Preliminary Analysis of Wind Parameters at the Planned Construction Sites of Wind Generators

To understand why the long-term preliminary measurements of wind are so important, let us consider situation in Germany.

The author has created prototypes of two instruments for measuring these parameters: a long-acting anemometer and a brazmometer. Plese see the Figure 1 and 2. The name “Brazmometer” is firstly introduced here. This word comes from Latvian brāzma – gust. Both devices can be suspended on a one rope between two trees or be installed on 10 meter mast. The devices are purely mechanical and are very cheap to manufacture, which makes the experiments described below relatively cheap.

Fig. 1.

Fig. 2.

It makes no big sense to buy one expensive electronic anemometer, which records the strength of the wind every second and transmits data via satellite communication to a computer because it functions only at one location. It is much better for the same money to install dozens of cheap mechanical devices at different locations. Because for decision about installation of wind turbine no one needs the wind speed report every second. In reality, only two integral figures are needed after many years of observation: the average wind speed and the maximum possible wind speed. But these two numbers must be measured for many locations to choose between them.

A consumer, for example, a farmer who is considering construction of wind turbine in his field, will be able to install several such measurement devices at different ends of his field, and after many years of observation, he will be surprised to find that at some location the average wind blows stronger, and dangerous gusts of wind are weaker than elsewhere in the same field. That will give him the opportunity to get benefits from the correct placement of the wind turbine.

2.4.

The Beaufort scale

Fig. 3.

There was no official algebraic definition for the Beaufort wind scale, but it looks like following:

Wind speed (m/s) = 0.849 * Beaufort ^ 1.5, or Beaufort=1,12∗V(23)~F(13)\[\text{Beaufort}=1,12*{{V}^{\left( \frac{2}{3} \right)}}\tilde{\ }{{F}^{\left( \frac{1}{3} \right)}}\] where V – is the wind speed in m/s. This is only my estimation – the formulae which is not mentioned in the literature. But anyway it fits to our needs.

Possible decision is to make a mechanism where the angle of the hands on the dial is proportional to cubic root of the force F (wind pressure). Such special mechanism was invented. This technical solution is in the process of being patented.

The power produced by the wind generator is proportional to the cube of the wind speed. Therefore, the average wind speed is not exactly the parameter that is responsible for the power produced. However, it is known from practice [12] that the speed distribution obeys approximately the Weibull distribution, which is the same in all countries and continents. f(V)=KA(KA)K−1=(−(VA)K). \[f\left( V \right)=\frac{K}{A}{{\left( \frac{K}{A} \right)}^{K-1}}=\left( -{{\left( \frac{V}{A} \right)}^{K}} \right).\]

This probability distribution in this formula is already normalized: ∫0∞f(V)dV=1. \[\int_{0}^{\infty }{f\left( V \right)dV=1.}\]

Here A – is the Weibull scale parameter in m/s: a measure for the characteristic wind speed of the distribution. K – is the Weibull form parameter. It specifies the shape of a Weibull distribution.

Fig. 4.

Here coefficient A is proportional to the mean wind speed V_average. The coefficient of proportionality slightly depends on K but approximately A = 1.13 * V_average.

The total anemometer reading will be proportional to the value: Average(V)=∫0∞V∗f(V)dV \[\text{Average}\left( V \right)=\int_{0}^{\infty }{V*f\left( V \right)dV}\]

At the same time, the total energy production will be proportional to the value: Average(V3)=∫0∞V3∗f(V)dV \[\text{Average}\left( {{V}^{3}} \right)=\int_{0}^{\infty }{{{V}^{3}}*f\left( V \right)dV}\]

Having worked a little bit with math, it is possible to get: Average(V)=0.716∗(Average(V3))13atK=1.5Average(V)=0.806∗(Average(V3))13atK=2.0Average(V)=0.859∗(Average(V3))13atK=2.5 \[\begin{array}{*{35}{l}} \text{Average}\left( V \right)=0.716*{{\left( \text{Average}\left( {{V}^{3}} \right) \right)}^{\frac{1}{3}}}\text{at}\,K=1.5 \\ \text{Average}\left( V \right)=0.806*{{\left( \text{Average}\left( {{V}^{3}} \right) \right)}^{\frac{1}{3}}}\text{at}\,K=2.0 \\ \text{Average}\left( V \right)=0.859*{{\left( \text{Average}\left( {{V}^{3}} \right) \right)}^{\frac{1}{3}}}\text{at}\,K=2.5 \\ \end{array}\]

According to the study [12], in majority of locations, 1.8 < K < 2.2 and never goes beyond the interval 1.5 < K < 2.5. According to observations in Riga region made by Latvian Meteorological Center, K = 2 very precisely.

This makes it possible to compare the average wind speed and cubic root from the cubic average wind speed.

Roughly, Average(V)=0.8∗(Average(V3))135%\[\text{Average}\left( V \right)=0.8*{{\left( \text{Average}\left( {{V}^{3}} \right) \right)}^{\frac{1}{3}}}5%\] possible mistake which comes from variation of K.

This means that instead of the hard-measurable average cubic wind speed responsible for mean power, it is possible to use the mean wind speed easy measurable by a simple anemometer. And then, multiply it by a coefficient 1/0.8 = 1.25 and raise it to the third power.

However, it should be understood that such a simple pattern does not apply to dangerous gusts of wind. The fact is that dangerous gusts of wind occur only during hurricanes. And hurricanes come from far away. They are formed thousands kilometers away above the oceans, and the local topography does not affect them as strong as the local topography affects the mean wind. Even if some dependence between two kinds of averaging still exists, it makes no sense to analyze it because during hurricanes the wind turbines do not function.

Therefore, hurricane statistics should be kept separately and not squeezed into the tail of the general Weibull distribution. For this reason, two devices are needed, but one is not enough.

Most people intuitively believe that the average wind is exactly the same at two similar points that are less than 1 kilometer apart. However, there is reason to hypothesize that this is not always the case, and the difference can be sufficiently large. This hypothesis is based on three arguments:

4.2.

Large-scale argument

Nowadays, only large-scale statistics is available in internet, for example, in Ref. [19] you can find statistics for many years 1980–1999 in USA. Gif files for different 5-year periods show that tornado and thunderstorm come again and again in the same place. In language of mathematics, they can be called tornado-attractors. But if this repeating in large scale, we can hope to find similar repetition in small scale, because according to Mandelbrot Fractal theory, what happens in large (national wide) scale takes place also in small scale (about 1 km).

Fig. 5.

4.3.

Argument based on formation of geographic relief

It is known that the whirlwinds of tornado roam the same routes from year to year: for centuries, they destroy the forest in the same places, which leads to the formation of natural forest glades. During thousands of years, the wind blows sand to the same places, resulting in the formation of hills. It is not yet clear: in what natural conditions such an uneven distribution of gusts and average winds takes place. Seif dune (barkhan) is the easiest visible result of such wind’s unevenness. Please see Figure 6.

Fig. 6.

So far, no one has made direct precise measurements to confirm or refute this hypothesis in scale of 1 km. To confirm it, hundreds of wind measurements and their averaging over a long period over a small practically uniform area are required. If this hypothesis will be confirmed, then the instruments will show different average wind and different maximal wind for a long observation period in places that are less than 1 km apart from each other. This difference may occur the same from year to year, which means that it must be taken into account when choosing the location of wind turbines.

I hope that the cheap mechanical devices described above in this article can be useful to those who plan to build wind generators and want to find the best place.

The idea of mean wind speed which is devaluated by hurricanes (name proposed: “Bojat wind speed”).

A wind generator has its own cost, which is an investment. The cost price is proportional to the weight of materials spent on construction. And the weight of the materials used is proportional to the required strength. The dangerous breaking pressure that some gust of wind can exert is proportional to the square of the wind speed V.

Therefore, it can be argued that the cost of building a generator is proportional to the square of the maximum possible speed of a gust of wind in a given location. Some estimation of real average price can be found here [13], [18], [4] and [14], but it is clear that there are different models in the market: some of them can hold very strong hurricane the other not, and it is reason why they cost differently.

In this economic game, underestimation leads to the loss of all money as a result of the destruction of the wind turbine, and overestimation leads to unnecessary construction costs.

If the maximal possible speed is estimated correctly, then the investment can be considered proportional to the square of the maximal wind speed.

On the other hand, it can be considered that the income received is proportional to the power of the wind generator, which is proportional to the average cube of wind speed or cube of average speed.

Therefore, the income from investments in wind energy will be proportional to one parameter: VBojat=Vaverage3Vmax2 \[{{V}_{\text{Bojat}}}=\frac{V_{\text{average}}^{3}}{V_{\max }^{2}}\] where V_average is the average wind throughout the year and V_max is the maximum possible dangerous gust of wind during a storm. V_Bojat is one integrated parameter which helps us to evaluate suitability of the place for wind power.

Accordingly, the payback period for investment in wind energy will be inversely proportional to this parameter.

This parameter has big advantage because it is purely physical. It is nothing else but some effective speed measured in meters per second by a purely physical method using the two instruments described above in this article. The name is suggested: average wind speed devalued by hurricanes. Let us name it here as the Bojat speed (from Latvian language Bojats – spoiled, devaluated). It will always be significantly less than the real average wind speed. In theoretical ideal case if the wind is constant, the formulae gives: V_Bojat = V_average = V_max = Constant, but in reality V_Bojat < V_average.

For economic calculations and for making final decision: to build or not to build, it should be multiplied by all kinds of economic coefficients such as the cost of electricity, profitability of a bank loan, demand for electricity and so on. How to take in to consideration all this non physical parameters and reuce all of them to one value can be found here [3]. But in our article we discussed only physics.

Information on average wind speed devalued by hurricanes (Bojat speed) is a basic physical parameter for wind energy and should be measured before any consideration of future construction begins.

From two places, the one with the best V_Bojat parameter should always be chosen.

Unfortunately, till now, only the average wind is available in internet. This leads to erroneous conclusions, for example:

Looking at these maps, you might think that Latvia and Mexico have roughly the same opportunities for wind energy development because both have areas marked by brown color which symbolize average wind speed V_average = 8 m/s. But it is incorrect conclusion. Hurricane Patricia south of Mexico, in 2015 achieved its record peak intensity with maximum wind gust of V_max = 96 m/s [17]. On the other hand, for Latvia, the maximum wind gust was in 1967, V_max = 48 m/s [16].

Using above formulae, we calculate: V_Bojat = 0.055 m/s for Mexico and V_Bojat = 0.22 m/s for Latvia.

So, Latvia has the areas which are four times more profitable for wind energy than Mexico! Not because the mean wind is stronger but just because there is no need to make the Latvian wind turbines as hurricane-proof as in Mexico; so, they can be made four times cheaper.

The essence of proposed method is that two parameters are reduced to just one integrated parameter. This idea is not unique: for example, Baseeret al. [3] introduce some “site suitability index” which integrates many physical and economical parameters. This index represents final easy to read evaluation for decisions makers. It is interesting to look on their map for Saudi Arabia:

Note that this map looks very similar to a fractal (as also the map of Latvian average speed of wind on Figure 7 does). It can be assumed that if going to smaller scales <1 km, such fractal scatter will continue to look very chaotic.

Fig. 7.

Fig. 8.

The advice to measure the average wind speed during at least a year not only in the region but in exact place can be found in many publications, for example here [11]. In addition to this advice the necessity to measure not only average speed, but maximal speed in hurricane should be emphasized. So two devices will be needed.

At [8] and [9] big discussion about wind dependence from height can be found. But as soon as our goal is to simplify things – for all measurements one standard height should be chosen. The height 10 meters is traditionally recommended for measurements by anemometer and brazmometer. It is clear, that real planned wind turbine will work at higher height, and some corrections will be needed. The good discussion about optimal size of wind turbine is here [1].

The average wind speed devalued by hurricanes V_Bojat is much more informative than just average wind speed. It would be nice in future to see map colored by V_Bojat.

Sets of two described devices are suitable to measure the V_Bojat. Hundreds or such devises should be produced and used for further study of wind in any country.