Performance parameters of a bi-directional wind turbine

Journal of Wind Engineering and Industrial Aerodynamics, 7 (1981) 111--128

111

© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

PERFORMANCE PARAMETERS OF A BI-DIRECTIONAL WIND TURBINE

R.E. CHILCOTT and V.G. SMYTH

Department of Agricultural Engineering, Lincoln College, Canterbury (New Zealand) (Received April 20, 1980; accepted in revised form June 17, 1980)

Summary In many regions, a combination of meteorology and topography produces a wind regime which is nearly bi-directional. In such a regime, a wind turbine with fixed horizontal axis, accepting wind from opposite directions by means of reversible-pitch blades, may he appropriate. Depending upon how nearly bi-directional the wind regime is, the loss in overall power output will be offset by greater simplicity of construction and control mechanism. In this paper, performance parameters for a simple model of a fixed-axis wind turbine are evaluated in various model and real wind regimes. The turbine is found to produce up to 88% of the energy output of a similarly rated fully-orienting turbine.

Notation

A

a(oj) b c

c(o~) CB

CE CF eF t CF 2

Cp

f, f~j

k k(Oj)

turbine disk area proportion o'f time the wind direction is in the sector Aflj proportion of observations registering calm Weibull scale parameter Weibull scale parameter for sector A0~ biaxial mean power factor, the ratio of the energy output of a fixed biaxial turbine to that of a similarly rated fully-orienting one mean power conversion efficiency, the ratio of the mean turbine output power to mean input wind power plant capacity factor, the ratio of the mean output power to the rated power primary plant capacity factor (see eqn. (3a)) secondary plant capacity factor (see eqn. (3b)) rated power coefficient, the ratio of turbine output power to wind input power at the rated wind speed directional response of a biaxial wind turbine number of observations within AVi number of observations within A U i and 40 i Weibull shape parameter WeibuU shape parameter for the sector A0j

112

ruth normalized moment about the origin total number of observations during a survey period n number of wind speed intervals P(v ~ Vo) cumulative probability function, the probability of an observation of windspeed greater than v0 probability of an observation within the wind speed interval Avi Pi probability of an observation within Avi and A0i P ~j p(v) windspeed probability density function p(v,O) wind speed and direction probability density function V wind speed mean wind speed rated wind speed VR AV i ith wind speed interval W turbine power output W mean turbine power output mean power in the wind p a s s ~ through the turbine disk ww wR rated power output w(v) specific power output function direction of the p m v ~ g ~ d in a bidirectional wind ~ e orientation direction o f a fLxed-axis turbine r gamma function (F (1 + x ) = f o e - t t x d t) 0 wind direction aoj i th wind direction sector p air density MS N

1. Description of a directional wind regime 1.1 The wind speed probability distribution A standard method of summarizing hourly wind speed and direction data is in the form of a speed and direction frequency table. This approach readily lends itself to a mathematical repzesentation as follows. Considering initiallyjust the speed, partition the wind speed range into n intervals Avl, Av2 .... , Av,. Suppose thereare fi observations of wind in the interval A v~, and N is the total number of observationa taken duzing the survey period. Then the probability of an observation in A vi is Pi = fi/N

If the set of fi contain all the obselwations, then the set o f values pl, p2 . . . . . Pn forms a discrete probability function which can ustmtiy be closely mpzesented by a continuous function p(v) where n

p, = f p(v)dv and ~ P i Av i

i=1

=

f p(v)dv = 1 o

In this case, if p(v) represents the data well, the following statisties relating to the observations can be calculated:

113 The mean: oo

= fvp(v)

dv

0

The mth normalized m o m e n t a b o u t the origin:

0

However, most sets of wind data contain readings of calm (in some cases up to 40% of the readings). Some of these will be due to genuine periods of calm, the rest to wind speeds below the starting threshold of the anemometer (often 1--2 m s-l). If the proportion of calm periods is b, then ~fi = (1 - b) N. Thus to obtain a probability distribution ( ~ ' = l P i = 1) we must define p~ = f j ( 1 -

b)N

This is equivalent to taking the distribution p(v) and its statistics to represent only the period when the wind is above the threshold. To obtain true averages for the whole period, they must be reduced by the factor (1 - b). Takle and Brown [13] represented the calm periods with a weighted Dirac delta function p ' ( v ) = bS(v) + ( 1 - b ) p ( v )

which reduces the statistics automatically. But it seems simpler to correct for calms after the mean values have been calculated as usual with the calms excluded, and that is what is done here. The presence partly of "genuine" calms and partly of "sub-threshold" calms leads to some ambiguity when fitting the function p ( v ) to the probabilities p~; this problem is covered in the Appendix. 1.2 T h e Weibull p r o b a b i l i t y distribution

An analytical probability function which has been found to represent a wide range of wind regimes is the Weibull function (see Hennessey [6] and Justus et al. [10] ). This has the form p(v)

= ( k / c ) ( v / c ) ~-1 e -(~/~)h

The cumulative function is P(v~>vo) = 1-e

-(%/c)

For this function,

•

f

vmp(v)dv

0

= cmr(1 + m / k )

114 SO

= c r ( 1 + I/k) and Mm = r ( 1 + m / k ) / £ m ( 1 + l / k ) ~a

where F(1 + x) = f e - t ~ d t is the gamma function. 0

Commonly the Weibull probability function is f i n to wind data (see for example, Kolodin [11], Justus et al. [9] ) by using the mean wind speeds and third moments. F r o m the data, N

= (l/N) ~ [~vi (where v~ is the mid,point of Avi) i=I

and

i---1

These values are used to find appropriate values of the Weibu/~ p~rameters k and c. Generally speaking c is from about 1.12~ to 1.13~ and k takes values from 1.2 for variable wind regimes to 2.8 for persistent winds. Kolodin [11] found, in a wind survey of Turkmenistan, that sheltered, open, and exposed zones could be brosdty characterized b y Weibuil k values of 1.0, 1.5, and 2.0, respectively. Davenport [4] points out that a simple model of l~ge-scale atmospheric motion causing wind is provided by two.dimensional normally distributed turbulent motion. If the normal distribution is intesrated around all directions, a Weibull distribution with k = 2 results. Davenport quotes Weibull shape parameters in the range k = 1.5--2.5. For the continental United States, Justus et al. [10] found Weibull shape parameters in the range 1.2--2.3. Figure 1 shows a Weibull distribution ~ t o wind speed data from Christchurch Airport (New Zealand) for t h e ~ 1960--1977. E ~ # and M3 gave the values k = 1.85 and c = 10.5 kt. 1.3 Wind directional characteristics When wind data are analyzed into directional sectors as well as speed intervals, a strong directional dependence is usually evident in the distribution. Let fu be the number of observations of wind in the ith speed interval &vi and in the jth direction sector &0j, N being the total number. Then the set of Pu = f u / N

forms a bivariate probability distribution. Representation of this distribution by a bivariate probability function, such that

115

p(v) {

g

(i)

°10t Oi),.~ O'09t(,i,)/~ QO 0081~ 0.06

q ~

Q

0.02~/ °°Ij 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 v(kt) Fig. 1. Wind speed probability distribution for Christchurch A i r p o r t ( 1 9 6 0 - - 1 9 7 7 ); p ( v ) = fitted Weibull distribution. Curves (i), (ii), and (iii) are Weibull distributions fitted

by replacing the 17.1% calms observed by 7.1%, 12.1%, and 17.1% respectively of observations of 2-kt wind.

~0 i ~v i

is generally complicated. Davenport and Jandali [ 5] suggested that the distribution from each direction sector A0i be represented by P(v > vo, 0 ~AOj) = a(0i)[1 - exp{-- (vo/c(0i))k(eJ))]

with a(0j), c(01), and k(O;) represented by finite Fourier series. Fitting Weibull distributions to the data from each 10 ° sector individually gives a range of values of a, c and k (e.g. see Figs. 2--4). Typically all three parameters have peaks at predominant wind directions indicating strong, persistent winds. However, the dominating variation is in a (0;). This suggests use of the simple model obtained by holding c and k constant and varying only a(Oj). Using a continuous function a (0) to represent a (0j), three possible forms of a {0) are as follows: Model (i). For an isotropic wind regime, a(O) = 1/2~, i.e. constant

Model (ii). For an axially symmetric wind regime aligned in the direction a(O) = ( 1 / 2 ~ ) [ 1 + c o s 2 ( 0 - ~ ) ]

Model (iii). For a more pronounced axially symmetric wind regime in

116 which there is no wind from two quarters, I (1/2) cos 2(0 - ~) when/> 0 a(O) = 0 otherwise a (6,;

Ic

!17

16

k 2.5 2.4 2.3 2.2 2.1 2.0 1.9 18

15

-14 •12

•I1 .10 ,9 8

/

\ ,

/

,

.]

~

O OB ,"

"(

k

007 0.06 0.05 004

.7

17

6

1.6 1.7 1.(~

41

D03 . .i

.

I. II,

.

/-I

.

.

I

I

~I

iI..

'.

I

II,.,

/

I

~I

. •

::).01

1 0°

3.02

-

i...

30"

~"

i

,

i

,,

0

90" 1~o" 16o" 18o" 2~o' 2:~0, 2~0, 300' 330" 3e:) °

Fig. 2. W e i b u l l p a r a m e t e r s a, c, a n d k f i t t e d t o e a c h 10 ° s e c t o r : A u c k l a n d A i r p o r t 1 9 6 8 - 1 9 7 7 . a ( ..... ), c ( - - - ) , k (~).

c

k 22 2.8 21 2.7

t-~', it" ~1

19~

2.6 18~ 2.5i 2.4 2.2 15~

'!l

2.; 2.1 13 12 2,0 1,£

~,

A

?:

j

c /

i i /

a(e) 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09

'OOa i0.07 I0.06

1.5 1.7 1.(~

3.05 3.04

1.~ 14

3.03 3.02 3.01

1.3! 0°

30 °

60"

90,

120,

150, 180' 210" 240" 270' 300, 330" 360" 8

Fig. 3. W e i b u l l p a r a m e t e r s a, c, a n d k f i t t e d t o e a c h 10 ° s e c t o r : W e l l i n g t o n A i r p o r t 1 9 6 0 1 9 7 7 . a ( ..... ), c ( - - - ) , k (--).

117 l a(O) 0.09

k 1

a~J

::

0.08

24

~,

23 13 22 1

k

1c 9l c

/ .:/'

0.07

i -'-..---,

0.06 .05

i ""\ 3.04

1E1~1~4t717

~

i

l'~

.:

3.03

"" .. -....

•"

;.,.

3.02

'., .,

.,..

.,,

:

" ' " i ' " " " ' " i" ' " "

0°

•

..... " 3.01 i

i

,

,

30 ° 6'0 ° 90 ° 120 ° 150° 180° 210° 240 ° 270 ° 300 ° 330° 360 ° @

Fig. 4. W e i b u l l p a r a m e t e r s a, c, a n d k f i t t e d t o e a c h 10 ° s e c t o r : C h r i s t c h u r c h A i r p o r t

1960--1977. a (..... ), c ( - - - ) , k (--). 2. Performance parameters of a bi-directional wind turbine Depending on its degree of sophistication and the intended use, a wind turbine may be equipped with either fixed- or variable-pitch blades, and operate at either constant or variable tip-speed. A bi-directional turbine accepts wind from two opposing directions and so, if it has fixed-pitch blades, the aerofoil must be symmetrical about both principal axes. Also, the generator must operate when rotated in either direction. This is probably acceptable for simple, small-scale battery-charging applications. However, a more effective design has blades with conventional symmetrical aerofoils which can be feathered so that the same leading edge is always into the wind, whichever direction it comes from. In this case the turbine always rotates in the same direction, and the advantages of variable pitch can be employed. Therefore only this type of turbine is considered here. The analysis is similar to that of Chilcott [2], and Sterne and Fragioan~s [12].

2.1 Mean output power It is assumed that the power response to a wind speed v, at a yaw angle 0, can be written W = Waw(v)f(o)

where Wa is the rated (maximum) power which can be delivered from the generator, w(v) is taken to be a simple ramp function (Fig. 5), which is quite similar to most variable-pitch turbine o u t p u t functions, e.g. Clausnizer [3] and Hutter [8] (for a more detailed study, see Holme [7] ). It is further as-

118 1.C

w(v)

(w/wR) O , =_

0

0.5

1.0

I','5 '

'

Fig. 5. Wind turbine power o u t p u t function w (v).

sumed that Che cut.in speed is half the rated wind speed VR (the speed at which the turbine first reaches its rated power), and that between the cut.in and rated speeds the response is linear: (vR/2 < v ~ va)

w (v) = 2v/vR - 1

(la)

For wind speeds greater than VR, (lb)

w(v) = 1

There would normally be a furling speed above which w (v) = 0, but this is assumed to he high enough not to affect calculations, and so it is ignored. The directional response to off-axis wind is represented by f(O) =

I cos(20) w h e n ~ 0 0 otherwise

(lc)

In practice this function depends on the pitch angle of the blades, tending towards a cos(0) response for zero pitch angle. However, the cos(2#) function is a conservative representation of the average response. Using a probability density function p(v,O) to represent the wind regime, the average power output can be calculated from 27f

w =

oo

f 0

0

2.2 Plant capacity factor An important performance parameter is the plant capacity factor: 2~"

CF = W/WR = f 0

oo

f w(v)f(O)p(v.O)dvdO

(2)

0

This is the ratio of the average power to the rated power and is a measure of how fully the turbine u ~ s the ~ d r e g i m e . It is ~ to cons~er separately the periods when the turbi_ne is w o r ~ at part c ~ i W and full capacity. Define the p ~ p ~ t capacity factor as

119

2~f

CF, = f o

w(v)f(e)p(v,O)dvde

(3a)

vR

and the secondary plant capacity factor as 27r vR

CF~ = f f w(v)f(e)p(v,8)dvdO 0

(3b)

0

Examples of total, primary and secondary capacity factors for various values of the Weibull shape factor k are given as functions of v R ~ i n Figs. 6 and 7. Obviously, as VR/~ decreases, a greater proportion of the wind is above the rated wind speed, so CF increases.

0

10

2.0

3.0

4.0

1.C~

/'~CE/CP

o.~--\ \ \ \ \

//"~

0

1.0

k-3.o

2.0

3.0

vR/~

Fig. 6 (left). Primary (C F 1) and secondary (CF~) c o m p o n e n t s of C F in Weibull wind regimes (no calms). Fig. 7 (right). Parameters C F and C E / C P in Weibull wind regimes.

2.3 Power.conversion efficiency Define the rated power coefficient as

Cp = WR/(1/2)pv~A where (1/2)pv~A is the power in the wind flowing through the turbine disk area A at the rated wind speed. Then from the equation

4.0

120

W = (ll2)pv~tACpw(v)f(O) it is clear that a low va, although giving a high plant capacity factor, will give a low o u t p u t power. A more useful parameter for optimization is the mean power conversion efficiency. This is the ratio of the mean turbine power o u t p u t W to the mean power in the wind flowing through A in the absence of a turbine. The mean power in the wind is --

2~

oQ

Ww = ( l l 2 ) p A

f

f v3p(v,O)dvdO = (lt2)pA(O)3M3

0

0

where M3 is the normalized third m o m e n t of p (v, 0 ) (see Section I), Hence we can define the mean power conversion efficiency as CE = WIWw = C F C p

(vR/~)31M3

A useful ratio is

CE/C e = CF(VR/O)3/M3

(4)

Examples of CE/C P for various Weibull shape factors are given in Fig. 7. Optimization of Cz with VR will return t h e m a x i m ~ mean power f r o m a given turbine area. However, a compromise in the value of VR m a y be nece~ sary to give a sufficiently large capacity factor.

2.4 Bi-dlrectional mean p o w e r factor In order to quantify the relative merits of fully-orienting a n d bi,directional wind turbines, the bi-directional mean power factor is defined: CB = W/Wo where Wo is the mean power o u t p u t from a fully-orienting turbine in the same wind ~ e (f ( O) - 1). In the case o f the simple-model directional Weibull wind regime, where (Section 1) p (v,O) = a(O)[(k/c) (vtc) k-~ e- (~14' ]

then 21r

CB = f

a(O - a ) f ( O -

fl)dO

0

where the predominant wind direction is a and the ~ i n e with respect to some reference direction. Thus: for Model (i),

CB

=

1/~

axis direction is

121

for Model (ii), Ca = ( l / u ) + ( 1 / 4 ) c o s 2 ( a - ~) for Model (iii), CB = (1/2)[(u/2) + a - ~]cos 2(a - ~) + (1/4)sin 2(a - ~) These functions are given (with ~ = 0) in Fig. 8.

"~~) 0.4

(i)

0°

10 °

20 °

30 °

40 °

50 °

60 °

70"

80 °

90 °:~

Fig. 8. Biaxial mean power factor C B as a function of biaxial turbine alignment with wind regime. Curve (i), CB(a ) = 1 / . ; c u r v e (ii), CB(a) = ( 1 / . ) + (1/4)cos 2a; curve (iii), C B (a) = (1/2) [(~/2) + ~] cos 2a + (1/4) sin 2a.

3. Bi-directional wind turbine performance in empirical and Weibull wind regimes

3.1 Case study using real data Hourly wind speed and direction data from Auckland, Wellington and Christchurch Airports (Table 1), when viewed as speed--direction frequency TABLE 1 Data collection sites Site Auckland Airport Wellington Airport Christchurch Airport

Period

v (kt)

M3

Calms (%)

k

c (kt)

1968--1977

11.3

1.85

12.6

2.07

12.8

1960--1977

15.6

1.89

2.02

17.6

1.85

10.5

1960--1977

9.36

2.07

7.02 17.1

122

Oc

0.~

~.....CE/cp

Q8 0.7

07

0.6

0.6

Q5

Q5

Q4

OA

Q3

O~

02

0.1

0.1 0

1.0

2.0

3.0

4..0 V~I~

0

1.o

2.0

3.0

4.0

Fig. 9 (left). Parameters CF and Cz/Cp for a fully-orienting turbine: Auckland Airport (the broken lines are from a fitted Weibull distribution with k ffi 2,07, c = 12.8 kt, 12.6% calms). Fig. 10 (right). Parameters CF and CE/CP for a fully-orienting turbine: Wellington Airport (the broken lines are from a fitted Weibull distribution with k = 2.02, c ffi 17.6 kt, 7.02% calms).

o.e~ 0.7 O.e

,/--"-.,

'~c~

/~,.

~,

/

/

~

~

\

q/c~

,, ,

,

,

"

',,

0.5

0.4 Q3 02

0

1.0

2.0

3.0

4..0 VR/P

Fig. 11. Parameters CF and C~.]Cp for a fully-orienting turbine: Chrimtchurch Airport ( t h e b r o k e n lines are f r o m a fitted Weibull distribution w i t h k = 1 . 8 5 , c = 1 0 . 5 kt, 17.1%

calms).

123 tables, show the characteristic bi-directional wind regime which suggests suitability for a bi-directional wind turbine. Initially, for the sake of comparison, performance parameters for a fullyorienting turbine were calculated using eqns. (1), (2), (3) and (4) (with f(O) = 1). Graphs of CF,, CF2, CF and CF./Cp are given in Figs. 9--11. In order to test the appropriateness of the Weibull distribution for this application, it was fitted to the data from each site and used to calculate the same parameters. In most cases the agreement is good. Divergence will result from combining regimes from different directions with different Weibull characteristics into a single distribution. In order to find the optimum orientation direction for a bi-directional turbine, CF was calculated using eqns. (1) and (2) for various values of VRf~. The results are shown in Figs. 12--14. Once the optimum direction was found, the parameters CF~, CF~, CF, CE/Cp, and CB were calculated at each site as functions of Vsf~. The corresponding graphs are given in Fig. 15. The interesting CF

0.5 04 O3

i

0°

30 °

60 °

i

90 °

1 2 0 ° /~

i

150 °

180 =

Fig. 12. Capacity factor C F for a biaxial turbine as a function of orientation angle ~: Auckland Airport. C~

0.8J 0.7

Oe 05

~ z f 1.5

////

04 03 O2

/ /

01 0°

30 °

6G

90 °

120 ° ~

150 °

18G

Fig. 13. Capacity factor C F for a biaxial turbine as a function of orientation angle ~: Wellington Airport.

124 CF Q6 0.5

1.5

0.3 ~ 0.~ 0J i

O°

I

BO°

,

~o°

80'~'

i

120 ° ,~

__

150°

180'

Fig. 14. Capacity factor CF for a biaxial turbine as a function of orientation angle fl: Christchurch Airport. QC Q e . . . . ""

0.7 Q6

i

,

,"

",

-..-~----~'~CB

.....................

Q5

'

'

.

.

.

.

.

.

0.4 ". '

0.2

'-?,

X

/ .... ...

¢

•

o~ 0.1

0

q/c~ / Y

, ..

1.0

'=)\~

....~:,

2.0

3.0

VR/F

40

Fig. 15. Performance parameters CB, C F and CE/C P of a biaxial turbine: Auckland Airport ( - - ) ; Wellington Airport (-- ); Christehurch Airport ( . . . . . ).

parameter in this case is Cs. Wellington Airport is clearly an ideal location since CE/C p and CB peak together, and the maximum value of Cs is 0.88, 3.2 Model .Weibull wind regimes The most important use of an analytical wind speed and direction p~obability function is at a site where the complete frequency distribution is not available from the data. The parameters k and c for the Weibull probability function can be estimated from the site mean wind speed and terrain type (Cherry [1] ). To give an order-of-magnitude fizst estimate of the performance of a bidirectional wind turbine, one o f the directional models given in Section 1 can be chosen knowing the topography and meteorology of the site. For example, both Auckland and Christchurch Aizports ate well e x t x ~ e d

125

to all directions and have bi-directional wind regimes, suggesting Model (ii). Wellington Airport, on the other hand, is set in a valley aligned with the prevailing winds, suggesting the more directionally constricted Model (iii). Using k = 1.8 for Auckland and Christchurch, k = 2.0 for Wellington, and (Section 1) calculating c from c = ~/F(1 + 1/k) the probability function p (v,0) is fully determined. Figure 16 shows how these models compare with the empirical data discussed in the previous section. In each case the capacity factor CF is underc 0.8, 0.7

~

0.6

0.4 0.3

~>.

"

0.1 0

to

2.0

3o

VR/~ 4.0

Fig. 16. Capacity factor C F calculated from biaxial wind regime models. Curve (i), C F from Model (ii) ( - - ) compared with C F calculated from data for Auckland (-- -- --) and Christchurch ( . . . . . ) Airports; curve (ii),C F from Model (iii) ( - - ) compared with C F calculated from data for Wellington Airport ( . . . . . ).

estimated by - I 0 - - 1 5 % in the useful range VR/-~ = 1.5--2.0. The main reason for this is the simple separation of variables implied in the assumption p(v,0) = a(O)p(v) In reality, the higher wind speeds, which are useful to a wind turbine, come from a much narrower direction range than the lighter winds. 4. Conclusions Both the power conversion efficiency and the plant capacity factor of a wind turbine are dependent upon the rated wind speed, which in turn is determined by the size of load and the pitch control. For maximum usefulness the turbine needs to be carefully designed to suit its operating environment. This means matching the plant capacity to available storage or need for firm

126

power, and at the same time o p ~ i n g t h e power conversion ~ i e n c y from the wind ~ e t o the generator. ~ ~ ummlly imply the use of a rated wind speed of between one and two times the sitemean wind speed. In three simple-model bi-directionalwind r ~ e s , a fixed-directionb~ial wind turbine would produce up to 32%, 57% and 79% of the average power output of a fully-orientingturbine (Fig. 8). At Auckland, Christchurch and Wellington Airports, a bi-directionalturbine, when ah'gned with the optimum direction and rated for m a x i m u m power conversion efficiency (Cz), would produce 65%, 54% and 88% respectively of the fully-orien~ turbine output. The simplicity and cheapness of a bi-d~tional turbine COuld justify itsuse in a wind regime like that of Wellington Airport. Appendix

Fitting a probability function to data containing calm periods The data used in this report were taken from Munro generating anemometers which have a starting threshold of ~ 3 kt (see Fig. A1). The actual readings

Fig. A1. Munro generating anemometer output at low wind speeds.

are 10-rain means taken just before the hour. Since the scale is non-linearat the low end, and the firstgradation is 5 kt, readings of 5 kt and lesswill be of low accuracy (a measure of the subjectivity of the scaling is the marked bias towards multiples of 5 kt in Fig. 1). Certainly many of the zero readings are simply from periods during which the wind did n o t exceed 3 kt. For the direct calculation of turbine performance parameters such as CF and Cz/Cp, this inaccuracy in the lower values is of no consequence, since this range is normally below the startingthreshold of the turbine. However, the mean and third m o m e n t of the wind speed d i S ~ b u t i o n are proportional to ( 1 - b ) a n d 1 / ( 1 - b) 2 respectively, where b is the proportion of calms. So a WeibuU function fitted to the distn~bution will be dependent on b. It would be a very useful property if the fit to the upper part of the ~ t i o n (v > ~ at least) was reasonably insensitive to the pt~mnce of "sub-thnmhold" calms.

To test this, increasing proportions of the ~ Airport were replaced with 2-kt wind speed ~ .

recorded at C h r ~ u r c h The value o f 2 kt was

127

chosen fairly arbitrarily but is reasonable. The new statistics and Weibull parameters are given in Table A1. The Weibull functions are plotted in Fig. 1 with the empirical distribution. Since the empirical distribution is calculated from Pi = f i / N

where N is the total number of wind speed observations, the probabilities Pi should be recalculated with a larger N as more 2-kt readings are added. However, to show all the curves on the one graph, the Weibull curves have been proportionally increased instead. Except in the region v < 5 kt, where there are "missing data" as in the empirical distribution, the fit appears to improve as T A B L E A1

Replacement of calms with 2-kt readings: Christchurch Airport calms (%)

v (kt)

M3

k

c (kt)

17.1 10 5 0

7.76 7.90 8.02 8.11

2.07 2.31 2.47 2.65

1.85 1.69 1.61 1.52

10.59 9.89 9.42 9.00

i

10

CF ~f(iv) 0.9 \~'{v ' ~ o(iii) !) 0.8

\'~

//"~",,.,X/dii)

0.7 06 0.5 0.4 0.~ 0.2 0.1 0

1.0

2.0

3.0

4.0 ~/~

Fig. A2. Parameters C F and CEICP calculated from the Weibull distribution fitted to the Christchurch Airport data. Curves (i), (ii), (iii), and (iv) are fitted with 17.1% (as observed), 10%, 5%, and 0% calms, respectively.

128

increasing proportions of calms are replaced. But ~ i f i c a n U y , from v = 9 kt upwards, there is not much difference. CF and C~/Cpwere calculated from the various Weibull functions and are plotted in Fig. A2 against VRf0 (each ~ being adjusted according to the proportion of calms and added 2-kt readings, as given in Table A1). At the rated wind speed for maximum conversion efficiency (VR/~ = 2.6), CF is reduced from 0.81 to 0.75 (~18% and 7%, respectively). A comparison with Fig. 11 shows that the empirical curves follow ~ e s with less than the observed proportion of c ~ s , On the other hand, Figs. 9 and 10 show that the Auckland and Wellington data agree well with the Weibutl distribution fitted when the observed number of calms is used. The only conclusion is that this ambiguity is a source of uncertainty (possibly 10--20%) which must be added to the already known sources (bias during the observation period, anemometer over-run, etc.).

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