduced by choosing the sample length as long as possible, i.e. minimum one-year

data sets. There are different ways for trying by methodological extensions to

circumvent or reduce the effects of the mentioned deficits which is discussed

extensively in the corresponding literature, e.g. [28].

Since the presented MCP method is a purely formal-statistical procedure, the

physical conditions like orography, plant cover or building are not considered.

For the reliability of the correlation investigation it is important to choose

either a suitable sample length to represent the annual wind cycle, or at best the

correlation of an entire annual wind record of the reference and planning site.

The disadvantage of the method is that the calculated wind speed is valid only

for the individual measuring height exactly at position of the measuring mast. A

transformation of the long-term corrected wind regime to other points than the one

measured (in measuring height at the mast position) is only possible by physical

models which consider the effects of the local site conditions on the flow.

4.5 Prediction of the wind regime

166

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1320(EN), 2002, Risø National Laboratory, Roskilde Denmark

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[16] Stull R.B.: An Introduction to Boundary Layer Meteorology, 1988, Kluwer Acad.

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Wind Turbine Power Performance Verification in Complex Terrain and Wind Farms,

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trägen (Technical Guidelines for Wind Energy Converters, Part2: Determining the

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[19] IEC 61400-1, Ed.3: Wind Turbine Generator Systems – Part 1: Safety Requirements,

2005

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(Guideline for the Certification of Wind Turbines), Hamburg 2003

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[22] Kaiser K., Langreder W.:

Site Specific Wind Parameter and their Effect on Mechani-

cal Loads, Proceedings EWEC, Copenhagen, 2001

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4 The wind

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[29] Thomsen K., Madsen H.A.: A new simulation method for turbines in wake – applied to

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R. Gasch and J. Twele (eds.), Wind Power Plants: Fundamentals, Design, Construction 168

and Operation, DOI 10.1007/978-3-642-22938-1_5, © Springer-Verlag Berlin Heidelberg 2012

5 Blade geometry according to Betz and

Schmitz

With the help of the Betz or the Schmitz (Glauert) theory [1, 2, 7], designing a

wind turbine is relatively straightforward. These theories provide the blade chord

and the blade twist relative to the radius, after the design tip speed ratio, the aero-

dynamic profile and the angle of attack or the lift coefficient have been specified.

The Betz theory only takes into account the axial downstream losses. Schmitz,

additionally, takes into account the downstream swirl losses, due to the rotational

wake. For rotors with a low tip speed ratio (design tip speed ratio of Ȝ

< 2.5), this

results in a blade geometry significantly different from that determined by the

Betz theory. Profile losses and losses resulting from the air flow around the blade

tips are ignored in both theories. They have to be accounted for by an additional

reduction in the turbine’s power. The number of blades is not an issue as this only

affects the tip losses.

5.1 How much power can be extracted from the wind?

In accordance with section 2.3.1 the kinetic energy of a mass in motion is

vmE . (5.1)

The power in the wind flowing through a control area A can be calculated as

wind

vAvmEP



 (5.2)

since the mass flow rate is m



A dx/dt =

A v

, see Fig. 5-1.

As already illustrated in chapter 2, the wind’s power cannot be extracted com-

pletely. Betz [1] analysed a heavily idealised wind turbine that extracts the power

from the wind at an ‘active plane’ by retarding its speed, without any losses. The

physical process that extracts the kinetic energy from the air flow is not yet con-

sidered at this stage.

As shown, in Fig. 5.2, Betz assumed a homogeneous air flow v

that is retarded

by the turbine to the velocity v

far downstream. That means that, for the reasons

of continuity, he assumes a stream tube with divergent streamlines

U

 v

A =

v

3 .

(5.3)

5 Blade geometry according to Betz and Schmitz

169

Fig. 5-1 Air flow in a stream tube of velocity v through a control area A (from chapter 2)

Fig. 5-2 Air flow through an ideal Betz wind turbine (from chapter 2)

As the changes in the air pressure are minimal, the air density

can be assumed to

be constant.

The extracted kinetic energy E

is upstream energy minus downstream energy





1ex

vmE . (5.4)

The power extracted from the wind is therefore





1ex

vmE



. (5.5)

If the wind were not retarded at all (v

= v

), no power would be extracted. If the

wind is retarded too much, the mass flow rate m



becomes very small. Taken to the

5.1 How much power can be extracted from the wind?

170

extreme ( m



= 0), this would lead to a ‘congestion’ of the stream tube (v

= 0)

such that once again no power can be extracted. There must be a value between v

= v

and v

= 0 for maximum power extraction. This value can be determined if

the velocity v

at the rotor plane is known. The mass flow rate will then be



A v

. (5.6)

At this point, the plausible assumption



(5.7)

is made, which will be proved later on (Froude-Rankine Theorem). If the mass

flow rate of equation (5.6) and the rotor plane velocity v

of equation (5.7) are

inserted into the power-extraction equation (5.5), we obtain





vAE



(5.8)

Power in the wind

Power coefficient c

Thus, the power in the wind is multiplied by a factor c

which depends on the ratio

/ v

The maximum power coefficient is

P, Betz

= 0.59 . (5.9)

It occurs when the wind velocity v

(upstream of the rotor) is retarded to v

= (1/3)

60% of the power in the wind is extractable by an ideal wind turbine! In this case,

the velocity in the rotor plane is 2v

/3 and far downstream v

/3.

The diagram, in Fig. 5.4, illustrates how much power can be extracted in Betz’s

ideal case of c

P, Betz

= 0.59. Of course, it depends on the wind velocity and the

diameter of the wind turbine, Due to additional losses which will be analysed later

on, the actual power of modern wind turbines is somewhat smaller. However,

values up to c

§ 0.50 are being achieved by some modern wind turbines.

(downstream). This can be found either by drawing the curve, Fig. 5-3, or,

mathematically, by setting the first derivation of equation (5.8) to zero. Nearly

5 Blade geometry according to Betz and Schmitz

171

0.6

0.5

0.4

0.3

0.2

0.1

1.00.80.60.40.20.0

0.0

0.5 1/3 1/4 a 0.0

P.max

= 16/27

0.6

0.5

0.4

0.3

0.2

0.1

1.00.80.60.40.20.0

0.0

0.5 1/3 1/4 a 0.0

P.max

= 16/27

Fig. 5-3 Power coefficient c

versus the ratio of wind velocity v

far downstream and the up-

stream wind velocity v

; c

P,max

= 0,593 at v

= 1/3, additional axis: induction factor a

Now, using the principle of linear momentum, we can determine the thrust acting

on the wind turbine at the rotor plane when maximum power is extracted. The

thrust equation

T =







vvm

(



= 

A 



(

(5.10)

yields for the given ratio

/ v

= 1/3

T = c





A ; c

= 0.89 . (5.11)

The term in brackets is the dynamic pressure on the area A. Comparing this result

with the drag D of a circular disk in the wind flow

D = c





A ; c

= 1.11, (5.12)

it is found that in the case of optimum power extraction the thrust is nearly 20%

lower than that of a circular disk.