# Introduction

Rules of thumb for the selection of a site for a wind turbine or a wind park contain simplified variations of the basic equations of wind power and wind energy production. As professional wind measurement is expensive and needs a long period of time to gain reliable data, especially for the siting of small wind turbines it could be sufficient to use available data and to estimate wind speeds in higher altitudes and related energy outputs by rules of thumb.

# Estimating Annual Wind Energy Output

The forecasting of annual energy generated by a single wind turbine or a wind park at a specific site is a very complex task requiring very much work in advance for wind measurement and the following site evaluation. However, some basic relationships like the increase of wind power with wind speed, the increase in wind speed with height, the dependency of wind power towards air density and general assumptions about the wind speed distribution can be used to create a rough estimation of the annual wind energy output.

## The Relationship of Wind Speed to Wind Power

The power $P$ of a wind stream with the speed $v$ is given by

$P=1/2\cdot \rho \cdot A\cdot V^{3}$ where $\rho$ is the density of air and A is the swept area by the rotor of the wind turbine. Using air density at sea level $\rho$ =1,225 kg/m3 the relationship can be expressed as $P=0,6125AV^{3}$ . The power in the wind stream is influenced by the wind speed with a cubic exponent. This means even a small increase in wind speed substantially raises the power in the wind and stresses the need for a good estimation of wind speeds choosing a new site for a turbine. The increase in power caused by a certain increase in wind speed can easily beestimated: Leaving the influence of air density and rotor area apart, assume a site with a wind speed $v_{1}$ of 10 m/s and another site with an average wind speed $v_{2}$ of 12 m/s. The difference in wind speed is only 20 % or $v_{2}/v_{1}=12/10=1.2$ . Nevertheless this relatively small increase in wind speed results in bigger increase of wind power:

${\frac {P_{2}}{P_{1}}}=({\frac {V_{2}}{V_{1}}})^{3}$ $P_{2}=1,2^{3}\cdot P_{1}=1,73\cdot P_{1}$ The wind power at the site with an average wind speed of 12 m/s is almost 70 % higher than the wind power at the other site.

## The Increase in Wind Speed with Height

Near the ground the wind speed is influenced by any obstacle, because the flow of the wind is disrupted and therefore slowed down. At sites with very rough terrain often a dramatically increase in wind speed with height can be observed. For that reason wind speed data for a specific site should always contain information about the height the wind speed was measured. If nothing is stated concerning the height of measurement, usually a height of 10 meters can be assumed. The standard height set in the German Renewable Energies Law within the definition of a reference site is 30 m above the ground.

A rough but conservative and easy method of wind speed estimation in higher altitudes is the power law method using the following simple equation

${\frac {V}{V_{0}}}=({\frac {H}{H_{0}}})^{\alpha }$ where $V_{0}$ is the wind speed at the measured height $H_{0}$ , $V$ is the wind speed at the 'wanted' height $H$ and $\alpha$ is parameter characterizing the roughness of the terrain. The appropriate values for the exponent $\alpha$ have been set by experience and a couple of typical roughness exponents are shown in table 1.

Table 1: Typical roughness exponents

Surface roughness exponent $\alpha$ Water or ice
0,1
Low Grass or Steppe
0,14
Rural with obstacles
0,2
Suburb and Woodlands
0,25

Thus an estimate of wind speed $V$ in the height $H$ can be generated by

$V=({\frac {H}{H_{0}}})^{\alpha }\cdot V_{0}$ As an example, a typical farmland in a rural area with a couple of obstacles is considered: According to Table 1 a value of 0,2 can be chosen for $\alpha$ . If a wind speed of 5 m/s at a height of 2 meters is measured, how great will the wind speed be at a typical small wind turbine height of 30 m? The answer is given by

$V=({\frac {30m}{10m}})^{0,2}\cdot 5m/s\approx 6,22m/s$ .

Thus in a rural area the wind speed increases about 25% between the height of 10 to 30 meters. At an altitude of 60 m the wind speed is approximately 43% higher than at 10 meters height.

## The Increase of Wind Power with Height

Combining these first rules of thumb it can easily be understood why wind turbines are typically placed on high towers. Because of the cubic relation between wind speed and wind power the increase in wind speed with height has an very important impact on the increase of wind power with height. Using the previous example the wind speed increases by 25% between the heights of 10 m and 30 m. We compare the power at the two wind speeds using the first rule of thumb:

$P=({\frac {6,38m/s}{5m/s}})^{3}\cdot P_{0}=2,08\cdot P_{0}\qquad$ The power contained in the wind stream more than doubles with an increase in height of 20 m.

## The Dependency of Wind Power on Air Density

As the initially mentioned equation for wind power shows, the air density $\rho$ has an influence on the wind power. Air density changes with temperature, thus comparing two sites with equal average wind speeds, wind power will be higher at the site with lower temperatures. Generally wind turbines generate more power in the winter than in the summer months.

Air density decreases with elevation: The less dense air on a mountain top can cut power production by 10-20 % compared to sea level conditions. As a conclusion the higher the considered site is located above sea level and the higher are the average temperatures at the site, the smaller will be the energy which can be generated by a wind turbine installed at this site. Compared to the increases in wind speed with height, changes in air density have a relatively small impact on the wind energy generation.

## The Dependency of Wind Power Generation on the Swept Rotor Area

Until now only variables influencing the wind speed and the power contained in a wind stream have been described. The area $A$ swept by a wind turbine influences the wind power generation in a linear way. Doubling the swept rotor area means doubling the power available in the wind stream covered by the rotor. As the rotor area $A$ is given by $A=\pi R^{2}$ a small increase in the radius $R$ of the rotor has significant impact on the power which can be generated by the turbine. As an example increasing the length of the rotor blades from 10 to 12 meters results in an increase of the swept area by 44%:

$A_{2}=(12/10)^{2}\cdot A_{1}=1,44\cdot A_{1}$ Caused by the linear relation of wind power and swept rotor area, the wind power increases by 44%, too. The important effect of small increases in rotor blade length on wind power generation makes the rotor size a central characteristic of a wind turbine. Unlike the rated power value of a wind turbine, which depends on the related wind speed and other conditions, the size of the rotor is a clear indicator for the power that can be generated by a specific wind turbine.

#### The Power Coefficient $c_{p}$ of Wind Power Production

Considering the previous paragraphs all variables necessary for estimating the wind power contained in a wind flow with a certain speed have been explained and the influence of the wind turbine characteristics (height of the tower, rotor area) have been introduced.

The power that can be used by a wind turbine is limited by Betz Law, which should not be explained in detail here. As principle idea the power contained in a wind stream can not be extracted completely by a wind turbine, because for this purpose the wind stream would have to be slowed down to a wind speed of zero. As a result the air masses would not move out behind the turbine and there would be no space for the air masses following. The maximum efficiency of a wind turbine found by Betz is a value of 59%. This efficiency is used as the power coefficient $c_{p}$ in the function for power generated by a wind turbine:

$P={\frac {1}{2}}\cdot \rho \cdot A\cdot V^{3}\cdot c_{p}$ The value of $c_{p}=0,59$ is a theoretical limit, which is not reached under real conditions even by modern wind turbines. Under good site conditions these new and highly refined wind turbines have a $c_{p}\approx 0,5$ .Small wind turbines in comparison seldom reach a $c_{p}>0,3$ . The majority of medium and bigger sized wind turbines range between these values depending on site conditions.

## The Importance of Wind Speed Distribution for Wind Energy Production

Since our objective is estimating the wind energy output of a wind turbine during a hole year the average wind speed does not contain sufficient information even for a rough estimation: Owing to the exponential relationship between wind speed and wind power, periods of time with high wind speeds have a far greater influence on the energy produced than periods of time with a lower wind speed. Thus it will be of central importance, if an average wind speed of 6 m/s consists of a few month with very high winds or is reached by continuously blowing low velocity wind streams. The distribution of wind speeds can be described by mathematical approximations used by meteorologists. For temperate climates the Rayleigh Distribution is a good approximations. For the purpose of the estimation of wind energy production, the Energy pattern factor or Cube factor of these distributions are important. Including the cube factor allows considering the additional information about the shares of high wind speeds in the estimation function. A Rayleigh Distribution typically has a cube factor of 1,9.

## The Swept Area Method for Estimating Annual Wind Energy Production

Having explained the basic rules of thumb the estimation of annual wind energy production (AWP) is just combining all variables in one function:

$AWP={\frac {1}{2}}\cdot \rho \cdot A\cdot V^{3}\cdot Cubefactor(1,9)\cdot c_{p}\cdot 8760h/yr\cdot 1kW$ For the sake of simplicity one variable missing in this equation often related to as the swept area method: The so-called Power curve of a specific wind turbine is a function of the power produced by the turbine at different wind speeds. But similar to the cube factor summarizing the effect of a Rayleigh wind distribution on power production, the effect of the wind turbine efficiency at a specific wind speed can be included in a simplified manner assuming a linear relation between wind speed and power production.

# Site Selection

## Roughness

Describing the characteristics of a terrain the roughness is mostly stated as a roughness class or roughness length. This parameter describes the height above the ground for which wind speed is zero. Roughness is influenced by the ground structure and by the size and number of obstacles in the landscape. The most important planning information is the roughness of the surface upwind in the prevailing wind direction and directly below the wind turbines.

Even long grass increases the roughness value significantly, thus it is sometimes stated that 'sheep and wind turbines are best friends'. Bushes, shrubs, trees and especially buildings have a much greater influence on terrain roughness, because besides friction effects between wind stream and ground surface, obstacles like rock formations or buildings cause turbulence. The turbulent zone caused by an obstacle sometimes extends the height of an obstacle by three times. The slow down of the wind speed behind an obstacle increases with the length, the height and a parameter named porosity of the obstacle. The term porosity refers to the degree the wind can pass through the obstacle: A tree in winter time for instance, has a high porosity, while a rock formation has a porosity of zero.

While for the planning process of a wind park a detailed description of landscape parameters is needed, for the installation of small wind turbines this may most often be a too big financial effort. For this purpose 30-feet rule of thumb can be used:

To avoid effects of turbulence caused by obstacles a wind turbine should be installed at least 10 meters above any obstacle within a distance of 100 meter.

A very cheap and simple, but clarifying tool to get a first hint about the character of a wind stream at a site for as small wind turbine is a kite. Threads can be tied to the cord of a kite every few meters. The behaviour of the threads while flying the kite gives a clue to develop an idea of the turbulence in different heights at the site. Certainly this could not be the basis for an greater investment, but it offers a possibility to gain first information about the quality of a wind site without any significant financial effort.

## The Hill Effect

Placing a wind turbine on a hill combines the positive effect of reducing influence of surrounding obstacles with some dynamic advantages of wind stream passing hills with low or moderate slopes: At the upwind side of a hill, the wind stream is compressed, while the pressure behind the hill is generally lowered. If the slope of a hill is not to steep, the compressed air can flow to the top of the hill where it expands and is accelerated at the same time. This hill effect causes additionally benefits for the wind turbine placed on the top of the hill.

In contrast to this positive effect escarpments can cause intense turbulence and require the utilization of higher towers to avoid the negative effects of the disturbed wind stream on wind energy production.

## The Park Effect

In most cases wind turbines themselves are by far the biggest obstacles in the surrounding of other wind turbines. If a greater number of wind turbines should be installed, the placement of the machines will always be guided by the objective of enlarging the distance between the turbines as much as possible. As the distance is always limited by the available area for the wind park, today wind turbines are commonly placed in a distance of 5 to 9 rotor diameters in the prevailing wind direction and 3-5 rotor diameters in the direction perpendicular to this.