1

Wind Energy Systems MASE 5705

Spring 2014, L3+ L4

Review of L1 + L2

2

3

(2.75) p. 63 , Second Edition

4

(2.74. p/ 63, 2nd edition

Dynamic

Power

5

Wind power

Grid

Mechanical – Electrical Conversion Chain Efficiency

(based on “Wind Turbines” Erich Hau, Springer, 2006)

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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.

We wish to calculate the wind-energy

production potential.

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Two Widely Used Methods

1. Tabular approach or Method of Bins

2. Closed-form expressions for wind-

speed probability distribution

functions.

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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 and Raleigh distribution functions.

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1. As for the coverage of probability and

statistics, we follow the text; the

emphasis is on applications and not on

precision.

2. 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.

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For the present assume:

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Given N independent measurements of wind

speed: Ui, i= 1 , …, N

Mean wind speed = average wind speed = expected wind speed

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……..……..(2.39)

p.54

(A)

(B)

13

……..……..(2.40)

p.54

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.

14

Notational Differences (2.20) p. 40

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Total mean Turbulent

or random

16

17

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.)

18

The data were continuously measured every hour for one year ≡ 8760

measurements. The site refers to MOD-2 (Boeing) 91.4 m dia turbine; cut-

in speed=6.25 m/s and cut-out speed=22.5 m/s.)

19 Bin ≡ wind velocity interval, specifications often require 0.5 m/s bins.

A Sample of Database Arranged into Bins

BIN Duration

Min(m/s) Max(m/s) Δ t k (h/y)

1 0 6.25 2147

2 6.25 6.75 416

3 6.75 7.25 440

4 7.25 7.75 458

5 7.75 8.25 468

6 8.25 8.75 470

7 8.75 9.25 466

8 9.25 9.75 453

9 9.75 10.25 435

10 10.25 10.75 410

11 10.75 11.25 381

12 11.25 11.75 349

13 11.75 12.25 314

14 12.25 12.75 278

15 12.75 13.25 242

16 13.25 13.75 208

17 13.75 14.25 175

18* 14.25 >22.4 648

19 >22.4 2

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 (22.5-14.25 = 8.5 m/s)

is not known, we will revisit the

data base.

20

Similarly

21

(A)

(C)

N–1 j=1 N j=1 (2.47) p. 55

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)

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Experiment:

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24

We also need

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An illustration:

wind speed histogram

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i U i m i f(u i ) F(U i )

1 0 0 0 0

2 1 0 0 0

3 2 15 0.071 0.071

4 3 42 0.199 0.270

5 4 76 0.360 0.630

6 5 51 0.242 0.872

7 6 27 0.128 1.000

N= 211

(ref: Wind Energy Systems, G. L. Johnson, Prentice Hill, NJ, 1985, p. 53.)

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Formally,

Wind speed variations represent continuous

random functions, that is, continuous

mathematical functions. Bypassing

mathematical details we accept the next set

of definitions:

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Cumulative distribution function

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(2.56, p.58)

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U

f(U)

Probability Density Function f(U)

U

F(U)

Cumulative Distribution Function F(U)

1

0

0

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

See page 58, 2nd Ed.

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)

Power

Density W/m2 Speed m/s (mph)

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)

4.4-5.1 5.1-5.8 200-300 5.6-6.4

2 100-150 (9.8-11.5) 160-240 (11.4-13.2) (12 5-14.3)

5.1-5.6 5.8-6.5 300-400 6.4-7.0

3 150-200 (11.5-12.5) 240-320 (13.2-14.6) (14.3-15.7)

5.6-6.0 6.5-7.0 400-500 7.0-7.5

4 200-250 (12.5-13.4) 320-400 (14.6-15.7) (15.7-16.8)

6.0-6.4 7.0-7.4 500-600 7.5-8.0

5 250-300 (13.4-14.3) 400-480 (15.7-16.6) (16.8-17.8)

6.4-7.0 7.4-8.2 600-800 8.0-8.8

6 300-400 (14.3-15.7) 480-640 (16.6-18.3) (17.8-19.7)

7.0-9.4 8.2-11.0 800-2000 8.8-11.9

7 400-1000 (15.7-21.1) 640-1600 (18.3-24.7) (19.7-26.6)

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Wind Data Distributed by Bins for Hypothetical

Site (average wind speed 5.57 m/s or 12.2 mph)

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* fi = observed bin probability or probability density function for the i-th bin

(fi is also referred to as frequency)

BIN hi fi (0bserved)

Ui

Min(m/s) Max(m/s) hours/year (pct) Bin Avg.

Speed Σfiui

hi/8760 (m/s) 1 0 0 80 0.91% 0 0.0000 2 0 1 204 2.33% 0.5 0.0116

3 1 2 496 5.66% 1.5 0.0849 4 2 3 806 9.20% 2.5 0.2300 5 3 4 1211 13.82% 3.5 0.4838

6 4 5 1254 14.32% 4.5 0.6442 7 5 6 1246 14.22% 5.5 0.7823 8 6 7 1027 11.72% 6.5 0.7620 9 7 8 709 8.09% 7.5 0.6070

10 8 9 549 6.27% 8.5 0.5327 11 9 10 443 5.06% 9.5 0.4804 12 10 11 328 3.74% 10.5 0.3932

13 11 12 221 2.52% 11.5 0.2901 14 12 13 124 1.42% 12.5 0.1769 15 13 14 60 0.68% 13.5 0.0925

16 14 No upper

bond 2 0.02% -

N=8760 5.572

Wind Speed

Ū=

34

Bin

Pro

babili

ty f

(Ui)

Use of the closed–form

expressions for probability

distribution functions of wind

speeds.

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By definition (2.60)(2.61) p. 59, 2nd Ed.

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Weibull

Observed wind speed distributions for three New England locations and

the Weibull with the parameters k = 1.8 and c = 11. 5 mph

(From: Atmospheric Turbulence, H. Panofsky and J. Dutton, Wiley, 1984)

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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)

Pro

babili

ty D

ensity f

unction

Pro

babili

ty D

istr

ibution f

unction

We accept: (2.53) p. 58, 2nd Ed.

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(2.54) p. 58, 2nd Ed.

An example

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Repeat

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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 .

a) What is the average velocity at this site?

b) Estimate the number of hours per year that the wind

speed will be between 5.5 and 7.5 m/s during the year.

c) Estimate the number of hours per year that the wind

speed is above 16 m/s.

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Pr. B (2.6) p 618, (pp. 58-59)

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(2.61) and (2.62) pp. 59-60

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54

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56

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58

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

59

60

k and c

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Cf(

u)

For c=1

U/C = 0.7 = U

Cf(U) = 0.858 = f(U)

for c=10

U = 7 m/s

f(U) = 0.0858

Finding Weibull Parameters k and c

1. Least squares fit for the observed values

of bin frequency fi (e.g. subroutine:

lsqcurvefit), (not in the text.)

2. Analytical–Empirical Approach

(pp. 60-61)

3. Graphical Approach (p.61)

4. Closed-form approach

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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.)

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BIN hours/year 0bserved bin Bin Avg. Speed (Ui) Σfiui Fi

No. Min(m/s) Max(m/s) frequency fi (m/s) Cumulative Frequency

1 0 0 80 0.91% 0 0.0000 0.009

2 0 1 204 2.33% 0.5 0.0116 0.032

3 1 2 496 5.66% 1.5 0.0849 0.089

4 2 3 806 9.20% 2.5 0.2300 0.181

5 3 4 1211 13.82% 3.5 0.4838 0.319

6 4 5 1254 14.32% 4.5 0.6442 0.462

7 5 6 1246 14.22% 5.5 0.7823 0.605

8 6 7 1027 11.72% 6.5 0.7620 0.722

9 7 8 709 8.09% 7.5 0.6070 0.803

10 8 9 549 6.27% 8.5 0.5327 0.866

11 9 10 443 5.06% 9.5 0.4804 0.916

12 10 11 328 3.74% 10.5 0.3932 0.954

13 11 12 221 2.52% 11.5 0.2901 0.979

14 12 13 124 1.42% 12.5 0.1769 0.993

15 13 14 60 0.68% 13.5 0.0925 1.000

16 14 14> 2 0.02% 14 0.000 1.000Total no. of

hours or

measurements

N= 8760

5.571804 1.000

Wind Speed

Least-Squares Approach

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(2.66) p. 60 2nd Ed.

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Analytical-Empirical Approach

pp. 59-60, 2nd Edition

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Empirical-Analytical Method

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Hypothetical Site

Parameters k c

least squares 2.117 6.293

empirical-analytical 2.066 6.291

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The next graph shows how Weibull fi from these two methods and

also from Rayleigh (k=2) compare with the data or observed values

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.05

0.1

0.15

BIN Number

BIN

Fre

qu

en

cy

data

Weibull(lsq)

Weibull(empirical)

Homework

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.

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Homework

77

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K

f(k)

Graphical Method (Hints)

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y = a x + b Finally,

k = a and c = exp (-b/k)

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): 132-145.

(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.

79

Please submit all the homeworks and

projects in two separate files.

HW file: Weekly on Tuesdays

Project file: April 26, 2012

80

81

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Set or

Set

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(2.62), p. 60, 2nd Edition

84 (2.64) p. 60 2nd Edition

85

86

Most probable or most frequent wind speed

and

Energy Pattern Factor (EPF)

and

The closed-form expression for the Weibull

Case

87

88

89

f(u)

0.854

c=1

Energy Pattern Factor

90

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(2.69) p. 61 2nd Edition

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Table 2.4 p. 61, 2nnd

Edition

Raleigh distribution

A special case of Weibull

Distribution with

shape factor k = 2

# 2.4.3.3 (pp. 58 – 61)

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(2.61) p. 59, 2nd Edition

Rayleigh Distribution

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(2.59) p. 59, 2nd Edition

(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.

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Pr. 2.7. , P. 618

97

For Raleigh distribution

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(2.59) p. 59, 2nd Edition

99

100

Application of Raleigh and

Weibull to a specific database

101

Consider the earlier-treated database

for “Hypothetical Site.”

Wind Data Distributed by Bins for Hypothetical

Site (average wind speed 5.57 m/s or 12.2 mph)

102

BIN hiFrequency

(0bserved)Ui

Min(m/s) Max(m/s) hours/year (pct)Bin Avg.

Speed Σpiui

hi/8760 (m/s)

1 0 0 80 0.91% 0 0.0000

2 0 1 204 2.33% 0.5 0.0116

3 1 2 496 5.66% 1.5 0.0849

4 2 3 806 9.20% 2.5 0.2300

5 3 4 1211 13.82% 3.5 0.4838

6 4 5 1254 14.32% 4.5 0.6442

7 5 6 1246 14.22% 5.5 0.7823

8 6 7 1027 11.72% 6.5 0.7620

9 7 8 709 8.09% 7.5 0.6070

10 8 9 549 6.27% 8.5 0.5327

11 9 10 443 5.06% 9.5 0.4804

12 10 11 328 3.74% 10.5 0.3932

13 11 12 221 2.52% 11.5 0.2901

14 12 13 124 1.42% 12.5 0.1769

15 13 14 60 0.68% 13.5 0.0925

16 14No upper

bond2 0.02% -

8760 5.572

Wind Speed

103

= 0.136

104

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.05

0.1

0.15

BIN Number

BIN

Fre

qu

en

cy

data

Rayleigh

105

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.05

0.1

0.15

BIN

BIN

Fre

qu

en

cy

data

Weibull(lsq)

Rayleigh

Weibull(empirical)

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.

106

Limitations of Closed-form expressions

for wind-speed distributions.

107

Ui

f(U

i)

1. As expected Weibull is better than Rayleigh.

2. The Weibull approximation is not satisfactory.

(Wind Energy Systems, Ibid.)

108

f(U

i) (from: Wind Energy

Systems, Ibid.)

The data shows double-peaked behavior The approximation is not

satisfactory.

109

f(U

i)

(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.

Double-Peaked bi-Weibull Distribution

110

Conclusions 1. 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.

2. 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.

3. It is always advantageous to explore the

feasibility of finding such closed-form

expressions.

111