WIND ENERGY SYSTEMS - Cd3wd
WIND ENERGY SYSTEMS - Cd3wd WIND ENERGY SYSTEMS - Cd3wd
Chapter 2—Wind Characteristics 2–57 Service stations will be mentioned briefly here. Its three-cup rotor drives a gear train which momentarily closes a switch after a fixed number of revolutions. The cup speed is proportional to the wind speed so contact is made with the passage of a fixed amount of air by or through the anemometer. The anemometer can be calibrated so the switch makes contact once with every mile of wind that passes it. If the wind is blowing at 60 mi/h, then a mile of wind will pass the anemometer in one minute. The contact anemometer is obviously an averaging device. It gives the average speed of the fastest mile, which will be smaller than the average speed of the fastest 1/10th mile, or any other fraction of a mile, because of the fluctuation of wind speed with time. But one has to stop refining data at some point, and the fastest mile seems to have been that point for structural studies. The procedure is to fit a probability distribution function to the observed high wind speed data. The Weibull distribution function of Eq. 30 could probably be used for this function, except that it tends to zero somewhat too fast. It is convenient to define a new distribution function F e (u) just for the extreme winds. Weather related extreme events, such as floods or extreme winds[11], are usually described in terms of one of two Fisher-Tippett distributions, the Type I or Type II. Thom[23, 24] used the Type II to describe extreme winds in the United States. He prepared a series of maps based on annual extremes, showing the fastest mile for recurrence intervals of 2, 50, and 100 years. Height corrections were made by applying the one-seventh power law. National Weather Service data were used. The ANSI Standard[1] is based on examination of a longer period of record by Simiu[21] and uses the Type I distribution. Only a single map is given, for a recurrence interval of 50 years. Other recurrence intervals are obtained from this map by a multiplying factor. Tables are given for the variation of wind speed with height. Careful study of a large data set collected by Johnson and analyzed by Henry[12] showed that the Type I distribution is superior to the Type II, so the mathematical description for only the Type I will be discussed here. The Fisher-Tippett Type I distribution has the form F e (u) =exp(− exp(−α(x − β))) (68) where F e (u) is the probability of the annual fastest mile of wind speed being less than u. The parameters α and β are characteristics of the site that must be estimated from the observed data. If n period extremes are available, the maximum likelihood estimate of α may be obtained by choosing an initial guess and iterating Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001
Chapter 2—Wind Characteristics 2–58 α i+1 = 1 n∑ x j exp(−α i x j ) j=1 ¯x − n∑ exp(−α i x j ) j=1 until convergence to some value ˜α. The maximum likelihood estimate of β is β =¯x − 0.5772 ˜α (69) (70) If F e (u) = 0.5, then u is the median annual fastest mile. Half the years will have a faster annual extreme mile and half the years will have a slower one. Statistically, the average time of recurrence of speeds greater than this median value will be two years. Similar arguments can be made to develop a general relationship for the mean recurrence interval M r ,whichis which yields M r = 1 1 − F e (u) (71) u = − 1 α ln ( − ln(1 − 1 M r ) + β (72) A mean recurrence interval of 50 years would require F e (u) = 0.98, for example. Example At a given location, the parameters of Eq. 68 are determined to be γ =4andβ =20m/s.What is the mean recurrence interval of a 40 m/s extreme wind speed? From Eq. 68 we find that [ ( ) ] −4 40 F e (40) = exp − =0.939 20 From Eq. 67, the mean recurrence interval is M r = 1 1 − 0.939 =16.4 At this location, a windspeed of 40 m/s would be expected about once every 16 years, on the average. If we have 20 years or more of data, then it is most appropriate to find the distribution of yearly extremes. We use one value for each year and calculate α and β. If we have only Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001
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Chapter 2—Wind Characteristics 2–58<br />
α i+1 =<br />
1<br />
n∑<br />
x j exp(−α i x j )<br />
j=1<br />
¯x − n∑<br />
exp(−α i x j )<br />
j=1<br />
until convergence to some value ˜α. The maximum likelihood estimate of β is<br />
β =¯x − 0.5772<br />
˜α<br />
(69)<br />
(70)<br />
If F e (u) = 0.5, then u is the median annual fastest mile. Half the years will have a faster<br />
annual extreme mile and half the years will have a slower one. Statistically, the average time<br />
of recurrence of speeds greater than this median value will be two years. Similar arguments<br />
can be made to develop a general relationship for the mean recurrence interval M r ,whichis<br />
which yields<br />
M r =<br />
1<br />
1 − F e (u)<br />
(71)<br />
u = − 1 α ln (<br />
− ln(1 − 1 M r<br />
)<br />
+ β (72)<br />
A mean recurrence interval of 50 years would require F e (u) = 0.98, for example.<br />
Example<br />
At a given location, the parameters of Eq. 68 are determined to be γ =4andβ =20m/s.What<br />
is the mean recurrence interval of a 40 m/s extreme wind speed?<br />
From Eq. 68 we find that<br />
[ ( ) ] −4 40<br />
F e (40) = exp − =0.939<br />
20<br />
From Eq. 67, the mean recurrence interval is<br />
M r =<br />
1<br />
1 − 0.939 =16.4<br />
At this location, a windspeed of 40 m/s would be expected about once every 16 years, on the<br />
average.<br />
If we have 20 years or more of data, then it is most appropriate to find the distribution<br />
of yearly extremes. We use one value for each year and calculate α and β. If we have only<br />
Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001