The determination of the annual energy yield and as well the assessment of the wind turbine loads are mostly based on measured 10-min averages of the wind speed; seldom is the averaging period a 1-min or 1-hour interval.
Additionally, the wind direction is measured during the whole measuring period using a wind vane, and all the values are recorded in time series.
Apart from the 10-min average wind speed, the maximum and minimum value as well as the standard deviation of the averaging period may be stored in the data series as well, cf. Fig. 4-30.
The measuring period should be long enough, especially for the calculation of the predicted energy yield at a site. Depending on the local climate, the wind speed may undergo strong seasonal fluctuations, so the measuring period should be an integer multiple of a year.
Investigations show that the annual energy content of the wind may vary in a range of up to r25% [5]. Fig. 4-17 shows for a period of 100 years the energy con- tent averaged over five-year periods and normalized with the 100-year average.
The strong fluctuations are clearly visible.
On one hand, the long-term measuring data should be stored carefully, (“at best on a CD ROM in a safe”). On the other, it has to be processed using statistics.
Fig. 4-17 Relative wind energy, averages over 5-year periods in Laeso, DK [5]
4.2 Atmospheric boundary layer 132
Frequency distribution – Histogram of wind speeds
For the wind turbine design as well as for the assessment of the expected energy yield, the pure time series of possibly several years of measurement are quite impractical. One possibility for attaining a compressed representation of the wind conditions is the generation of a frequency distribution of the wind speed. The wind speeds are sorted into classes and summed up, Fig. 4-18. This way, the tem- poral share of the individual wind speed class of the entire considered time period is determined. The width of the wind speed class is typically 1 m/s.
Fig. 4-18 Left: Diurnal time series of hourly averaged values; right: corresponding histogram of the relative frequency hi of the respective day
The relative frequency of the individual wind speed class vi is hi = ti /T, e.g. hours per 24 hours in Fig. 4-18. Of course, the sum of the relative frequencies must be exactly 1.0, resp. 100%. Fig. 4-21b shows such a frequency distribution of the Tauern wind farm (Austria) with a wind speed class width of 1 m/s. In section 4.3 it will be presented how to calculate the expected energy yield with such a wind speed histogram and a given wind turbine power curve, cf. Fig. 4-25.
Wind speed distribution function
The measured frequency distribution is mostly “compressed” into a mathematical description using the Weibull distribution function which is quite flexible due to its two parameters
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The scaling factor A is a measure for the characteristic wind speed of the consid- ered time series. The shape factor k describes the curve shape. It is in the range between 1 and 4, and the value is roughly characteristic for certain wind climate:
k § 1: Arctic regions
k § 2: Regions in Central Europe k § 3 to 4: Trade wind regions
If there are small fluctuations around the mean wind speed v the value of k is high whereas large fluctuations give a smaller shape factor k, Fig. 4-19.
Fig. 4-19 Example of Weibull distributions for a mean wind speed v = 8 m/s and different shape factors k
Table 4.2 shows for some sites in Germany the Weibull factors A and k as well as the corresponding mean wind speed v .
Given Weibull factors allow an estimation of the mean wind speed v [33]
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Both Weibull factors A and k change with the height above ground, Fig. 4-20.
4.2 Atmospheric boundary layer 134
Table 4.2 Weibull factors for different sites in Germany, measuring height 10 m, from [36]
Site k A in m/s v in m/s
Helgoland 2,13 8,0 7,1
Hamburg 1,87 4,6 4,1
Hannover 1,78 4,1 3,7
Wasserkuppe 1,98 6,8 6,0
Fig. 4-20 Change of Weibull factors with height [1]
The Rayleigh distribution function, Fig. 4-21 (a), is a special, simplified case of the Weibull distribution function for the shape factor k = 2.
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It is very straightforward since it depends only one parameter: the mean wind speed v , e.g. the annual mean wind speed which is roughly known for many sites.
In general, the reference energy yield given by wind turbine manufacturers in the data sheets is calculated based on a Rayleigh distribution function.
Fig. 4-21 (b) shows for the site of the Tauern wind farm a measured wind speed frequency distribution and the fitted curves for both, the Weibull and the Rayleigh distribution function. The differences between the measurement results and the analytic description are strong.
If the analytic approximations by the distribution functions are used for the yield calculation the following criteria should be considered:
- The calculated wind energy of the analytic and measured frequency distri- bution should be approximately equal.
- For wind speed classes higher than the measured mean wind speed the frequencies should be identical.
- The sum of the frequencies should be used as checksum and always be 1.00, else the relative frequencies have to be weighted with the used wind speed class width (e.g. 0.5 m/s).
Moreover, the analytic description should represent correctly the classes which contain the maximum energy. The determination of the Weibull factors for a given measured frequency distribution is done e.g. by least square fitting, eventually it is beforehand necessary to take twice the logarithm [7, 34].
Note: If the measured frequency distribution, Fig.s 4-18 and 4-21, (discrete rep- resentation) is transformed into a distribution function h(v) (continuous represen- tation) or vice versa, it has to be considered that hi = h(v) dv § h(vi) 'vi , since the distribution function has the dimension 1/(m/s). A different wind speed class width 'vi causes other relative frequency values hi , which is obvious: the larger the class width of the distribution function the more events are found within this class, hi = ti /T.
The Weibull distribution function with its two factors is quite useful to describe the histograms. But even if measuring the wind at a site for several years, extreme events like the 50-year wind speed are perhaps not recorded because they did not occur. So they have to be represented separately, cf. chapter 9.
Moreover, the analytic distribution functions are not very suitable to represent the range of calms, since the functions always start with the value h (v = 0 m/s) = 0.
Frequencies of less than 1%, i.e. 10-min average values with less than 525 events per year, may not be estimated with a Weibull distribution. Therefore, if the calms statistics are required, e.g. for wind pumping and stand-alone systems, they are recorded separately.
4.2 Atmospheric boundary layer 136
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20
0 5 10 15 20 25 30
Windgeschwindigkeit v in m/s Relative Họufigkeit hR (v) in 1/(m/s)
v = 4 m/s
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Windgeschwindigkeit v in m/s Relative Họufigkeit hR (v) in 1/(m/s)
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Wind speed vin m/s Relative frequency hR(v) in 1/ (m/s)
0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20
0 5 10 15 20 25 30
Windgeschwindigkeit v in m/s Relative Họufigkeit hR (v) in 1/(m/s)
v = 4 m/s
6
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0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20
0 5 10 15 20 25 30
Windgeschwindigkeit v in m/s Relative Họufigkeit hR (v) in 1/(m/s)
v = 4 m/s
6
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Wind speed vin m/s Relative frequency hR(v) in 1/ (m/s)
Wind speed vin m/s Relative frequency hR(v) in 1/ (m/s)
a)
0 5 10 15 20 25 30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Wind speed class in m/s
Relative frequency in %
Measuring height: 10 m Measuring period: 03/97 to 09/02 Height above sea level: 1835 m Temperature: 8 °C
Measurements: 2038 days (2935090 1 min-values )
Rayleigh curve for v = 6,76 m/s
Weibull curve for A = 7,44 m/s and k= 1,46 Measurement
b)
Fig. 4-21 a) Frequency distribution according to Rayleigh for different mean wind speeds;
b) measurements in the “Tauernwindpark” in Austria, wind speed histogram and the fitted Weibull and Rayleigh frequency distribution functions [www.tauernwindpark.com]
10.0 12.5
SSE NNW
ESE ENE
WSW WNW
NNE
SSW
SSE NNW
ESE ENE
WSW WNW
NNE
SSW
SSE NN
W
SEE NEE
SWW NWW
NNE
SSW
0 - 5 m/s 5 - 10 m/s 10 - 15 m/s 15 - 20 m/s 20 - 40 m/s
a 2
5 7.5
Fig. 4-22 Wind roses: Frequency rose, rose of mean wind speed and energy rose
For siting wind farms with more than one wind turbine it is important to know the frequency distributions of the wind speed for the different wind directions in order to arrange the wind turbines sophistically to prevent that the turbines shade each other, cf. section 4.3.5. This sectorised wind information is represented in a wind rose, Fig. 4-22. Three types of roses are important: the wind speed rose, the fre- quency rose and the energy rose.
The wind speed rose shows the mean wind speed in the individual wind direc- tion sectors. In Fig. 4-22 the highest mean wind speeds are observed in the sectors NNE and WNW, while the Eastern wind is the smallest. In contrast, the wind fre- quency rose, shows a clear dominance of the Southern wind directions. But since the energy is proportional to the cube of the wind speed, and the Southern wind di- rections show relatively small mean wind speeds, the North-western sectors in the energy rose provide the largest energy share. These considerations serve for find- ing an optimum wind farm layout where, among others it is important to prevent that the turbines shade each other which reduces the energy yield and increases the loads due to wake effects, Fig.s 4-28, 4-29 and 4-39.
4.2 Atmospheric boundary layer 138