Posted: July 24th, 2015
Distribution of wind speed:
Distribution of wind speed:
When building a wind farm, it is very important to describe the variation of wind speeds at the prospective farm site. Engineers need the information to optimize the design of the turbines to be used at the farm. This allows them to minimize generating costs. Just as importantly, potential investors in the wind farm need the information to estimate the potential income from electricity generation.
When these data are presented graphically, the result is called a Weibull distribution. In this laboratory, you will prepare a Weibull distribution based on observations of wind speeds and wind directions collected by the Oklahoma Mesonet.
When wind speeds throughout a year are examined, it becomes clear that days with strong winds, and days with no wind, are rare. Days with moderate winds are much more frequent. Thus, there is a distribution of wind speeds for each possible wind farm site. If a wind farm is to be built, the power generating potential of the wind for the site must be determined. The result determines whether or not it is economically feasible to build the farm.
One might erroneously conclude that the power generating potential can be determined from knowing the average wind speed and the efficiency of the wind turbine at that wind speed, but this is not the case. To calculate the power generating potential it is important to know the actual wind distribution. Usually the wind distribution is not symmetrical, that is, there is a difference between the average wind speed and the median wind speed. The median wind speed is that speed for which half of the observed wind speeds are greater than while the other half is less than. Another important statistical term to know is the “mode”. The mode is the wind speed that occurs most frequently at the site.
In previous laboratories you have been working with the average wind speed. This is the speed obtained by taking the sum of the daily average speed and dividing it by the number of days. Average wind speeds were used to identify potential wind farm sites. To determine the energy production potential of the site, the distribution of wind speeds around the average speed will be needed. Thus it is necessary to construct a graph showing the wind distribution for your potential wind farm site.
The Weibull Distribution:
The wind distribution diagram you will generate is called a “probability density distribution”. In a probability density distribution the area under the curve is exactly 1 unit. To find the probability density distribution for a particular site, you must count the number of days at each average wind speed. This was performed using data from the Perkins site from 2011 and included in a table on the next page.
2011 Wind Distribution for Perkins
Wind Speed (mph) Number of Days
3 6
4 13
5 29
6 43
7 37
8 50
9 31
10 37
11 24
12 22
13 20
14 11
15 11
16 12
17 9
18 2
19 2
20 2
21 2
22 1
Total Days 364
To calculate the probability density distribution, the probability of the occurrence of each speed must be determined. This is done by taking the number of days that that a particular speed is observed and dividing by the total number of days. The result is a plot of the probability of the wind speed as a function of the wind speed. The plot for Perkins in 2011 resulted in the plot below:
If the probabilities for all of the wind speeds were summed, the result would equal 1. It can be seen that for this site, the most common wind speed is 8 miles per hour and that particular wind speed will be observed at about .14 or 14% of the time.
Another factor that is considered when building a wind farm is the predictability of the wind speeds. Potential investors would certainly not want to build a farm on a site where they are unsure of the wind speeds. While a high standard deviation implies that winds for future days would be difficult to predict with much certainty, a low standard deviation (less than 4) suggests that the wind speeds are fairly consistent, or close to the mean. In an exercise below, we will compute the standard deviation of the wind speeds of your site, which will be a measure of the consistency of the wind speeds.
The Power Distribution of the Wind:
Recall from the previous laboratory that the energy producing potential of the wind varies as the cube of the wind speed. For example, a day with a wind speed of 10 miles per hour can generate eight times as much power as a day with a wind speed of 5 miles per hour. Although high winds are rare, they generate much, much more energy. Thus, sites that have the most very windy days have a higher energy generating potential.
To determine what wind speeds produce the most wind power at a site, the power distribution must be measured. This is obtained by multiplying the probability of a particular wind speed by the cube of the wind speed for each wind speed. The power distribution for Perkins is shown below.
Notice that the shape of this graph is very different than that for the wind distribution, but this graph is much more important. It suggests that even though the average and median wind speeds are about 9 miles per hour, the average power production of the wind at Perks is at 13 miles per hour! Another way of thinking about this is that 50% of all of the energy that can be produced by the wind in Putnam is produced in only 51 (14%) of the 364 days.
Questions and Exercises:
Submit both a Word Document and an Excel Spreadsheet for this lab that answers the following:
1. Generate a Weibull distribution for your wind farm site. You should be able to start with the same Excel Spreadsheet that you used for Lab #3. To find the probability density distribution (Weibull distribution) for your site, round the average wind speed for each day to the nearest integer by using the “ROUND” function. You will use the format, “=Round(Cx,d),” where Cx is the cell you wish to round, and d is the number of decimal places you wish to show. Since you are seeking integers, the value of d should be zero. Fill this function down so that all wind speeds are rounded to the nearest integer. Then, sort the data in this column, and count the number of days corresponding to each average wind speed. The table should look like the one for Putnam, Oklahoma shown in this lab. For each wind speed, calculate the fraction of days that the wind speed blows at that speed. This is done by taking the number of days that that speed is observed and dividing by the total number of days in the data set. To create the distribution diagram you plot the probability of the wind speed as a function of the wind speed. Copy the plot, and paste it into a Word document. Identify the average wind speed, the median wind speed and the modal wind speed. Comment on the graph. Is it symmetrical? In a couple of sentences, interpret the graph.
3. Compute the standard deviation of the unrounded wind data (from Lab #3). To do this, type “=STDEV(X)” where X highlights all of the wind speeds. Interpret your value. Does your site appear to have consistent or inconsistent wind speeds?
5. Determine at what wind speed half of the total power potential is produced. Comment on the difference between your power potential plot and the Weibull distribution plot.
Question 1
Average Wind Speed Rounded Wind Speed Sorted Wind Speed Wind Distribution
[mph] [mph] [mph] Wind Speed Number of Days Probability
16 16 16 3 [mph] [-] [-]
9.6 9.6 10 3 1 0 0
9.3 9.3 9 3 2 0 0
8 8 8 4 3 3 0.008333333
5.2 5.2 5 4 4 9 0.025
9.8 9.8 10 4 5 36 0.1
11 11 11 4 6 43 0.119444444
14.8 14.8 15 4 7 44 0.122222222
9.7 9.7 10 4 8 53 0.147222222
9 9 9 4 9 35 0.097222222
9.6 9.6 10 4 10 41 0.113888889
14.3 14.3 14 4 11 26 0.072222222
8.6 8.6 9 5 12 26 0.072222222
12.6 12.6 13 5 13 19 0.052777778
4.8 4.8 5 5 14 14 0.038888889
12.4 12.4 12 5 15 5 0.013888889
17.2 17.2 17 5 16 2 0.005555556
7.5 7.5 8 5 17 2 0.005555556
7.5 7.5 8 5 18 2 0.005555556
5 5 5 5 19 0 0
8 8 8 5 20 0 0
16.2 16.2 16 5 21 0 0
7.3 7.3 7 5 22 0 0
6.1 6.1 6 5 23 0 0
4.5 4.5 5 5 24 0 0
12.2 12.2 12 5 25 0 0
11.8 11.8 12 5 26 0 0
4.9 4.9 5 5 27 0 0
7 7 7 5 28 0 0
6.8 6.8 7 5 29 0 0
5.3 5.3 5 5 30 0 0
7.4 7.4 7 5 Total Days: 360 1
8.5 8.5 9 5
14.7 14.7 15 5 Average Wind Speed
14.1 14.1 14 5 [mph]
10 10 10 5 8.772222222
11.5 11.5 12 5
11.1 11.1 11 5 Median Wind Speed
8.4 8.4 8 5 [mph]
9.2 9.2 9 5 8
8.1 8.1 8 5
14.7 14.7 15 5 Modal Wind Speed
18.1 18.1 18 5 [mph]
8.7 8.7 9 5 8
9.4 9.4 9 5
9.7 9.7 10 5
11.1 11.1 11 5
11.9 11.9 12 5
6.8 6.8 7 6
8.9 8.9 9 6
6.2 6.2 6 6
7.5 7.5 8 6
10.5 10.5 11 6
5.1 5.1 5 6
12.1 12.1 12 6
10.4 10.4 10 6
13.3 13.3 13 6
7.7 7.7 8 6
12 12 12 6
9 9 9 6
6.3 6.3 6 6
8.3 8.3 8 6
13.9 13.9 14 6
8.8 8.8 9 6
10.8 10.8 11 6
7.6 7.6 8 6
13.1 13.1 13 6
12.3 12.3 12 6
7.3 7.3 7 6
5.9 5.9 6 6
7.5 7.5 8 6
10.3 10.3 10 6
9.4 9.4 9 6
7.1 7.1 7 6
9.4 9.4 9 6
6.7 6.7 7 6
9.6 9.6 10 6
5.5 5.5 6 6
5.7 5.7 6 6
4.2 4.2 4 6
11.3 11.3 11 6
9.8 9.8 10 6
11.4 11.4 11 6
6.2 6.2 6 6
10 10 10 6
5.2 5.2 5 6
8.6 8.6 9 6
7.7 7.7 8 6
5.7 5.7 6 6
8.9 8.9 9 6
6.7 6.7 7 6
6.4 6.4 6 7
9.9 9.9 10 7
10.2 10.2 10 7
3.6 3.6 4 7
4.2 4.2 4 7
5.9 5.9 6 7
6.3 6.3 6 7
7.6 7.6 8 7
10.3 10.3 10 7
6.3 6.3 6 7
9 9 9 7
7.8 7.8 8 7
8 8 8 7
7.7 7.7 8 7
6.3 6.3 6 7
6.3 6.3 6 7
4.5 4.5 5 7
4.6 4.6 5 7
6.1 6.1 6 7
4.7 4.7 5 7
7.6 7.6 8 7
7.3 7.3 7 7
4.8 4.8 5 7
7.8 7.8 8 7
8.4 8.4 8 7
7.8 7.8 8 7
5.6 5.6 6 7
5.6 5.6 6 7
6.8 6.8 7 7
6.5 6.5 7 7
7.9 7.9 8 7
8.2 8.2 8 7
5.8 5.8 6 7
4.6 4.6 5 7
9.6 9.6 10 7
11.8 11.8 12 7
8.2 8.2 8 7
7.2 7.2 7 7
10 10 10 7
6.1 6.1 6 7
9.7 9.7 10 7
6.3 6.3 6 7
5.5 5.5 6 7
6.3 6.3 6 7
12.3 12.3 12 8
8.4 8.4 8 8
5.3 5.3 5 8
8 8 8 8
10.4 10.4 10 8
9.9 9.9 10 8
9.6 9.6 10 8
12.1 12.1 12 8
6.6 6.6 7 8
10.6 10.6 11 8
4.1 4.1 4 8
5.4 5.4 5 8
5.8 5.8 6 8
7.4 7.4 7 8
7.7 7.7 8 8
5.8 5.8 6 8
8.6 8.6 9 8
4.5 4.5 5 8
9.1 9.1 9 8
5.7 5.7 6 8
6.3 6.3 6 8
8.1 8.1 8 8
7.1 7.1 7 8
7.3 7.3 7 8
7.6 7.6 8 8
8.2 8.2 8 8
5.2 5.2 5 8
5.2 5.2 5 8
4.7 4.7 5 8
6.5 6.5 7 8
5.7 5.7 6 8
5.2 5.2 5 8
6 6 6 8
8.4 8.4 8 8
4.6 4.6 5 8
7 7 7 8
5.1 5.1 5 8
5.2 5.2 5 8
4.9 4.9 5 8
7.3 7.3 7 8
6.9 6.9 7 8
5.8 5.8 6 8
8.3 8.3 8 8
11.7 11.7 12 8
4.9 4.9 5 8
11.3 11.3 11 8
7.5 7.5 8 8
4.2 4.2 4 8
3.4 3.4 3 8
7.1 7.1 7 8
10.2 10.2 10 8
4.9 4.9 5 8
10.3 10.3 10 8
11.6 11.6 12 9
10.8 10.8 11 9
12.5 12.5 13 9
11.7 11.7 12 9
7.7 7.7 8 9
12.6 12.6 13 9
10 10 10 9
14.1 14.1 14 9
13.7 13.7 14 9
7.8 7.8 8 9
12.4 12.4 12 9
13.6 13.6 14 9
6.9 6.9 7 9
11 11 11 9
16.5 16.5 17 9
14.1 14.1 14 9
8.9 8.9 9 9
10 10 10 9
10.8 10.8 11 9
11.3 11.3 11 9
8.1 8.1 8 9
12.9 12.9 13 9
9.3 9.3 9 9
7.4 7.4 7 9
6.3 6.3 6 9
9.4 9.4 9 9
14.4 14.4 14 9
13 13 13 9
12.1 12.1 12 9
7 7 7 9
8.3 8.3 8 9
8.7 8.7 9 9
8.6 8.6 9 9
9.5 9.5 10 9
8.9 8.9 9 9
8 8 8 10
10.6 10.6 11 10
8.7 8.7 9 10
9.5 9.5 10 10
4.2 4.2 4 10
10.3 10.3 10 10
9.9 9.9 10 10
10.3 10.3 10 10
7.3 7.3 7 10
10.4 10.4 10 10
9.3 9.3 9 10
9.3 9.3 9 10
9.3 9.3 9 10
6.9 6.9 7 10
6.1 6.1 6 10
11.3 11.3 11 10
10.7 10.7 11 10
4.5 4.5 5 10
5.7 5.7 6 10
5.6 5.6 6 10
9.6 9.6 10 10
13.4 13.4 13 10
4.8 4.8 5 10
8.3 8.3 8 10
12.2 12.2 12 10
4.3 4.3 4 10
6.5 6.5 7 10
8.8 8.8 9 10
7.1 7.1 7 10
8.4 8.4 8 10
7.4 7.4 7 10
10.6 10.6 11 10
14.2 14.2 14 10
12.5 12.5 13 10
9.3 9.3 9 10
9.9 9.9 10 10
18.1 18.1 18 10
11.3 11.3 11 10
12.7 12.7 13 10
13.3 13.3 13 10
13.9 13.9 14 10
13.9 13.9 14 11
12 12 12 11
8.1 8.1 8 11
7.1 7.1 7 11
7.2 7.2 7 11
10.2 10.2 10 11
12.6 12.6 13 11
11.6 11.6 12 11
11.6 11.6 12 11
13.6 13.6 14 11
13.4 13.4 13 11
7.2 7.2 7 11
9.8 9.8 10 11
13.6 13.6 14 11
8.6 8.6 9 11
7.7 7.7 8 11
6.2 6.2 6 11
10.2 10.2 10 11
8.9 8.9 9 11
7.3 7.3 7 11
9.5 9.5 10 11
8.3 8.3 8 11
4 4 4 11
7.2 7.2 7 11
10.8 10.8 11 11
10.5 10.5 11 11
12.9 12.9 13 12
9.9 9.9 10 12
11.1 11.1 11 12
10.6 10.6 11 12
11.7 11.7 12 12
11 11 11 12
10 10 10 12
8.6 8.6 9 12
14 14 14 12
6.1 6.1 6 12
10.7 10.7 11 12
12 12 12 12
12.2 12.2 12 12
5.3 5.3 5 12
10.4 10.4 10 12
13 13 13 12
8.2 8.2 8 12
8.2 8.2 8 12
15 15 15 12
7.3 7.3 7 12
5.9 5.9 6 12
6.9 6.9 7 12
4.2 4.2 4 12
6.8 6.8 7 12
5 5 5 12
12 12 12 12
12.5 12.5 13 13
4.6 4.6 5 13
7.5 7.5 8 13
6.7 6.7 7 13
7.6 7.6 8 13
4.7 4.7 5 13
5.4 5.4 5 13
5.3 5.3 5 13
13.2 13.2 13 13
6 6 6 13
6 6 6 13
7.5 7.5 8 13
9.9 9.9 10 13
3.3 3.3 3 13
6 6 6 13
5.1 5.1 5 13
6.5 6.5 7 13
5.6 5.6 6 13
8.4 8.4 8 13
12.5 12.5 13 14
4.8 4.8 5 14
10.9 10.9 11 14
8.1 8.1 8 14
10.8 10.8 11 14
12.2 12.2 12 14
6.8 6.8 7 14
8.3 8.3 8 14
8.2 8.2 8 14
7.9 7.9 8 14
6.7 6.7 7 14
13.4 13.4 13 14
14.9 14.9 15 14
10 10 10 14
12 12 12 15
9.4 9.4 9 15
6.1 6.1 6 15
7.5 7.5 8 15
3 3 3 15
5.9 5.9 6 16
7.1 7.1 7 16
6.9 6.9 7 17
6.1 6.1 6 17
9.2 9.2 9 18
9 9 9 18
Question 3
Standard Deviation
[mph]
2.934581593
Question 4
Power Distribution
Wind Speed Probability “Power Potential” Normalized Potential
[mph] [-] [-] [-]
1 0 0 0
2 0 0 0
3 0.008333333 0.225 0.000245873
4 0.025 1.6 0.001748432
5 0.1 12.5 0.013659626
6 0.119444444 25.8 0.028193469
7 0.122222222 41.92222222 0.045811351
8 0.147222222 75.37777778 0.082370583
9 0.097222222 70.875 0.077450082
10 0.113888889 113.8888889 0.124454374
11 0.072222222 96.12777778 0.105045562
12 0.072222222 124.8 0.13637771
13 0.052777778 115.9527778 0.12670973
14 0.038888889 106.7111111 0.116610713
15 0.013888889 46.875 0.051223599
16 0.005555556 22.75555556 0.024866591
17 0.005555556 27.29444444 0.029826553
18 0.005555556 32.4 0.035405752
19 0 0 0
20 0 0 0
21 0 0 0
22 0 0 0
23 0 0 0
24 0 0 0
25 0 0 0
26 0 0 0
27 0 0 0
28 0 0 0
29 0 0 0
30 0 0 0
Total “Power Potential”: 915.1055556 1
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