The Physics of Energy, page 116
Figure 28.16 Wind frequency distributions from two Austrian Alpine stations. After [182]. Fits to Weibull distributions are shown, as are the fit values of the Weibull parameters and k. (a) Aineck station; (b) Großeck Station. The relatively small values of k reflect the relatively high variability of the winds at these mountain locations. (Credit: http://windharvest.meteotest.ch)
Systematic wind speed measurements form the basis of an evaluation of a site for wind power production. The wind speed measurements can be binned in convenient intervals and compiled into a histogram like the one shown in Figure 28.15. For effective wind power evaluation at a given location, measurements are typically taken over several years. With sufficient data, a histogram such as Figure 28.15 averages out to a smooth curve. If the distribution is rescaled so that the area under it is one, then the result is the wind frequency distribution for the site. For each value of the speed, v, the function measures the fraction of the time that the wind speed lies between v and .
The wind frequency distribution can be interpreted as a probability distribution
(28.7)
Assuming that other factors – e.g. climate, land use, vegetation – do not change, then gives the probability that at the site in question at a random time, the wind speed will be observed to be between v and . The integral of f from to v gives the probability that the wind speed is less than v,
(28.8)
which is known as the cumulative function of the probability distribution.
The average wind speed and mean cube wind speed are given by moments of the wind frequency distribution,
(28.9)
Examples of wind frequency distributions for two mountain locations in Austria are shown in Figure 28.16.
28.2.2 Modeling Wind Speed Variations – Weibull Distributions
It is often desirable to find a simpler characterization than the complete set of data in the histogram giving the function . This is generally done by using a small set of numerical parameters that capture the important features of the wind speed distribution at a given location. This makes it easier to compare the wind power potential at different sites and to estimate the power that a specific turbine design can harvest at a particular site. For many types of probability distributions, there are simple functions that capture the important features of the distribution in the limit of a large number of samples. Examples include the Poisson distribution, introduced in Box 15.1, which describes radioactive decays, and the Gaussian distribution , which characterizes many probability distributions.
Experience has shown that wind frequency distributions can usually be fit quite well by a distribution function known as the Weibull distribution. The Weibull distribution is a simple probability distribution, normalized to one in the range , which depends on two parameters, a scale parameter , and a shape parameter k
(28.10)
The parameter sets the scale of the wind speed: a larger value of indicates a windier location. k, on the other hand, determines the variability of the wind: the smaller the value of k, the more variable the wind. The need to describe both the scale and variability of the wind explains why it is necessary to have two parameters; a one-parameter distribution will not suffice. Sometimes, though, when only a crude approximation is needed, or when data are not sufficient to fit both parameters, the parameter k is fixed to . In this case the Weibull distribution reduces to the Rayleigh distribution, which is closely related to the Gaussian distribution (Problem 28.8). The choice implies strong assumptions about the way that wind varies at a given site, and in many cases does not give a good description of the distribution of wind measurements. Weibull functions that fit empirical data on wind frequency distributions are shown in Figure 28.16.
Figure 28.17 Examples of Weibull distributions. (a) Distributions with the same mean wind speed m/s and varying shape parameter k. (b) Distributions with the same shape parameter and various values of the scale parameter (in m/s).
One reason that Weibull distributions are convenient is that can be written as a derivative of a simple function, as in eq. (28.10). Therefore the cumulative function takes a particularly simple form,
(28.11)
Note that as , so that the Weibull distribution is properly normalized. Examples of Weibull distributions are shown in Figure 28.17. Figure 28.17(a) shows the variation with k of Weibull distributions all of which share the same value of . It is clear that small k indeed corresponds to greater variability and implies a longer tail out to large values of v. Figure 28.17(b) shows distributions with the same value of k and different values of .
Weibull Distributions
Wind speed distributions at any given location are generally modeled well by the Weibull distribution
The parameter characterizes the scale of typical wind speeds, while k characterizes the variability at the specific site (smaller k implies greater variability)
Another advantage of the Weibull distribution is that the average and root mean cube wind speed can be expressed simply in terms of the special function known as the Gamma function (Appendix B.4.3). The mth moment of the Weibull distribution is given by (Problem 28.10),
(28.12)
Note that the mth moment is proportional to , in accord with the statement that sets the scale of the wind distribution.
Both and can be expressed in terms of Gamma functions:
(28.13)
The ratios , and
(28.14)
are graphed as functions of k in Figure 28.18. The significance of the shape parameter in determining the power available in the wind is most clearly visible in the ratio plotted in Figure 28.18(b). measures the power available in the Weibull distribution compared to a steady wind with the same average wind speed. The relative importance of the average wind speed and of its variability are easily estimated with the use of the Weibull distribution parameters. An example is given in Example 28.2.
Figure 28.18 Average and root mean cube wind speeds, both scaled by the Weibull scale parameter , and the ratio , all as functions of the Weibull shape parameter k.
Example 28.2 Comparing Wind Power at Two Austrian Sites
It is clear from a glance at Figure 28.16 that Aineck Station (A) in Austria is a better location for wind powerthan Großeck Station (G). How much better? The Weibull distribution parameters – given in the figure – enableus to answer this quantitatively.
The average wind speeds at the two locations are obtained from eq. (28.13),
If the power in the wind was proportional to the cube of the average wind speed, Aineck Station would have times as much power available. The average cubes of the wind speeds, however, are
and the actual ratio of powers available is . Thus, as expected, Aineck Station is a better wind power site, but the relatively frequent occurrence of wind speeds greater than the average, indicated by the small value of the Weibull parameter k, makes the Großeck site a somewhat better location than the average wind speed alone would suggest.
28.2.3 Wind Direction Distributions
Another feature that is important in siting and designing a wind farm is the direction of the prevailing winds, or, more precisely, the weighted distribution of wind directions. As described below, wind turbines have significant downstream shadows. An array of wind turbines cannot be arranged so that the turbines each avoid one another’s shadows independent of the wind direction, so finding the optimal layout for a given directional distribution of wind is an important consideration. The correlation between the direction and intensity of the wind is important: if the winds blow 70% of the time gently from the southwest, but 30% of the time strongly from the northwest, then despite the prevalence of southwest winds, the wind farm should be aligned to best capture winds from the northwest.
Figure 28.19 Wind rose for Klamath Lake, Oregon from July 2005 through September 2015. The wind speeds are measured at a height of 10 m. The average windspeed over this period was 3.6 m/s. The concentric circles represent increments of 5% probability for the wind coming from the given direction with speed in a specific range. (Credit: US Geological Survey, Oregon Water Science Center)
Information on the direction and intensity of the wind is usually summarized in plots known picturesquely as wind roses. The compass is divided up into (usually) 8, 12, 16, or as many as 36 wedges, and the probability of observing the wind from each direction is displayed. Usually the data is further subdivided to display the probability as a function of both direction and speed. The example shown in Figure 28.19 is a form that is particularly easy to interpret. For each compass direction, the length of the wedge indicates the probability that the wind will be found in a given direction with the speed indicated by the color. Thus, for example, the wind at the Klamath Lake site on average blew from the west 21% of the time and it both came from this direction and had a speed between 6 and 8 m/s approximately 6% of the time. Integrating the information in a wind rose over all directions provides a rough estimate of the wind speed frequency distribution, which is given in the key to the figure (Problem 28.14).
28.2.4 Wind Power Classes and Wind Atlases
The wind resource characteristics described in the previous sections must be considered in detail when evaluating any particular site for wind power potential. On a larger scale, however, it is convenient to use a less detailed classification. Governments collect and publish data on wind energy. At the meta-level, the US government groups the information into wind power classes ranging from 1 to 7. The classes are defined in Table 28.2. Other governments and international organizations use similar schemes. In some cases, only wind power data is reported, and in some cases only average wind speed at a particular height is reported. The US scheme for conversion between power and average wind speed assumes a Rayleigh () distribution. Data are typically presented at heights of 10 and 50 meters using the rule (see Footnote 2) to relate the two. According to the US National Renewable Energy Laboratory (NREL), areas designated as wind power class 3 or above are generally suitable for utility-scale wind turbine applications, class 2 is marginal.
Table 28.2 Classes of wind power density at 10 m and 50 m [185]. The speed–power relationship assumes a Rayleigh distribution. The height variation of the wind between 10 m and 50 m is assumed to follow a law.
Data on wind resources from many sites can be combined to create a wind atlas, which makes it possible to visualize wind power potential over large areas. Wind atlases are now available or under development for many regions throughout the world. Figures 28.20 and 28.21 give an overview of the wind power resources available in the US and Europe. Many more detailed atlases are available online. European wind resources (Figure 28.21) are concentrated on its northwest coast and in the British Isles. Wind power potential is relatively low in southern Europe, though local resources in the Pyrenees Mountains and in the South of France (where the Mistral blows) are prominent. The US (Figure 28.20) is relatively rich in wind capacity, though some areas such as the Southeast coastal plain and the Southwest deserts have little potential. Much of the resource is in sparsely populated areas such as the upper plains (Oklahoma through the Dakotas), and the Rocky Mountain peak areas. There are, however, some exceptionally felicitous areas where high wind potential coincides with significant population density. Examples include the region just to the East of the San Francisco Bay Area, where the Altamont Pass windfarms were developed many years ago, and (just to the Southeast of MIT) in the shallow waters surrounding Cape Cod, Martha’s Vineyard, and Nantucket off the coast of Massachusetts. This local resource can be seen clearly on the more detailed atlas in Figure 28.22.
Wind Power Classes
In the US, wind atlases and other large-scale surveys of wind power potential use a simplified system, wind power classes, to classify wind power potential. Wind power classes range from 1 to 7, based on the power in the wind referred to a nominal height of 10 or 50 m. Class 3 ( W/m at 50 m) and above are considered suitable for utility-scale wind power development.
Figure 28.20 An annual average wind atlas of the continental US [185]. Note the absence of wind power in the Southeast.
Figure 28.21 An annual average wind power atlas for much of Western Europe. The key gives wind power in W/m at a height of 50 m over an open plain. Note that the data for Finland, Sweden, and Norway are designated differently. The power is greater for coast, open ocean, hills, and ridges, less for sheltered terrain. See [186] for a more detailed key. (Credit: windpower.org)
Figure 28.22 An estimated annual average wind power atlas for the east coast of the State of Massachusetts, for points within 50 nautical miles for shore (1 nmi = 1.852 km). (Adapted from map by US NREL)
28.3The Potential of Wind Energy
We conclude this chapter with a brief survey of local constraints on the exploitation of a wind power resource and an estimate of the global potential for human use of wind power. Where it is abundant, wind energy is a considerably more concentrated resource than solar energy. Unlike solar energy, however, wind energy is not a completely two-dimensional resource distributed across the planet’s surface. A given packet of air flowing horizontally over Earth’s surface carries energy over many kilometers of distance. Extracting energy from the wind at one location thus depletes the energy supply for a substantial distance downstream. This places limits on wind energy extraction both locally and globally. Locally it limits the density of placement of individual turbines in a wind farm. When the required spacing between wind turbines is included, wind and solar power are roughly comparable resources in terms of power produced per unit of land area. On a larger scale, the total energy available in atmospheric wind flow limits the power that can be extracted from winds near Earth’s surface. Also, the complexity of modeling energy transport in the atmosphere has made it difficult to estimate the world’s total wind power resources accurately.
28.3.1 Local Constraints on Wind Power Density
Table 28.2 indicates that at class 6, the power in the wind at a height of 50 meters above ground level is over 600 W/m2. The available power is diminished by efficiency of the wind turbine, which is limited by 16/27 (the Betz limit) and may be as large as 40% under ideal conditions (we discuss wind turbine efficiency in more detail in the next two chapters). Taking 40% efficiency with a wind power of 600 W/m2 at a height of 50 m, the recoverable power is W/m2. This power per unit area is almost an order of magnitude greater than that from solar photovoltaics (~250 W/m2 averaged over the year in good locations, gathered with 15% efficiency), but unlike solar collectors, wind turbines cannot be densely placed across a two-dimensional plain.
Figure 28.23 (a) Turbulent wakes at the Horns Rev offshore wind farm in Denmark made visible by unusual weather conditions that created condensation in the turbulence. (b) A simulation of power output from five turbines separated along the flow direction by [187].
In a wind farm, wind turbines must be sited at some distance from one another in the direction along the flow of the wind to prevent the disturbance in air flow from one turbine from affecting others downstream. Behind each turbine there is a wind shadow in which the wind speed is reduced. This effect is illustrated in Figure 28.23(a). The effect is quantified in Figure 28.23(b), which shows the result of a simulation in which five 5 MW rated turbines with 120 m rotor diameters have been separated along the direction of flow by three times their rotor diameter. Only the first turbine performs at its rated power. The power output of the other four is suppressed by a factor of ~1/2. Thus, wind turbines must be placed at some distance behind one another so that the wind field has space to recover by bringing energy down from higher altitudes so that the wind turbines that are further downstream get the full benefit of the wind power. As a rule of thumb, losses from turbine interactions are believed to be less than 10% when turbines with rotor diameter are placed at a distance of – apart in the direction perpendicular to wind flow, and on the order of apart along the direction of flow [178, 179], though recent work that includes land and turbine cost considerations suggests that greater distances (~15D in the flow direction) may avoid inefficiency due to excess shadowing effects [188]. If, for example, turbines are separated by in the crosswind direction and in the downwind direction, then each turbine with blade covering an area is allocated a land area of . Thus the overall power produced per unit of land area is reduced by a factor of , so that in the example of the previous paragraph the power output would be roughly 5 W/m2, somewhat smaller than could be realized for a solar field of similar area in a desert location. Thus, the land area needed for large-scale wind farms even in a high-quality (wind power class 6, for example) location is larger than that needed for solar power. In a wind power class 3 or 4 location, the area needed is several times that needed for a comparable solar power plant. On the other hand, solar arrays require covering the allocated land area densely with solar collectors, whereas most of the wind farm area is left undeveloped. Thus land used for wind turbines can simultaneously be used for other purposes such as agriculture, while total ground coverage by solar thermal mirrors or photovoltaic panels makes other uses more difficult or impossible.
Constraints on Wind Power Density
Wind turbines disturb the atmosphere in their wake, reducing the power available to downstream turbines in a closely spaced wind farm. Although more research needs to be done, the present state of knowledge indicates that turbines should be spaced by 10 to 15 times their rotor diameter in the direction of prevailing winds and four times their rotor diameter in the crosswind direction.
