1. Wind Energy Systems MASE 5705
Spring 2017, Feb. 1,6: L5 + L6

2. 1. Key points of L3 +L4
2a. WT output estimation for ideal conditions and for Weibull- distribution
2b. HW2
3. Bin Width, and Calculation of bin frequency or probability and the corresponding pdf
4. Estimation of AEO* of a proposed site
5. AEO Estimate of a Specific WT in a proposed site
6. Review of Ch. 2 through applications to a realistic database (Mod MW WT)
7. Comments on HW 1 involving graphical solution of Weibull Parameters
HW. 4
*AEO ≡ Annual Energy Output (kWh/y)

3. 1. Key points of L3 + L4

4. Weibull Distribution p. 60 (2.60-61)

5. 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 (Lect. 3-4, pp ) (Lect. 5-6, pp )

6. p. 60 (2.66)
Eq. (2.66) page 60, 2nd edition

7. Energy Pattern Factor
(2.69) p. 61
Eq. (2.69), p. 61, 2nd Edition

8. Eq. (2.69), p. 61, 2nd Edition

9. p. 61
Rayleigh
p. 61 in 2nd Edition

10. Variation of parameters with
Weibull k shape factor
Table 2.4 p. 61 k Ke
1.2
0.837
3.99 2
0.523
1.91 3
0.363
1.40 5
0.229
1.15

11. Why bother about σU/Ū and Ke?
σU/Ū = Turbulence intensity ( L7 + L8, ) = a measure of turbulence impact on 1) extracted power from a proposed site, 2) random loads of WT Ke = Energy pattern factor = very useful in calculating annual energy Output (AEO) of a proposed site (an example follows)

12. Rayleigh distribution A special case of Weibull Distribution with shape factor k = 2 # (p. 59)

13. Eqs. (2.59), (2.66), pp , 2nd edition

14. Rayleigh Distribution
( ) p. 59
Eqs. ( ) p. 59, 2nd Edition

15. 2a. WT Output Estimation for Ideal Conditions and for Weibull Distribution

16. PROJECT #1 Spring 2017 (Due Feb. 20) A
PROJECT #1 Spring (Due Feb. 20)   A. For each of the two data bases for Dodge City and for Kansas, compute the following from the data only: 1.) Mean wind velocity, U-bar 2.) Standard deviation of wind velocity, U 3.) Mean of the wind velocity cubed (NOT the cube of the mean!). 4.) k and c for Wiebull distribution using the approximate analytical method. 5.) The Wiebull estimation of the mean of velocity cubed. B. Assume that the first bin (1 m/sec) has a range of 0 to 1.5 m/sec. Assume that bins 2-27 have a range of V-mid +/- 0.5 m/sec, and assume that bin 28 has a range of 27.5 m/sec to infinity. With that assumption, for each of the two data sets: 1.) Compare the fi(U) of each bin with the prediction from Wiebull. 2.) With your U-bar, U, k, and c for the Wiebull distribution––and assuming a COP of 40%, a cut-in speed of 2.5 m/sec, and a cut-out speed of m/sec––compute the estimate of power that you will generate in one year at each site.

17. Dodge City Eastern Kansas (m/sec, m)

18. 1. Ideal Conditions:
1a. Maximum WT efficiency (Betz factor)
mechanical . electrical =  = 1 (ideal transmission)
(physics of Betz factor Cp,max ; derivation in ch.3)
1b. Carlin’s one-two-three equation (2.82), p. 64
1c. Wind speed that gives maximum power for
Weibull distribution.

19. p. 63 (2.75)
2nd Edition, (2.75), page 63
1/2 𝜌𝐴(𝑈 ̅ )^3

21. U2= U3 U4 U1=Ū (1) (2) (3) (4) U1=Ū U2= U3 U4 = far wake velocity (4)
U2= U3 U4
U1=Ū
(1)
(2)
(3)
(4)
U1=Ū
U2= U3
U4 = far wake
velocity
Force on disk
Force on fluid
(4)

22. In Chapter 3, we will show that for optimum conditions

23. = 2A2

24. Protor = Pextracted by the rotor = Pinput – P“output”
lost in far wake
Protor = P1 – P4

26. Ideal Power

27. Power =

28. An Ideal Machine
η (2.72) P. 63

31. Carlin’s 1-2-3 Equation (2.69, p. 61)

32. Carlin’s One-Two-Three Equation (2.82 p. 64)
(2.5.10)
p. 61

33. B.2.8 Annual Power Estimation Problem-Betz Type Machine (p. 618)
Estimate the annual production of a 12 m diameter horizontal axial wind turbine which is operating at the standard atmospheric conditions ( = kg/m3 ) in a 8 m/s average wind speed regime. You are to assume that the site wind–speed probability density is given by the Rayleigh density distribution.

34. Pr. 2.8 , p. 618
(2.82), p. 64
(2.5.10)
p. 61

35. = 40.14 x 24 x 365 = 40.14 x 8760 = 352,000 kWh/y =352 MW/year year*
kWh
year
= x 24 x 365
= x 8760
= 352,000 kWh/y =352 MW/year

36. 2b. Homework 2

37. 2b. Homework (not 2017) Find:
Use:

38. Hints
(not really!)
and solve:
1/k

39. k should be 1/k (show that it is true)

40. In other words:

41. 3. Bin Width, and Calculation of Bin probability and the corresponding pdf

42. The probability density function f(U) has the following fundamental properties

45. 4. Estimation of Average Energy Output (AEO) of a proposed site

46. Estimation of AEO (kWh/y) of a proposed site with wind speed Bin Data or Wind Speed Histogram
Bin width (e.g. 0.5 m/s)
For each bin, Umax, Umin and number of hours in that bin are measured.

47. Wind Data Distribution for Hypothetical Site
0.9998
We used this database earlier, in estimating Ū for the site.
Now, we use the same database for site AEO estimation, neglecting bin 16 ( = 1.15 kg/m3). (From: Energy Systems Engineering, F.M. Vanek and L.D. Albright, McGraw Hill, 2008, with permission)

48. Method of Bins for site AEO estimation
We will compute available power (η = 1).

50. Remarks
The preceding method of bins is the most reliable and it is always applicable. This method, however, requires a comprehensive database involving hourly measurements for at least one year for each bin: Umax, Umin and number of hours. For several sites only Ū is available.

51. When only Ū is available, the Rayleigh approximation of the number of hours in each bin is found to be satisfactory for several sites. In this case, we create our own bins and estimate the number of hours for each bin from Rayleigh.

52. 2nd Edition page 61

53. Let us apply these two approximate methods for the same earlier-treated hypothetical site (Ū = 5.57 m/s) .

54. The analyst-created Bin Data for Hypothetical site when only Ū = 5
The analyst-created Bin Data for Hypothetical site when only Ū = 5.57 m/s is known,  = 1.15 kg/m3.

55. AEO Estimation for Hypothetical Site,

56. Explanation
Consider bin 6: Umin = 4 m/s , Umax = 5 m/s, Ū6 = 4.5 m/s Calculate the probability that the wind-speed measurements belong to this bin, and then the number of hours in that bin.

58. * More accurate calculations follow
Comparison of Annual Output from Observed Bin Data and from Rayleigh-Estimated Bin Hours
Observed (kWh/m 2 )
Estimated (kWh/m 1
0.0 3
1.0
1.2 4
7.2
8.5 5
29.9
27.9 6
65.7*
62.4 7
119.2
108.2 8
162.2
155.9 9
172.0
194.2 10
193.9
214.1 11
218.4
212.2 12
218.3
191.2 13
193.3
158.0 14
139.3
120.5 15
84.9
85.1
Total
1605.3
1539.4
* 65.7 = (52.4 x 1254)/1000
(Bin 16 is not included)
Annual Output
BIN
* More accurate calculations follow

59. Repeat : BIN # 1/2 Ū (hr/year) 1 80 2 0.07 204 1.94 496 4 8.98 806 5
Observed Dynamic Power
Observed
Duration
1/2 Ū 3
(hr/year) 1 80 2
0.07
204
1.94
496 4
8.98
806 5
24.65
1211 6
52.4
1254 7
95.67
1246 8
157.91
1027 9
242.58
709 10
353.12
549 11
492.99
443 12
665.63
328 13
874.5
221 14
124 15 60 16
8760

60. Observed and Estimated AEO of Hypothetical Site

61. AEO (kWh/m2)
Bin Number

64. Site AEO (kWh/m2):
Observed = 1605
Rayleigh-Estimated = 1539 (4.1%)
EPF-based = 1662 (3.6%)
For this hypothetical site, these two approximations provide excellent results. After all, yearly wind-speed averages show ±10% variations.

65. 5. AEO estimate of a specific WT in a proposed site

67. 5b. For a proposed site for which only Ū is
5b. For a proposed site for which only Ū is measured or known and the number of hours for each bin is estimated from Rayleigh distribution. (Homework 3)

68. A specific WT ≡ the power curve is known
(the most important officially issued document for that WT)

70. Measurements of site Bin Data and WT power
output data for Mod MW WT (300 ft diameter) *
* Reasons for such a wide bin width are not known. Lower values are more likely than higher values. Take Ū18= m/s

71. (Measured) Power Curve for Mod-2 2. 5 MW HAWT (Uci = 6
(Measured) Power Curve for Mod MW HAWT (Uci = 6.25 m/s , Uco = 22.4 m/s)

72. (1980) The 2.5 MW Mod-2 HAWT, the second generation turbine in the large-scale segment of the Federal Wind Energy Program.

73. What is Efficiency at 2.5 MW?
U = 14m/sec, ρ = 1.225kg/m3, D = 91.4m (1/2)(1.225)(π/4)(91.4)2(14)3 = 11,000,000 watts = 11,000 KW = 11.0 MW COP = 2.5/11.0 = 23% η = (27/16)(.22) = 0.38

74. Power Curve of Wind Turbine
Measured power output versus wind speed is referred to as power curve. “The power curve is a wind turbine’s official certificate of performance, which has to be guaranteed by manufacturer. This is why the accurate description and confirmation of the power curve is of special significance…”

75. “Wind turbines having a power curve based only on theoretical calculations are sold only in exceptional cases. As a rule, the power curve of all commercial wind turbines is measured and certified by independent institutions ...”
(from: Wind Energy, Erich Hau, Springer 2006.)

76. Ur
Uci
Uco

77. “ The development of German 3-MW Growian 100 m dia HAWT in 1982 represented the greatest technology leap of the times, as well as the highest technological risks. … It never operated satisfactorily and was dismantled after only limited testing time.”

78. AEO Estimate of a Specific WT
Method of Bins in a proposed site
Rayleigh Distribution (only Ū is known)

79. AEO of a Specific WT in a Proposed Site
AEO ≡ Annual Energy Output Specific WT ≡ WT with a specific power curve, and Uci and Uco are known Proposed Site ≡ Wind-speed histogram or bin data (Umax, Umin and number of hours in each bin) is known

80. Power Curve for a hypothetical case
(for illustration)
Power Curve for a 1.5 MW WT
Uci = 3m/s,
Uco=21 m/s,
Dia = D= 69 m
 (hub height) = 1.15 kg/m3
Power Output (kW)

81. AEO of a specific 1.5 MW WT in a proposed site (Ū = 8.4 m/s)
* See next figure

82. Power Curve for 1.5-MW turbine for wind speeds from 0 to 25 m/s

83. Method of Bins

84. Energy-Extraction Efficiency of the 1.5 MW Wind Turbine
(Depends on density.)

86. Left curve, windspeed histogram
Left curve, windspeed histogram. Center curve, power curve for Carter 25* . Right, energy in wind and hatched part is the energy captured by the wind turbine.
* 9.8 m–dia 25 kW machine
(From : Wind Characteristics , J. Rothatgi and V. Nelson, Alternative Energy Institute , West Texas A and M University, Canyon, TX, 1994)

87. AEO Estimate of a Specific WT in a Site for which only Ū is known.

88. Create your own bin data, and estimate the number of hours in each bin by using the Rayleigh distribution. (Similar exercises have been earlier treated for estimating the site potential when only Ū is known.)

89. Homework (not 2017)
For the preceding 1.5 MW WT with the given power curve in slide 82. (Include a cut-in speed of 3 m/sec and cut-out sped of 21 m/sec. Calculate the AEO for this turbine with Ū = 8.4 m/s, and a Rayleigh wind-speed distribution. Use the bin widths from slide 81.

90. 6. Review of Chapter 2 through applications to a realistic database (Mod-2 2.5 MW WT )

91. Measurements of site Bin Data and WT power
output data for Mod MW WT *
* Reasons for such a wide bin width are not known. Lower values are more likely than higher values. Take Ū18= m/s

92. 8760 bin Ūi Duration Observed 1 3.13 2147 2 6.5 416 3 7 440 4 7.5 458
468 6
8.5
470 9
466
9.5
453 10
435
10.5
410 11
381 12
11.5
349 13
314 14
12.5
278 15
242 16
13.5
208 17
175 18
17.25
648 19 30
8760

95. 7. Comments on HW1 involving graphical solution of Weibull parameters
“Graphical” = Electronically graphical (Using Excel)

96. Text, p. 59 Graphical : log-log plot
“Using this method, a straight line is drawn through a plot of wind speed, U, on the x-axis and F(U) on the y-axis of log-log paper. The slope of the straight line gives k. Then, the intersection of a horizontal line with F(U) = gives an estimate of c on the x-axis.”

97. The first sentence (“Using this method …) is ambiguous.
Details of F(U) = 0.632

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

99. y = m x b
For U = c,
F(U) = F(c) = 1 – 1/e = 0.632

100. Our approach

102. For the Hypothetical Site (Ū = 5.57 m/s , σU = 2.86 m/s)
Weibull Parameter
Least squares
empirical
graphical k
2.1170
2.0663
2.04
6.2925
6.291
6.125

103. Mod-2 2.5MW WT Database (Ū = 8.6655 m/s)
Weibull Parameter
Least squares
empirical
graphical k
2.6603
Not done
2.6763
9.7491 *

104. Homework (not 2017)
We have designed a HAWT and make the following claim: “Our turbine with a blade length of 40 meters achieves its rated output of 1.5 MW in winds of 20 MPH.” ( = 1.15 kg/m3) What is the flaw in this statement? Use appropriate calculations to show the flaw. (Based on energy Systems Engineering, ibid.)