Wind power statistics and an evaluation of wind energy density

~ )

Pergamon 0960-1481(95)00041-0

Renewabh, Energi,, Vol. 6, No. 5 6, pp. 623 628~ 1995 Copyright ~C; 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960 1481,'95 $9.50+0.0(/

W I N D P O W E R STATISTICS A N D A N E V A L U A T I O N OF WIND ENERGY DENSITY M . J A M I L , S. P A R S A a n d M . M A J I D I Materials and Energy Research Centre (MERC), P.O. Box 14155-4777, Tehran, lran

Abstraet~ln this paper the statistical data of fifty days' wind speed measurements at the MERC-solar site are used to find out the wind energy density and other wind characteristics with the help of the Weibull probability distribution function. It is emphasized that the Weibull and Rayleigh probability functions are useful tools for wind energy density estimation but are not quite appropriate for properly fitting the actual wind data of low mean speed, short-time records. One has to use either the actual wind data (histogram) or look for a better fit by other models of the probability function.

i. INTRODUCTION W h e n c h o o s i n g a n a p p r o p r i a t e installation site for wind energy convertors (WECs) a n d m a k i n g decisions a b o u t a p p r o p r i a t e sizes o f WECs, it is i m p o r t a n t to answer some essential questions before the design a n d erection o f a wind m a c h i n e ( W E C ) to generate electrical power.

where P(v) is the cumulative p r o b a b i l i t y function. By multiplication o f cumulative probability with 8760 (one year has 24 x 365 = 8760 hours) we get the total h o u r s of wind with speeds greater t h a n v*. In the same m a n n e r we can calculate the cumulative probability P(v, < v < v2): P(Vl < v < v2) =

I. W h a t are the characteristics (see Table 3) o f the wind in the p r o p o s e d region? 2. W h a t is the p r o b a b i l i t y of available wind for the given region? 3. H o w m a n y h o u r s in a year can one expect winds o f velocity v*? i.e. wind speeds between 5 a n d a b o u t 20 m/s.) 4. H o w m a n y h o u r s in a year does the region have winds of speeds between Vl a n d v2?

/p(t;)dv

= 1,

;0

p(v) dv =

2. THEORY OF WIND PROBABILITY DISTRIBUTION

The wind speed b e h a v i o u r o f a region is a function o f altitude, season a n d h o u r o f m e a s u r e m e n t s [1,2]. Generally, one year of records a n d weather w a t c h i n g is sufficient to predict the long-term seasonal m e a n wind speed to within a n accuracy o f 10% with a confidence level o f 9 0 % [3], Let v be the wind speed in m/s for a given site at a k n o w n altitude. Since v is a c o n t i n u o u s function of time t, its m e a n for a period T can be derived from the e q u a t i o n

(1)

p(v) dr,

(3)

provided t h a t the probability function p(v) is k n o w n for the region. In the literature two functions called Weibull a n d Rayleigh [1] are widely used for this purpose.

| ('T

where v is the wind speed in m/s. F r o m eq. (1) it follows : P(v > v*) = 1 -

(v)dv,

d~t

To answer such questions one has to k n o w the p r o b ability function of the region, which can be determined with the help of wind observations a n d statistical wind data. In this p a p e r we are going to investigate a m a t h ematical model for this p r o b a b i l i t y function considering all the statistical data of wind for the proposed region. This probability function, say p(v) has to satisfy the following e q u a t i o n :

f

i c2 p

(v) =

T|3ov(t) dt.

(4)

A n d for n different speed records the m e a n is easily given by

(2) 623

M. JAMIL et al.

624

Table 1. The wind speed measurement data at the M ERC-solar site (1994) i

v (m/s)

v,

f

p(vi) (%)

P(v)

p,(v) (%)

pR((vi) (%)

1 2 3 4 5 6 7 8 9 10

0-2 24 4~6 6 8 8-10 10-12 12-14 14~16 16-18 18-20

1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0 17.0 19.0

159 189 247 318 173 85 16 13 0 0

13.25 15.75 20.58 26.50 14.42 7.08 1.33 1.08 0.0 0,0

0.1325000 0.2900000 0.4958333 0.7608333 0.9050000 0.9758333 0.9891666 0.9999999 0.9999999 0.9999999

8.22 15.62 13.72 8.12 3.54 1.18 0.31 0.06 0.01 0.00

4.41 11.04 12.83 10.46 6.53 3.24 1.30 0.42 0.11 0.02

30 25 20 Frequency (or probability %) 15 10 5 0 1

3

5

7

9

11

13

15

17

Wind speed (m/s)

19

PI?I-A~B02

Fig. 1. The histogram of wind speed distribution for the MERC-solar site (14 October 1994).

N

=

~ i --1

N

fv,IY, f,,

(5)

i--I

w h e r e f is the frequency of each observed speed class. F o r the case where the probability function of the region is known, the mean has to be determined from

In the 5th column of Table 1 we have given the percentage probability for each wind class according to the relation p(v~) --

f

N

Zi/

=f-"(i = 1,2, n

"'''

N).

(8)

i=|

@> =

vp(v) dr.

(6)

As an example, we have arranged the wind data from a test region ( M E R C - s o l a r site, 1994) in Table 1. These are the data for 50 days of wind observation, 24 times a day, with the measurements in short statistical form. Figure 1 shows the histogram of the wind frequency distribution of the given wind data. The mean wind speed of the data in Table 1 calculated from eq. (5) is = 5.9 m/s and its standard deviation determined from eq. (7) is a = 3.13 m/s. O" =

[ N 1~ i="5 ~ f/(~)i--)

2

]1/2 "

(7)

The cumulative probability in the 6th column of Table 1 is determined from J

P(v/) = ~ p(v~)

(9)

i=1

W h e r e j ~< i and p(vl) is the probability of each velocity v~ for i = 1 , 2 , . . . , N . The probability of having all wind speeds will be unity, i.e. N

P(vN) = ~ p(v,) = 1.

(10)

i=l

F o r a continuous probability function the standard deviation has to be calculated from

Wind power evaluation

(7 ~ [fix'( V- < V>) 2p( V)d V] ''2.

625

k In (v) - k In (c) = In ( - In [1 - P(v)]).

(ll)

(18)

N o w let us suppose Now, we try to replace the h i s t o g r a m o f Fig. 1 with a c o n t i n u o u s function called a Weibull distribution, which is defined with two p a r a m e t e r s as follows :

(12) where c a n d k are two p a r a m e t e r s called the scale (m/s) a n d form parameters, respectively [1]. F o r k = 2 a n d c = 2 ( v ) / v / ~ = 1.12838 this will have the socalled form o f a Rayleigh distribution :

pR(V) = 2 2 exp

- g ~

x = In (t,), y = In ( - In [1 - P(v)]). In this way eq. (17) will have a linear form y = Ax + B, if A = k a n d B = - k i n ( c ) where C = e x p ( - - B / A ) . Therefore, the Weibull p a r a m e t e r s are related to the p a r a m e t e r s A a n d B of the line. A is the slope of the line a n d B is its intersection point o r d i n a t e with the 3'axis. The values o f x~ and y~ are collected for the data given in Table 2. The analytical calculation of A a n d B is possible with the help o f the least squares m e t h o d (LSQM), The formulae to find out the values of A and B with the help o f the L S Q M are :

(13)

~ ( x , - x)O',-y)

Both eqs (12) a n d (13) are useful relations to describe the actual wind speed distribution. We explain here a m e t h o d for evaluation o f the c a n d k values with the help of collected wind data (Table 1).

A =

p(v)dv =

p(v)dv+

p(v)dv = 1

N

and

B=f--AYc,

(19)

y, (x~- x)-'

i=1

where ,,~ a n d fl are m e a n s o f x j a n d Yi which have to be determined considering the frequency.£ f r o m

l N ~ [ixi --hi= I

Y:--

2.1. The cumulative probability method for c and k evaluation Justus [4] has s h o w n briefly five different m e t h o d s for estimation of the c a n d k parameters. He explained this m e t h o d as least squares fit to the observed distribution in 1978 a n d J o h n s o n has applied this m e t h o d for data collected f r o m K a n s a s City a n d D o d g e City. W e apply this m e t h o d for d a t a collected from the M E R C - s o l a r site. F r o m eq. (1) we have

i=l

and

1 ~ fYi. --hi= 1

p--

(20)

F o r the given data in T a b l e 2 these are worked o u t a n d their values are Yc= 1.57 a n d y = - 0 . 1 4 . W i t h the help o f e q s (19) we can then find k = A = 1.8691 and

(21)

(14)

These are the values o f the Weibull parameters. W i t h these values of the Weibull p a r a m e t e r s we have plotted the two model probability c o n t i n u o u s

or

f

~ p(v) dv = 1 - P(v).

(15)

After i n t e g r a t i o n o f p(v) from v to infinity we get

Table 2. The values of x, and y~ for data collected from the MERC-solar site i 1

This is the cumulative p r o b a b i l i t y function o f a Weibull distribution. To write this in a linear from we take the l o g a r i t h m o f e q . (16) twice, i.e. - - ( ~ ) k = l n [ l - P(v)],

(17)

2 3 4 5 6 7 8 9 10

& = In @3

).i = In ( - I n (1 -P(v3))

0.0

-

1. l 1.61 1.95 2.20 2.40 2.56 2.71 2.83 2.94

- 1.07 -0.38 0.36 0.86 1.31 1.51 2.81 2.81 2.81

1.95

f, 159

189 247 318 173 85 16 13 0 0

626

M. JAMIL et al.

ooo I

1-

•

Yi 0

-3 0.0

•

I 0.5

I 1.0

I 1.5

I 2.0

I 2.5

Xi

3.0 P171-A287614

Fig. 2. The linear relation between x, and Yi for determination of A and B, which are related to c = exp ( - B/A) and k = A.

functions according to relations (12) a n d (13) as claimed to be a p p r o p r i a t e tools of wind b e h a v i o u r description, but as is obvious from Fig. 3, they do not properly fit the actual wind data collected from the M E R C - s o l a r site. This m a y be because of the s h o r t recording period o f wind speeds or, after consultations with Professor J o h n s o n in the U.S.A., due to the nature of these two models which are a p p r o p r i a t e for fitting data for wind speeds with m e a n s of a b o u t 12 m/s or higher. This is also clear from the cases discussed in Ref. [1].

3. EVALUATION OF MEAN WIND ENERGY DENSITY One of the i m p o r t a n t wind characteristics is its m e a n energy density. The i n s t a n t a n e o u s wind energy is related with the third power of its speed according to the relation I

3

P , = ~pAv-

where p = 1.225 kg/m 3 is the air density, which depends o n altitude (air pressure) and temperature.

Frequency (or probability %)

1

3

5

(22)

7 9 11 13 Wind speed (m/s)

Fig. 3. Weibull chart (16 October 1994).

15

17

19

P171-A:~87812

Wind power evaluation

627

Table 3. Wind Characteristics of the MERC-solar site 122.88 W/m 2

1. Wind Energy Density / 2V ,,k Cmec= c'~l + ~)

2. Wind speed of maximum energy carrier

7.66 m/s

3. Most probable wind speed

3.45 m/s

4. Mean wind speed

4.59 m/s

5. Standard deviation of wind speeds

2.60 m/s

MERC-WEE Software package summer 1994 CLASS. FOR Wind speed frequency histogram

Program Class (V i , fi )

Program Jamil yi) and (e, k)

JP. FOR Linear curve

(X i ,

Program Weiray Pw(V) Weibuil PR(v) Rayleigh

WR. FOR

Probability graph

Program Test Evaluation of all wind characteristics

TEST. FOR Wind energy density and others

Fig. 4. MERC-WEE Software package summer 1994. We make it constant for measurements of wind speeds at 10 m height and normal ambient temperature. A is the area. To find the mean wind energy density one has to calculate (P./A) i.e. = 'zp(v 3) = ~p

v3p(v)dv.

(23)

After integration for the Weibull probability model it follows that

where c and k are the Weibull parameters and F is the usual g a m m a function. Other wind characteristics are also summerized in Table 3 using the Weibull model of the wind probability function, where Umpand Vmecare the most probable wind speeds of the region, and maximum energy carrier of wind speed for WECs, respectively. ( v ) and

M. JAMIL et al.

628

are the m e a n s a n d s t a n d a r d deviations of the wind speeds. These are all determined with the help of a software package developed d u r i n g passing m o n t h s at the M E R C c o m p u t e r d e p a r t m e n t . This package is called ' M E R C - W E E Software Package' a n d is created from a c o m b i n a t i o n of four short p r o g r a m s as demo n s t r a t e d in the flow c h a r t (Fig. 4). 4. CONCLUSION The Weibull a n d Rayleigh probability models are useful tools for wind energy density estimation b u t not quite a p p r o p r i a t e to fit actual wind speed data with low m e a n speeds. This is obvious from Fig. 3 a n d we would like to suggest either to use actual wind data (histogram) or look for a better fit by o t h e r models of the probability function.

Acknowledgements--We would like to express our kind thanks and gratitude to the Iranian Meteorological Organisation, Tehran for providing an apparatus of Anemometer Type SIAP (Bologna/Italy) for wind speed measurements.

REFERENCES

1. (3. L. Johnson, Wind Energy Systems. Prentice-Hall (1985). 2. H. Dilger, K. Nester and S. Vogt, Statistische Auswertungen des Wind-, Temperature-, und Feuchte Profils sowie der Strahlung und der Windrichtungsfluktuation am KerlTforschungszentrums Karlsruhe. KFK 2164 (1975). 3. R. Guzzi and C. G. Justus, Physical Climatology for Solar and Wind Energy. World Scientific, Singapore 1988 4. C. G. Justus, et al., Methods for estimating wind speed frequency distributions. J. App. Meteo. 17, 350 353 (1978).