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Energy generated by a wind turbine
The energy generated by a wind turbine can be treated statistically by knowing the power curve and the distribution of wind speeds at the rotor hub height. The energy calculation is usually performed annually to determine the capacity factor (CP) of the wind turbine. The energy and capacity factor can vary from year to year due to wind variability and site environmental conditions.
Preliminary Concepts
To calculate the annual energy produced by a wind turbine, the following concepts must be considered:
The power curve of the wind turbine (P(v)): This curve can be obtained from the manufacturer. It should be noted that the curve provided by the manufacturer is obtained under standard conditions (15°C and 1.01 bar), for which a wind density of 1,225 kg/m3 is obtained.
The Weibull distribution curve (p(v)): This curve is obtained by a statistical treatment of the wind speeds at the hub height of the wind turbine for one year. It is a versatile continuous distribution that can be used to mathematically model wind speeds. It can also be replaced by the probability distribution of wind speeds.
There are two ways to calculate the power produced by a wind turbine: discrete or numerical. Generally, the power curve (P-v) is not expressed in mathematical form, but is provided by the manufacturer in discrete form, i.e. in the form of a table with a series of points relating a given wind speed (m/s) to a given power (MW).
- To use the numerical method, it is necessary to obtain the mathematical equation of the power curve and the Weibull distribution curve for one year.
- To use the discrete method, it is necessary to obtain the manufacturer’s power curve in tabular form and the statistical treatment of wind speeds for one year.
For this reason, it is common to use the discrete method to calculate the energy produced by the wind turbine.
Energy Calculation (Discrete Method)
The annual energy production (Ea) of a wind turbine is calculated using the following equation:
Pi : Power at speed “i
fi : Relative frequency “i”.
T : Production time period (8760 hours).
Capacity Factor (CF)
It is defined as the ratio between the electrical energy (Ea) produced during a period of time (T) and the energy that would have been produced if the unit had been operating continuously at rated power (Pn) during that period. It is also known as the capacity factor. It can be defined using the following equation:
Application
You have the recorded wind speed data at a given location. The data have been grouped into speed packets centered on intermediate values (1, 2, 3, 4, …, 20) because the manufacturer typically provides power data for values centered on natural numbers. Table 1 shows the statistical treatment of wind speeds for one year.
Class center (m/s) | Range | Number of data |
0.25 | <0-0.5> | 0 |
1 | <0.5-1.5> | 12 |
2 | <1.5-2.5> | 22 |
3 | <2.5-3.5> | 105 |
4 | <3.5-4.5> | 264 |
5 | <4.5-5.5> | 646 |
6 | <5.5-6.5> | 1293 |
7 | <6.5-7.5> | 2021 |
8 | <7.5-8.5> | 1868 |
9 | <8.5-9.5> | 1266 |
10 | <9.5-10.5> | 664 |
11 | <10.5-11.5> | 315 |
12 | <11.5-12.5> | 172 |
13 | <12.5-13.5> | 73 |
14 | <13.5-14.5> | 32 |
15 | <14.5-15.5> | 16 |
16 | <15.5-16.5> | 9 |
17 | <16.5-17.5> | 2 |
18 | <17.5-18.5> | 2 |
19 | <18.5-19.5> | 2 |
20 | <19.5-20.5> | 0 |
Total data | 8794 |
The P-V power curve is obtained from the manufacturer Goldwind GW121/2500 for standard conditions.
Wind speed (m/s) | Power mechanical (MW) |
1 | 0.00 |
2 | 0.00 |
3 | 0.05 |
4 | 0.21 |
5 | 0.41 |
6 | 0.71 |
7 | 1.07 |
8 | 1.61 |
9 | 2.23 |
10 | 2.44 |
11 | 2.48 |
12 | 2.50 |
13 | 2.50 |
14 | 2.50 |
15 | 2.50 |
16 | 2.50 |
17 | 2.50 |
18 | 2.50 |
19 | 2.50 |
20 | 2.50 |
21 | 2.50 |
22 | 2.50 |
Figure 2 shows the power curve (P-V) from the manufacturer Goldwind.
The number of hours per year corresponding to a given speed is obtained by multiplying the frequency of occurrence (fi) by the number of hours in the year (8760 h). In this way, Table 3, which represents the Goldwind GW121/2500 power output of 2.5 MW under standard conditions, can be filled in.
Class center | Range | Number of data | Relative frequency | Power (MW) | Time (horas) | Energy (MWh) |
0.25 | <0-0.5> | 0 | 0.00 % | 0.000 | 0.00 | 0.000 |
1 | <0.5-1.5> | 12 | 0.14 % | 0.000 | 11.97 | 0.000 |
2 | <1.5-2.5> | 22 | 0.25 % | 0.000 | 21.94 | 0.000 |
3 | <2.5-3.5> | 105 | 1.20 % | 0.048 | 104.71 | 5.026 |
4 | <3.5-4.5> | 264 | 3.01 % | 0.207 | 263.28 | 54.499 |
5 | <4.5-5.5> | 646 | 7.35 % | 0.410 | 644.23 | 264.136 |
6 | <5.5-6.5> | 1293 | 14.72 % | 0.714 | 1289.47 | 920.68 |
7 | <6.5-7.5> | 2021 | 23.01 % | 1.072 | 2015.48 | 2160.593 |
8 | <7.5-8.5> | 1868 | 21.27 % | 1.614 | 1862.90 | 3006.714 |
9 | <8.5-9.5> | 1266 | 14.41 % | 2.228 | 1262.54 | 2812.941 |
10 | <9.5-10.5> | 664 | 7.56 % | 2.436 | 662.19 | 1613.085 |
11 | <10.5-11.5> | 315 | 3.59 % | 2.500 | 314.14 | 785.348 |
12 | <11.5-12.5> | 172 | 1.96 % | 2.500 | 171.53 | 428.825 |
13 | <12.5-13.5> | 73 | 0.83 % | 2.500 | 72.80 | 182.001 |
14 | <13.5-14.5> | 32 | 0.36 % | 2.500 | 31.91 | 79.781 |
15 | <14.5-15.5> | 16 | 0.18 % | 2.500 | 15.96 | 39.891 |
16 | <15.5-16.5> | 9 | 0.10 % | 2.500 | 8.98 | 22.439 |
17 | <16.5-17.5> | 2 | 0.02 % | 2.500 | 1.99 | 4.986 |
18 | <17.5-18.5> | 2 | 0.02 % | 2.500 | 1.99 | 4.986 |
19 | <18.5-19.5> | 2 | 0.02 % | 2.500 | 1.99 | 4.986 |
20 | <19.5-20.5> | 0 | 0.00 % | 2.500 | 0.00 | 4.986 |
21 | <20.5-21.5> | 0 | 0.00 % | 2.500 | 0.00 | 0 |
8784 | 8760 | 12390.92 |
The capacity factor of the wind turbine is calculated as follows:
A capacity factor (CF) of 56.5% indicates an excellent evaluation of the energy production of the wind turbine. This is only an example, as it depends on the choice of the machine.
Reference
[1] Goldwind GW121/2500 – Fabricantes y aerogeneradores – Acceso en línea – The Wind Power. (s/f). Thewindpower.net. Recuperado el 21 de enero de 2023, de https://www.thewindpower.net/turbine_es_1029_goldwind_gw121-2500.php.
