WIND ENERGY SYSTEMS - Cd3wd

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Chapter 2—Wind Characteristics 2–53 r = w∑ (ū ai − ū a )(ū bi − ū b ) i=1 (67) w∑ ∑ w √ (ū ai − ū a ) 2 (ū bi − ū b ) 2 i=1 i=1 In this expression, ū a is the long term mean speed at site a, ū b is the long term mean speed at site b, ū bi is the observed monthly or yearly mean at site b, andw is the number of months or years being examined. We assume that the wind speed at site b is linearly related to the speed at site a. We can then plot each mean speed ū bi versus the corresponding ū ai . We then find the best straight line through this cluster of points by a least squares or linear regression process. The correlation coefficient then describes how closely our data fits this straight line. Its value can range from r =+1tor = −1. At r = +1, the data falls exactly onto a straight line with positive slope, while at r = −1, the data falls exactly onto a straight line with negative slope. At r = 0, the data cannot be approximated at all by a straight line. Example The yearly mean wind speeds for Dodge City, Wichita, and Russell, Kansas are given in Table 2.6. These three stations form a triangle in central Kansas with sides between 150 and 220 km in length. Dodge City is west of Wichita, and Russell is northwest of Wichita and northeast of Dodge City. The triangle includes some of the best land in the world for growing hard red winter wheat. There are few trees outside of the towns and the land is flat to gently rolling in character. Elevation changes of 10 to 20 m/km are rather typical. We would expect a high correlation between the sites, based on climate and topography. There were several anemometer height changes over the 26 year period, so in an attempt to eliminate this bias in the data, all yearly means were extrapolated to a 30 m height using the power law and an average exponent of 0.143. As mentioned earlier in this chapter, this assumption should be acceptable for these long term means. A plot of Russell wind speeds versus Dodge City speeds is given in Fig. 23, and a similar plot for Russell versus Wichita is given in Fig. 24. A least squares fit to the data was computed using a hand held calculator with linear regression and correlation coefficient capability. The equation of the straight line through the points in Fig. 23 is ū b = 0.729ū a + 3.59 with r = 0.596. The corresponding equation for Fig. 24 is ū b = 0.181ū a + 11.74 with r = 0.115. Straight lines are drawn on these figures for these equations. It can be seen from both the figures and the correlation coefficients that Russell winds are poorly predicted by Dodge City winds and are hardly predicted at all by Wichita winds. Years with mean speeds around 14 knots at Wichita display mean speeds between 12.3 and 16 knots at Russell. This gives further support to the conclusion made by Justus that one year’s data with a 90 % confidence interval at a candidate site is just as good as extrapolating from adjacent sites. In examining the data in Table 2.6, one notices some clusters of low wind speed years and high wind speed years. That is, it appears that a low annual mean wind may persist through the following year, and likewise for a high annual mean wind. If this is really the case, we may select a site for a wind turbine in spite of a poor wind year, and then find the performance of Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001
Chapter 2—Wind Characteristics 2–54 Table 2.6 Average yearly speeds corrected to 30 m (100 ft) Year Dodge City Wichita Russell 1948 14.98 14.09 14.14 a 1949 13.53 13.35 13.08 b 1950 12.89 12.77 14.11 1951 13.83 13.47 13.07 1952 14.22 13.97 12.31 1953 15.57 14.45 c 13.37 1954 15.20 14.98 12.83 1955 14.83 15.22 14.89 1956 14.60 14.86 14.85 1957 14.71 13.60 15.38 1958 13.85 13.74 13.63 1959 14.91 15.21 14.88 1960 15.23 14.55 13.67 1961 12.37 13.29 12.45 1962 13.53 13.18 12.99 1963 14.90 12.74 15.20 1964 16.32 13.95 16.04 1965 15.17 13.46 14.71 1966 15.37 13.40 15.17 1967 15.26 13.78 15.20 1968 15.25 14.51 15.70 d 1969 14.19 13.33 14.52 e 1970 14.85 14.11 15.66 1971 15.04 13.97 15.01 1972 14.40 13.84 13.39 1973 15.35 13.87 14.41 a Estimated from 50, 51, 52 data by method of ratios. b Estimated from 50, 51, 52 data by method of ratios. c Estimated from 50, 51, 52 data by method of ratios. d Estimated from 66, 67, 70, 71 data by method of ratios. e Estimated from 66, 67, 70, 71 data by method of ratios. the wind turbine to be poor in the first year after installation. This has economic implications because the cash flow requirements will usually be most critical the first year after installation. The turbine may go ahead and deliver the expected amount of energy over its lifetime, but this does not help the economic situation immediately after installation. Justus[14] examined this question also, for 40 sites in the United States, and concluded that there is about a 60 % probability that one low wind speed month or year will be followed by a similar one. If wind speeds were totally random, then this probability would be 50 %, the same as for two heads in a row when flipping a coin. The probability of an additional low speed month or Wind Energy Systems by Dr. Gary L. Johnson November 20, 2001
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