1. Wind Energy Systems MASE 5705
Spring 2017, L3+ L4
Revised

2. Review of L1 + L2

3. (2.75) p. 63 , Second Edition

4. (2.74) p/ 63, 2nd edition
Second Edition (2.74) page 63

5. (based on “Wind Turbines” Erich Hau, Springer, 2006)
Mechanical – Electrical Conversion Chain Efficiency
Dynamic Power
Wind power
Grid
(based on “Wind Turbines” Erich Hau, Springer, 2006)

7. Site potential for Wind Energy Production
We have the database --- a large quantity of data on wind speed and its direction. For example, 8760 hourly measurements for one year. (24 x 365 = 8,760)
We wish to calculate the wind-energy production potential.

8. Two Widely Used Methods
1. Tabular approach or Method of Bins 2. Closed-form expressions for wind- speed probability distribution functions.

9. 1. Elements of probability and statistics
2. Method of bins (Bin means wind-speed interval; typically this interval is held constant. Each bin contains data for that interval)
3. Weibull (way’-bal) and Raleigh (rah’-lee) distribution functions.

10. As for the coverage of probability and statistics, we follow the text; the emphasis is on applications and not on precision.
In the beginning, the concepts of probability may seem abstract. With the solution of problems, our understanding and appreciation of these concepts and their utility get much better.

11. For the present assume:

12. Given N independent measurements of wind speed: Ui, i= 1 , …, N
Mean wind speed = average wind speed = expected wind speed
……..……..(2.39)
p.54

13. ……..……..(2.40)
p.54
(A)
(B)

14. Here after, only Eq. (A) will be used.
However, Eq. (A) is considered a ‘better’ estimator.
(Random Data, Analysis and Measurement Procedures, Bendat, and Piersol, Wiley 3rd Ed., 2000, p. 87.)
Here after, only Eq. (A) will be used.

15. Notational Differences (2.20) p. 40
Total mean Turbulent
or random

18. Wind-speed data are arranged into bins or wind-speed intervals, which typically span 0.5 m/s. ( These procedures are often governed by specifications from ASME, AWEA and such organizations.)

19. A Sample of Database Arranged into Bins
The data were continuously measured every hour for one year ≡ 8760 measurements. The site refers to MOD-2 (Boeing) m diameter turbine; cut-in speed=6.25 m/s and cut-out speed=22.5 m/s.)
BIN
Duration
Min(m/s)
Max(m/s) Δ t k
(h/y) 1
6.25
2147 2
6.75
416 3
7.25
440 4
7.75
458 5
8.25
468 6
8.75
470 7
9.25
466 8
9.75
453 9
10.25
435 10
10.75
410 11
11.25
381 12
11.75
349 13
12.25
314 14
12.75
278 15
13.25
242 16
13.75
208 17
14.25
175
18*
>22.4
648 19
Totals:
8760 = N
Wind Speed
[from: Wind turbine Technology, David A. Spera, editor , 1994, p.222, ASME publication]
*For Bin 18,
The reason for the much larger interval ( = 8.5 m/s) is not known, we will revisit the data base.
Bin ≡ wind velocity interval, specifications often require 0.5 m/s bins.

21. Similarly (for mi = mean bin speed) pp. 54-55, Eqs.(2.40) and (2.47)

22. A descriptive account with illustrations (for mathematical details see Bendat and Piersol, ibid.) 1a) Probability of observing Ui or p(Ui) for the discrete values of Ui 1b) (corresponding) cumulative distribution function F(Ui) 2a) Probability density function f(U) for the continuous values of U. 2b) (corresponding) cumulative distribution function F(U)

23. Experiment:

25. We also need

26. An illustration: wind speed histogram
(ref: Wind Energy Systems, G. L. Johnson, Prentice Hill, NJ, 1985, p. 53.)

27. Formally,

28. Wind speed variations represent continuous random functions, that is, continuous mathematical functions. Bypassing mathematical details we accept the next set of definitions:

29. Cumulative distribution function where x = U’, dummy variable
(2.56, p.58)

30. Probability Density Function f(U)
Probability Density Function f(U) 1
F(U) U
Cumulative Distribution Function F(U)

31. ( ) pp
See page 58, 2nd Ed.
The probability density function f(U) has the following fundamental properties

32. Wind Power Classes (for later reference ), p
Wind Power Classes (for later reference ), p. 67 (Class 4 or greater are suitable for electrical utility applications)
Hub Height
10 m (33 ft)
30 m (98 ft)
50 m (164 ft)
Wind Power class
Power Density W/m2
Speed m/s (mph) 1
0-100
0-4.4 (0-9.8)
0-160
0-5.1 (0-11.4)
0-200
0-5.6 (0-12.5) 2
( )
( )
( ) 3
( )
( )
( ) 4
( )
( )
( ) 5
( )
( )
( ) 6
( )
( ) 7
( )
( )
( )

33. (fi is also referred to as frequency)
Wind Data Distributed by Bins for Hypothetical Site (average wind speed 5.57 m/s or 12.2 mph)
BIN hi fi
(0bserved) Ui
Min(m/s)
Max(m/s)
hours/year
(pct)
Bin Avg.
Speed
Σfiui
hi/8760
(m/s) 1 80
0.91%
0.0000 2
204
2.33%
0.5
0.0116 3
496
5.66%
1.5
0.0849 4
806
9.20%
2.5
0.2300 5
1211
13.82%
3.5
0.4838 6
1254
14.32%
4.5
0.6442 7
1246
14.22%
5.5
0.7823 8
1027
11.72%
6.5
0.7620 9
709
8.09%
7.5
0.6070 10
549
6.27%
8.5
0.5327 11
443
5.06%
9.5
0.4804 12
328
3.74%
10.5
0.3932 13
221
2.52%
11.5
0.2901 14
124
1.42%
12.5
0.1769 15 60
0.68%
13.5
0.0925 16
No upper
bond
0.02% -
N=8760
5.572
Wind Speed Ū=
* fi = observed bin probability or probability density function for the i-th bin
(fi is also referred to as frequency)

34. Bin Probability f(Ui)

35. Wind and Power Duration Curves
2nd Edition pages 55 and 56 After filling up bins, one can plot wind speed versus the number of hours the wind will be greater than or equal to that speed. Then, based on the power generation curve (including cut-in and cut-out speeds), one can plot wind turbine power versus the number of hours the wind turbine will exceed that power output.

36. Wind Duration Curve

37. Power Duration Curve

38. CAPACITY FACTOR
The capacity factor of a wind turbine at a given site is the average power divided by the rated power. Thus, it is an measure of how often the turbine is operating at its design point. CF = PW/PR 2nd edition, (2.73), p. 63

39. Use of the closed–form expressions for probability distribution functions of wind speeds.

41. The Weibull function allows two free variables.

43. By Definition (2.60)(2.61) p. 59, 2nd Ed.
For Rayleigh, k=2 and c2 = 4Ū2/π

44. Weibull
Observed wind speed distributions for three New England locations and the Weibull with the parameters k = 1.8 and c = mph
(From: Atmospheric Turbulence, H. Panofsky and J. Dutton, Wiley, 1984)

45. Probability Density function
Probability Distribution function
Wind speed U m/s
Approximation of the measured wind distribution on the island of Sylt by Weibull functions. (From: Wind Turbines, by Erich Hau, Springer, 2006)

46. We accept: (2.53) p. 58, 2nd Ed where x = U for mean velocity.

47. (2.54) p. 58
(2.54) p. 58, 2nd Ed.

48. An example

50. Repeat

52. = 0.89
= – = 0.89

53. This means the hours of operation are: 0. 89 x 8760 = 7
This means the hours of operation are: 0.89 x 8760 = 7.8 hours If cut-out speed raised to 35 m/sec, extra hours of operation 25 < U < 35 are: P(35) – P(25) = – = , 8760 x = 40 minutes per year of extra running time. Not worth it.

54. Weibull Distribution Calculations (B2.6, p.618)
From an analysis of wind speed data (hourly interval average, taken over a one year period), the Weibull parameters are determined to be c = 6 m/s and k = 1.8 .
What is the average velocity at this site?
Estimate the number of hours per year that the wind speed will be between 5.5 and 7.5 m/s during the year.
Estimate the number of hours per year that the wind speed is above 16 m/s.

55. Pr. B (2.6) p 618, (pp )
(2.53) p. (2.53) (2.61) 2. (2.61) p. 59

56. ( ) pp
(2.61) and (2.62) pp
Γ(n) =(n–1)!

63. Revisiting Weibull (2.60) (2.61), p. 59, 2nd Ed.

64. function.
function

65. k and c

69. Cf(u) For c=1 U/C = 0.7 = U Cf(U) = 0.858 = f(U) for c=10 U = 7 m/s

70. Finding Weibull Parameters k and c
Least squares fit for the observed values of bin frequency fi (e.g. subroutine: lsqcurvefit), (not in the text.)
Analytical–Empirical Approach (pp )
Graphical Approach (p.61)
Closed-form approach

71. Although, not included in the text, the least-squares-fit approach is very powerful and perhaps the best. We will first apply this approach and then take up the other three. ( The database refers to the earlier-treated hypothetical site.)

73. Least-Squares Approach
(2.66) p. 60 2nd Ed.

76. Analytical-Empirical Approach pp. 59-60, 2nd Edition

77. Empirical-Analytical Method

78. empirical-analytical
Hypothetical Site
Parameters k c
least squares
2.117
6.293
empirical-analytical
2.066
6.291
The next graph shows how Weibull fi from these two methods and also from Rayleigh (k=2) compare with the data or observed values

79. fi

80. Homework (not in 2017)
For the hypothetical site, study: a.) the graphical method to find k and c. b.) use closed-form method with Γ(n): (σU/U)2 = Γ(1+2/k)/Γ2(1+1/k) – 1 c = U/Γ(1+1/k) Hint: plot function and see where = (σU/U)2.

81. Homework

82. Graphical Method (Hints)
y = a x b
Finally,
 k = a and c = exp (-b/k)

83. Recommended Reading Stevens MJM and Smulders PT (1979)
“The estimation of parameters of the Weibull speed distribution for wind energy utilization purposes.” Wind Engineering 3(2):
(If you need a copy, contact Gaonkar or Peters.)
The paper discusses five methods of calculating k and c. This includes empirical and graphical methods but not the least squares method. The article is written in an easy-to-read and descriptive style, and it address some other type of WE issues as well.

84. Graphical Method

86. Set or
Set

87. (2.62), p. 60, 2nd Edition
(2.62) p. 60

88. misprint in book (2.64) p. 60
(2.64) p. 60 2nd Edition

90. Peak of Wiebull distribution.

91. Most probable or most frequent wind speed and Energy Pattern Factor (EPF) The closed-form expression for the Weibull Case

93. f(u)
c=1
0.854

94. Energy Pattern Factor

95. (2.69) p. 61 2nd Edition
(2.69) p. 61

96. (p. 61)
Table 2.4 p. 61, 2nd Edition

97. Raleigh distribution A special case of Weibull Distribution with shape factor k = 2 # (pp. 58 – 61)

98. (2.61) p. 59
(2.61) p. 59, 2nd Edition

99. Rayleigh Distribution
( ) p. 59
(2.59) p. 59, 2nd Edition

100. (p.618) B.2.7 Rayleigh Distribution Calculations
Analysis of time series data for a given site has yielded an average velocity of 6 m/s. It is determined that a Rayleigh wind speed distribution gives a good fit to the wind data. a) Based on a Rayleigh wind speed distribution, estimate the number of hours that the wind speed will be between 9.5 and 10.5 m/s during the year. b) Using a Rayleigh wind speed distribution, estimate the number of hours per year that the wind speed is equal to or above 16 m/s.

101. Pr , P. 618

102. For Raleigh distribution
2 (2.59) p. 59
(2.59) p. 59, 2nd Edition

105. Application of Raleigh and Weibull to a specific database
Consider the earlier-treated database for “Hypothetical Site.”

106. Wind Data Distributed by Bins for Hypothetical Site (average wind speed 5.57 m/s or 12.2 mph)

107. =
= (compare 0.142)

108. fi

109. fi

110. Limitations of Closed-form expressions for wind-speed distributions.
Wind-speed data points are notoriously fickle. Probabilities calculated from Weibull (more so with Rayleigh) exhibit considerable scatter for some cases. We will take a look at this next.

111. f(Ui) Ui
As expected Weibull is better than Rayleigh.
The Weibull approximation is not satisfactory.
(Wind Energy Systems, Ibid.)

112. (from: Wind Energy Systems, Ibid.)
f(Ui)
The data shows double-peaked behavior  The approximation is not satisfactory.

113. (from: Wind Energy Systems, Ibid.)
f(Ui)
(k=2.11
c=11.96)
(from: Wind Energy Systems, Ibid.)
Actually, wind-speed statistics always exhibit some scatter. Thus the above approximation may be satisfactory.

114. Double-Peaked bi-Weibull Distribution

115. CAPACITY FACTOR
The capacity factor of a wind turbine at a given site is the average power divided by the rated power. Thus, it is an measure of how often the turbine is operating at its design point. CF = PW/PR 2nd edition, (2.73), p. 63

116. Conclusions
A generalized statement about the accuracy of the Weibull or Rayleigh probability density functions (pdf) is not possible. Always, Weibull is more accurate than Rayleigh but the latter is much simpler.
If we can find such a Weibull or Rayleigh pdf, the selection of a specific WT or the prediction of its power output becomes dramatically simplified.
It is always advantageous to explore the feasibility of finding such closed-form expressions.