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Chapter 2—Wind Characteristics 2–49 The monthly mean wind speeds at Dodge City for 1958 were 11.78, 13.66, 11.16, 12.94, 12.10, 13.47, 12.56, 10.86, 13.77, 11.76, 12.44, and 12.55 knots. Find the yearly mean (assuming all months have the same number of days), the standard deviation, and the wind speeds one and two standard deviations from the mean. What monthly mean will be exceeded 95 % of the time? What is the 90 % confidence interval? By a hand held calculator, we find ū =12.42 σ =0.94 The wind speeds one standard deviation from the mean are 11.48 and 13.36 knots, while the speeds two standard deviations from the mean are 10.54 and 14.30 knots. From Table 2.4 we see that F (u) = 0.05 (indicating 95 % of the values are larger) for q = -1.645. From Eq. 65 we find u =12.42 + (−1.645)(0.94) = 10.87 knots Based on this one year’s data we can say that the monthly mean wind speed at Dodge City should exceed 10.87 knots (5.59 m/s) for 95 % of all months. The 90 % confidence interval is given by the interval 12.42 ± 1.645(0.94) or between 10.87 and 13.97 knots. We would expect from this analysis that 9 out of 10 monthly means would be in this interval. In examining the original data set, we find that only one month out of 12 is outside the interval, and it is just barely outside. This type of result is rather typical with such small data sets. If we considered a much larger data set such as a 40 year period with 480 monthly means, then we could expect approximately 48 months to actually fall outside this 90 % confidence interval. We might now ask ourselves how confident we are in the results of this example. After all, only one year’s wind data were examined. Perhaps we picked an unusual year with mean and standard deviation far removed from their long term averages. We need to somehow specify the confidence we have in such a result. Justus, Mani and Mikhail examined long term wind data[14] for 40 locations in the United States, including Alaska, Hawaii, and Wake Island. All sites had ten or more years of data from a fixed anemometer location and a long term mean wind speed of 5 m/s or greater. They found that monthly and yearly mean speeds are distributed very closely to a normal or Gaussian distribution, as was mentioned earlier. The monthly means were distributed around the long term measured monthly mean ū m with an average standard deviation of 0.098ū m where ū m is the mean wind speed for a given month of the year, e.g. all the April average wind speeds are averaged over the entire period of observation to get a long term average for that month. For a normal distribution the 90 % confidence interval would be, using Table 2.5, ū m ± 1.645(0.098)ū m or the interval between 0.84ū m and 1.16ū m . We can therefore say that we have 90 % confidence that a measured monthly mean speed will fall in the interval 0.84ū m to 1.16ū m . If we say that each measured Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001
Chapter 2—Wind Characteristics 2–50 monthly mean lies in this interval, we will be correct 90 times out of 100, and wrong 10 times . The above argument applies on the average. That is, it is valid for sites with an average standard deviation. We cannot be as confident of sites with more variable winds and hence higher standard deviations. Since we do not know the standard deviation of the wind speed at the candidate site, we have to allow for the possibility of it being a larger number. According to Justus[14], 90 % of all observed monthly standard deviations were less than 0.145ū m .This is the 90 percentile level. The appropriate interval is now ū m ± 1.645(0.145)ū m or the interval between 0.76ū m and 1.24ū m . That is, any single monthly mean speed at this highly variable site will fall within the interval 0.76ū m and 1.24ū m with 90 % confidence. The converse is also true. That is, if we designate the mean for one month as ū m1 , the unknown long term monthly mean ū m will fall within the interval 0.76ū m1 and 1.24ū m1 with 90 % confidence. We shall try to clarify this statement with Fig. 22. Figure 22: Confidence intervals for two measured monthly means, ū m1 and ū m2 . Suppose that we measure the average monthly wind speed for April during two successive years and call these values ū m1 and ū m2 . These are shown in Fig. 22 along with the intervals 0.76ū m1 to 1.24ū m1 ,0.76ū m2 to 1.24ū m2 , and 0.76ū m to 1.24ū m . The confidence interval centered on ū m contains 90 % of measured means for individual months. The mean speed ū m1 is below ū m but its confidence interval includes ū m . The mean speed ū m2 is outside the confidence interval for ū m and the reverse is also true. If we say that the true mean ū m is between 0.76ū m2 and 1.24ū m2 , we will be wrong. But since only 10 % of the individual monthly means are outside the confidence interval for ū m ,wewillbewrongonly10%ofthe time. Therefore we can say with 90 % confidence that ū m lies in the range of 0.76ū mi to 1.24ū mi where ū mi is some measured monthly mean. Example Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001
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