Generation of autocorrelated wind speeds for wind energy conversion system studies

0038-092)(/84 $3.00 + .00 © 1985 Pergamon Press Ltd.

Solar Energy Vol. 33, No. 6, pp. 571-579, 1984 Printed in the U.S.A.

GENERATION OF AUTOCORRELATED WIND SPEEDS FOR WIND ENERGY CONVERSION SYSTEM STUDIES MARIE BLANCHARD and GILLES DESROCHERS lnstitut de recherche d'Hydro-Qu6bec (IREQ), Varennes, Qu6bec, Canada (Received

12 N o v e m b e r 1982; a c c e p t e d 28 J u n e 1983)

Abstract--This paper describes a new model and a new generator of hourly wind speeds which were obtained using the Box Jenkins method. All the steps leading to the determination of an autoregressive model are described. Tests were performed to verify the adequacy of the model and comparisons were made between generated and real series to check whether the wind speed behavior is faithfully reproduced. Good results were obtained. In fact, hourly wind speed data prove sufficient to reproduce the main statistical characteristics of wind speed: monthly mean, standard deviation, high hourly autocorrelation and persistence. This simple model is, therefore, easily adaptable to the study of any wind energy conversion system or to mixed power system planning and reliability studies.

I. I N T R O D U C T I O N

Much work has been dedicated to the mathematical representation of wind speed as being essential for wind energy conversion system studies or mixed power system planning and reliability studies. Therefore, the need for a wind-speed model that is adequate, simple and easily adaptable is obvious. The first statistical studies in this field began 30 yr ago [1]. Over this period, different distribution functions have been suggested to represent wind speed including the Pearson [I], Chi-2 [2], Weibull [2-4], Raleigh [5] and Johnson [6] functions among which the Weibull distribution function is the most commonly used in applications. Different methods have also been developed to evaluate the parameters of these distribution functions [3, 6, 7]. Goodness-of-fit tests have been performed on historical data in the case of the Chi-2, Weibull and Johnson functions. None of them passed the Chi-2 tests but in about half the cases the Weibull and Chi-2 functions passed the Kolmoghorov-Smirnov tests at the 5 per cent significant level. However, these results are questionable since the tests are applicable only if the data are statistically independent and, as mentioned by several authors [8-10], wind speed data are highly correlated. This wind characteristic (high correlation) is reflected in the wind persistence (run duration) phenomenon, which has also been modeled [2]. Simulation of hourly wind speeds has been the subject of several publications. For example, a technique for simulating the hourly wind speeds for randomly dispersed sites has been developed [11], but it has the drawback of not taking the high temporal correlation, or autocorrelation, into account. Later a method using first-order autocorrelation and the Weibull distribution function was established [9] which needs 24 previously estimated diurnal cycle factors per month. Yet another method has been proposed [10] based on the Raleigh distribution function. It involves 2 components: one representing the daily cycle (one 571

value for each hour of the day), the other representing the dependence on previous hours, both of which imply many calculations; moreover no discussion about the fitness of the model is presented. This paper describes the steps which have led to the development of a new wind speed model and generator. The main objectives were simplicity and adequacy. Based on historical data, the model takes account of the high autocorrelation and allows a time series to be generated which preserves all the main characterist i c s of these data. In addition it does not require any assumptions about the wind speed distribution and it allows to make updated forecasts. 2. METHODOLOGY The model was built using the methodology for time series analysis developed by Box and Jenkins [12], which allows dependent variables to be studied and takes account of the nature of that dependence. It also allows the user to single out, from an entire class of models ( A R I M A ) , one that would best represent the original data. An A R I M A ( p , d , q ) model can be written as follows (1

-

(oiB

...

+ (1 where

OiB--

~ b p B p ) V d X t = Oo ...

-

OqBq)a,

(1)

B backward-shift operator BX,x7 d

X, ,

backward-difference operator of order d vX,-X,-

Xt,

-

X,

V~A; = V ~ ~xTX, n number of observations t - 1 . . . . . n random variable representing the process at instant t. No distinction will be made between the random variable and its actual value. (jbl,...~bp autoregressive parameters constant term 0o 0 o = ( 1 - ~bl . . . . - - 4%)U where u is the mean of the actual values

572

M+ BLANCIIARD and G. DESROCHERS

G,..., Oq moving-average parameters a,, t - 1. . . . . n white noise process (uncorrelated random variables, with mean 0 and variance o-2,).

MODEL IDENTIFtCATION

In fact, a larger class of so-called seasonal models includes A R | M A models but, as Section 3 will explain, these do not need to be considered to represent the wind behavior on a monthly basis. As shown in Fig. 1, the model-building process comprises three main steps: identification, estimation and diagnostic checking. The first step consists in selecting a model which seems to be a good representation of the data. This is done by examining the original data and its statistical characteristics. In addition, rough estimates of the model parameters are calculated. In the second step, the chosen model is fitted to the data and the model parameters are evaluated. Diagnostic checking, the last step, tests the validity of the model. If it is judged inadequate, the 3 steps must be repeated until a suitable model is obtained. The following sections describe an application of this 3-step process to historical wind-speed data, which produced an A R I M A (2, 0, 0) model, also called an autoregressive model of order 2, having the following form X, = 4hXt , + q~2X, 2 + 00 + a,.

(3)

where = ~

OF THE CURRENT MODEL

NO

YES

l

if observed data available

Lx~ if not. From wind speeds observed at the Magdalen Islands in Qu6bec, from 1 January 1973 to 31 December 1976, the 1976 data were chosen for this study as being truly representative. The data were divided into 12 onemonth series on the basis that wind has a relatively homogeneous behavior within a month.

The identification step consists in the determination of the values of p, d and q of eqn (1) and the preliminary estimation of the autoregressive and moving-average parameters of the model which will be used in the next step. The first task, therefore, is to determine the following values: d is a degree of differencing needed to produce a stationary series, p is order of the autoregressive operator, and q is order of the moving-average operator.

!

To illustrate the methodology, a study of the data of one representative month (January 1976) will be presented. The data shape for the first week of this month is shown in Fig. 2. Since no seasonal pattern, i.e. no periodic behavior, is visible in these data, the research was restricted to non-seasonal models. The value of d is determined by studying the estimates of the autocorrelations of Xt, rk(k = 1,2 . . . . ). Figure 3 shows these values for the month studied. It can be seen that the autocorrelation function decays exponentially and not very quickly. This can mean either that the series is not stationary or that it is stationary, but near the nonstationarity boundary. For the purpose of this study, it was supposed that d is zero, i.e. that the series is stationary. The determination o f p and q calls for examination of the estimates of the partial autocorrelations 4~kk (k = 1,2 . . . . ) in addition to the autocorrelations. 20

3. IDENTIFICATION

EL UTILIZATION

Fig. I. Steps of the model-building process

(2)

With an A R I M A model, forecasting or generation of a series can be performed [12]. For an autoregressive model of order 2, the generated value at instant t + k, noted -~,+k, is obtained from the following equation ~'t+k = 4hY,+k , + 4hY,+k 2 + a,+~ + 00

PAPv%ME.TER . ESTIMATION

I

- - 44

I

I

I

1

I

I

I

/,

48

++

+, • i

!~

+..... k~,+

I/

4

~'1

2 II

l

J

J

O(

20

40

60

__

,

~

80 tOO HOUeS

+

,'

+20

,140

Fig. 2. Observed wind speeds: first week of January 1976.

Generation of autocorrelated wind speeds for wind energy conversion system studies rk

573

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

II

0.l

9

I0

11

12

13

14

15

16

17

18 k

Fig. 3. Estimates of autocorrelations of actual data: January 1976. (Standard notations are used to represent the autoregressive parameters and the partial autocorrelations, that is 4~k and q~kk respectively. The double indices distinguish the partial autocorrelations). For the month of January, Fig. 4 displays two partial autocorrelations significantly different from 0: ~ and 4~22;the value of q~,~is 0.91 and of~22 is 0.10. This fact and the shape of the autocorrelation function suggest an autoregressive model of order 2. Consequently the model presented here could be as described in eqn (2). The same remarks about the autocorrelation and partial autocorrelation functions can be made about all the months studied. This A R I M A (2, 0, 0) model (or simply AR(2)) suggests that the wind speed at instant t depends on the wind speeds at instants t - 1 and t 2, which, intuitively, seems plausible, knowing the wind persistence characteristic. The next task is the preliminary estimation of all the model parameters, namely q~t, ~bz, 0o and d]. These values are required for the second step, estimation. For the AR(2) model, the method of moments (statistical estimation method) and the Yule-Walker eqns (12) give the following estimates for q~ and q~2 - r~)

(4)

~2 = (r2 -- r ~ ) / ( l - r~).

(5)

~l = rl(1

r2)/(1

The method is simple but does not give accurate estimators, which is why a more efficient alternative is used, in the estimation step. The preliminary estima-

tions provide the initial values required by the alternative method. Before the estimation of 0o, a test of significance is carried out in order to verify that it is significantly different from 0. In this case, eqn (6) gives the estimation 00 = Y(1 - 4~ =-4~2)

(6)

where Y represents the mean of the original data. For all months studied 0o was different from 0. The last parameter to evaluate is ~] which is done in the following way [12] ^z

O" a

~

6.z(1

--

~1rl

_

(7)

~2r2)

where 62x is the estimate of the variance of the original data. All these calculations were performed on the original data and were used in the estimation step. 4.

ESTIMATION

This step involves the estimation of all the model parameters. For this purpose, the values assigned to the parameters ~bt, q~2, 0o are those that minimize the sum of the square of the a , , t = 1 . . . . . n (the a , here are called residuals)

)-~ a ~ :-1

= )-~ (iv, t-I

~,x,

~ -

4~x~

2 -

00) 2

(8)

574

M. BLANCttARDand G. DESROCHERS

'bkk

1.0

0.9

0.8

0.7

0.6

0. S

0.4

0.3

0.2

0.1

,

0.0

2

[

,.,I

6'

I

8

'~oI

1

I 12

0.I

Fig. 4. Estimates of partial autocorrelations of actual data: January 1976. Table 1. Values of the model estimators for each month MONTH

JANU~Y

FEBRUARY ,HARCH

APRIL

HAY

JUNE

JULY

AUGUST SEPTEHBER OCTOBER NOVEMBER DECEMBER

0.816

0.717

0.707

0.776

0.828

0,725

0.759

0.750

0.795

0.830

0.778

0.731

0.107

0.223

0.238

0.174

0.115

0.191

0.112

0.147

0.145

0.090

0.134

0.219

0o

0.822

0.465

0.455

0.471

0.453

0.639

1.011

0.734

0.502

0.788

0.837

0.580

Oa2

4.138

2.514

2.859

3.044

2.707

2.736

3.008

2,886

2.558

4.333

3.708

4.299

The estimators 4it, 4]2 and 0o are called the least-square estimators. A modified s t e e p e s t - d e s c e n t algorithm[13] was used for the calculations, which were performed for all 12 months. Interesting results were obtained, as seen in Table 1. The estimations 4it and 4]2 always have the same behavior: 4]~ is positive and relatively high, while 4]2 is also positive but smaller. If such a model is acceptable (as will be seen in the next section), it implies that the wind speed at instant t depends mostly on the wind speed at instant t - 1 and less on the wind speed at instant t - 2.

One last parameter remains to be estimated: ~2. The following estimation is used

d] =~'a,(~,, ~2, Oo)/(n 2).

(9)

t=l

Table 1 presents the results obtained. The stationarity of the model is also verified. In the case of an AR(2) model, this corresponds to the following conditions being fulfilled [12]

Generation of autocorrelated wind speeds for wind energy conversion system studies

575

"~2 1.0 ,'0.8

/

'\ \ \

/

\

0.6

/

/

\

0.4

\

/ /

MODEL

PA RANFTER,¢:;

0.2

/

/ !

2.0

1.6

0.8

i

i

/

I

I

0.8

0.4

0.4

\

\

0.2

/ /

1.2

-

2.0

\

d~1 \

0.4

/

1.6

N

/

\

0.6

/

\

/ 0.8

/ / g.

\ \

...1..0Fig. 5. Area of stationarity of AR(2) model

~1 + ~2 < 1.

(10)

~ 2 - ~t < 1.

(11)

-1

(12)

<~2<

l.

These conditions are represented in Fig. 5. The values of ~ and ~2 are in the shaded area inside the triangle for each month studied, indicating that all the models found are stationary, but are near the nonstationarity boundary. In view of this, the method was applied with a differentiation of order 1 applied to the data, an operation which should suffice to make the data stationary if they are not already so. As shown in the next section, the best results are obtained with an AR(2) model. 5. WIND MODEL VALIDATION

Acceptation of the AR(2) model The AR(2) model was obtained by minimizing the sum of the squares of the differences between the actual speeds and those given by the model; these residuals for the first week of January 1976 are shown in Fig. 6. If the fitted model is adequate, then the autocorrelations of the residuals rk(a) should be uncorrelated and normally distributed with mean 0 and variance n ~. The first 24 autocorrelations for the residuals of January 1976 are presented in Fig. 7 which also shows the standard error limits _+2n 05. If the assumption of normality is justified, then 95 per cent of the rk(a) should be within the ±2n 05 boundaries, as is the case here, since only one of the 24 is outside the bounds. The model is finally accepted if these residuals are found to be uncorrelated; a statistical test using the statistic Q is performed in order to

accept or reject this assumption. The statistic Q is given by the summation of the squares of the autocorrelations of lag 1 to 32 of the residuals. Assuming that the residuals are uncorrelated, this statistic is Chi-2 distributed with 30 degrees of freedom. The test is accepted, with a probability of 5 per cent of rejecting a true hypothesis, if the statistic Q is less than 43.8. In the identification step, several representative models were investigated and two were selected as being interesting: an autoregressive one of order 2 (AR(2)) and an autoregressive-integrated one (ARI M A (1, 1,0)). For these two models the variance of the residuals and the value of the Chi-2 statistic for the test are given in Table 2 for each month of 1976. Both models are rejected only 2 months out of 12, but the AR(2) model is deemed to be a better choice because the variance of the residuals is less for every month. In addition, it is conceptually simpler, which is certainly an advantage.

4° 601

r

t

T'

/

tt,, i

i

i

I

i

"4 -4

1 -I -1

60

I

t

iO

t00

i

420

1

44O

Fig. 6. Model residuals: first week of January 1976.

I

~O

M. BLANCHARD and G. DESROCHERS

576 rk(a) .12 ,10

CONFIDENCE

RECION

.08

.06

.04

.02

I

2

I

i

,21 14

i

I

201

1 22

I

iL-

2, I

k

.02

.04

.06

.08

.10

.12

Fig. 7. Estimates of autocorrelations of the residuals: January 1976.

Table 2. Variance of residuals and statistic Q for the AR(2) and AR1MA(I, 1,0) models HAGDALEN ISLANDS WIND SPEEDS OF 1976 JAN

FEB

PAR

APR

HAY

JUN

JUL

AUG

SEPt

oct

NOV

DEC

AR(2) VARIANCE RES.

O

ARIHA(I,ID0) VARIANCE RES.

Q

2.89 2.56 4.33 3.71 4.30 4.14 2.5 2.86 3.04 2.70 2.74 3.01 24.5 42.9 44.9* 51.3" 33.6 38.5 34.19 36.6 42.8 3 2 . 8 27.9 3 0 . 2

3.02 2.63 4.50 3.86 4.39 4.29 2.58 2.93 3.11 2.781 2.84 3.19 30.0 44.0* 40.3 50.8* 32.6 39.5 32.9 35.9 40.5 37.5 38.8 40.7

* : F:OR THIS MONTIt THF. MODEl. IS REJECTED (Q ~ 43.8)

O t h e r tests on the A R ( 2 ) model were conducted for a few months of 1973 and 1975, including October and N o v e m b e r which were rejected for the winds of 1976. For both years the A R ( 2 ) model was accepted for the 2 months it was previously rejected, and it was only rejected in I case, J a n u a r y 1975. On the whole, the A R ( 2 ) model was accepted 83 per cent of the time. it may be argued that since some autocorrelations of lags larger than 2 for the data seem significantly

different from zero than an autoregressive model of order larger than 2 should give better results. To investigate this, an A R ( 3 ) model was fitted to the 1976 data. This validation technique, called "overfitting," allows the user to ascertain that a more sophisticated model can not better represent the data. Table 3 list the values of the three autoregressive parameters with the variances of the residuals and the statistic Q. The AR(3) model gives similar results for the acceptation

Generation of autocorrelated wind speeds for wind energy conversionsystem studies

577

Table 3. Overfitting: AR(3) model HONTH

~l

~2

~3

VARIANCE OF RE5IUUAI-~

JANUARY FEBRUARY HARCH APRIL HAY JUNE JULY AUGUST SEPTEMBER OCTOBER NOVEHBER DECEHBER

.82 .71 .70 .80 .82 .71 .76 .76 .79 .83 .78 .73

.14 .20 .23 .27 .05 .15 .11 .18 .14 .14 .15 .22

-.05 .02 .01 -.11 .07 .06 .004 -.04 .01 -.06 -.02 .001

4,13 2.51 2.86 2.99 2.69 2.73 3.00 2.88 2.56 4.31 3.70 4.3

test while the variances of the residuals are very close to those listed in Table 2 for the AR(2) model but it should be stressed that the third parameter (4~3) is generally much smaller than the other two and, therefore, unnecessary for describing the data. Another model investigated, ARMA(2, 1), revealed that the additional moving-average parameter did not lead to any improvement at all. Some authors [9, 10] have proposed models which are based on diurnal cycle factors. In order to check whether a model which takes this characteristic into account will give better results, the AR(2) model was modified to integrate this cycle. A mean was calculated for each hour of the day and then subtracted

0

41.1 34,5 40,2 37,6 29.0 27.6 30,6 24.6 43.4 42.5 51.9 35.8

from the actual data. Later, the estimation, validation and generation steps were performed with this modified series. Finally, the hourly means were added to the synthetic data. For the Magdalen Islands wind speeds of the 1976, no improvement in the results was found. Without further work this result cannot be generalized to different terrains and climates.

Preservation of the statistical characteristics by the wind generator In addition to the acceptation procedure described above, a complete validation of the AR(2) model should include an analysis of the wind speeds generated by the model in order to ascertain that these

Table 4. Comparison of actual and synthetic series AUTOCORRELATION OF LAG VARIANCE 2

I

3

4

5

i FZ ERROR ON MEAN

Z ERROR ON IVARIANCE

Z ERROR ON Ol

JAN.

10.6400 110.2260

25.474 24.617

.91148 .90679

.84767 .84188

.77740 .7789[

.70932 .72135

.64742 .66887

3.89

3.36

0.51

FEB.

7.8117 7.3587

17.898 16.394

.91982 .90944

.87981 .86681

.83693 .81381

.79101 .76737

.75497 .72236

5.80

8.40

1.13

NAR.

8.2530 r21.689 7 . 5 0 3 5 122.304

.92670 .92202

.89238 .88414

.85269 .83983

.80746 .79706

.76047 .75698

9.08

2.84

0.51

APR.

9.3940 9.5039

26.659 25.099

.93822 .93245

.90067 .89090

.84781 .80214 .84715 . 8 0 4 9 4

.75203 .76513

1.17

5.85

0.61

NAY

7.8736 7.7350

21.822 20.407

.93448 .92967

.88761 .87977

.85059 .83055

.80535 .78338

.77202 .73744

1.76

6.48

0.51

JUN.

7.6231 7.6422

14.457 13.897

.89562 .88687

.83949 .82543

.7R951 .76201

.75141 .70224

.71123 .64789

0.25

3.87

0.98

JUL.

7.8882 7.9564

11.458 IO.879

.85604 .84845

.76246 .75043

.67577 .65972

.60637 .58001

.52359 .51135

0.86

5.05

0.89

AUG.

7.1442 7.0958

13.049 12.535

.87958 .87290

.80686 .796U1

.72586 .72027

.65912 .65323

.58381 .59002

0.68

3.94

0.76

SEP.

8.4590 8.4998

19.485 18.501

.93014 .92356

.88484 .87317

.84OI7 .82175

.79026 .77429

.73689 .72992

0.48

5.05

0.71

OCT.

9.8614 9.8619

26.074 Z$.319

.912(~ .90593

.R4708 .83731

.77499 .77317

.70731 .7115(l

.62810 .~53h7

0.0!

2.90

0.67

NOV.

9.5450 9.5270

19.887 1q.859

.89901 .89680

.833~7 .82763

.765~4 .7h172

.68983 .7UO4)

.61274 ,6424?

o. 19

U.14

0.25 •

36.352 31.650

.93430 .91948

.EqQH 1 .87865

.Hht)O7 .8175h .H3234 . 7 9 i 1 ~

.71oa5 .7So37

3.47

12.93

I.$9

DEC.

11.788 12.197

F[KST LINk: ACTUAL N|NI~ NECOND LINE: SYNTHETIC WINDS

578

M. BLANCH&RD and G. DESROCHERS

synthetic winds are probable ones. Several parameters allow a quantification of the equivalence between the generated winds and the actual ones, the four selected being the mean, the variance, the autocorre[ations and the persistence. The first three statistics are well defined but the fourth, the persistence, is defined here as follows: a set of representative levels is chosen for the wind speed and then for each level the number of consecutive hours spent below and above this level together with the number of times this happens are computed. The AR(2) model, each month having its specific parameters, was used to generate 25 yr of synthetic winds. For each month the mean, the variance and the first five autocorrelations were calculated and the results are listed in Table 4 together with the actual parameters for 1976. The mean error was calculated for these parameters: on the mean it is 2.3 per cent, on the variance it is 5.07 per cent and on the first autocorrelation it is 0.76 per cent. These results lead to the conclusion that the main statistical characteristics are preserved by the AR(2) model. The wind is a very unstable phenomenon characterized by a sequence of lulls and sustained speeds, and a good wind generator must be able to reproduce such sequences. To check the validity of the AR(2) model, the persistences were calculated for each month of 25

yr of synthetic winds, then averages computed for each month. In Tables 5 and 6, a comparison between the actual persistences of 1976 and the synthetic ones is presented for 1 winter month, February, and I summer month, July. Since the number of sequences in a month can vary substantially, the percentage of sequences for each speed level is given instead of the number of sequences itself. Here again the autoregressive model appears to be quite accurate despite the wide variability of the persistence phenomenon.

Updatedforecasts The proposed model was used in the previous sections as a wind generator. Here it is examined in its possible role as a forecasting tool. Knowing the values of the wind in previous hours, the problem consists in forecasting the next hour's wind. This was obtained from eqn (13)

£<,~,=4~,x,+4~2x, ,+00.

The adequacy of the model cannot be better demonstrated than by a graph presenting both acutal and forecast data; thus, a graph for the first 48 hr of the month of January appears in Fig. 8, leaving no doubt about the ability of the model to follow the actual data.

Table 5.Wind speed persistence: February ACTUAL WINDS LEVELS

SEQUENCES ABOVE

SYN'I~ETIC WINDS

SEQUENCES BELOW

SEQUENCF..~ ABOVI~

SEQUENCES

BELOW

(m/a)

I OF ~AN I OF .HEAN SEQUENCES DURATION SEQUENCESDURATION

2 4 6 8 10 12 14 16 18 20 22 24

8.7 14.8 14.8 19.7 15.8 9.8 8.2 4.4 2.2 1.1 0.5 0.0

40.0 20.5 15.6 7.9 6.2 6.3 4.5 2.8 4.0 2,0 1.0 0.0

8.7 14.4 14.4 19.C 15,4 9.7 8.2 4.6 2.6 1.5 1.0 0.5

Z OF WEAN Z OF SEQUENCES DURATION SEQUENCES DURATIOI~

1.9 4.2 9.0 I0.5 16.4 29.4 37.8 72.2 31.2

9.3 14.3 19.4 21.4 16.7 10.4 5.2 2.1 0.8 0.4 0.O 0.0

222.7

335.5 672.0

33.1 19.1 10.9 6.7 4.9 3.8 3.2 2.6 1.5 0.6 0.3 0.0

8.7 13.7 18.8 20.9 16.4 I0,4 5.5 2.5 1,3 0.8 0.5 0.5

3.5 4.7 6.0 8.3 14.9 28.3 60.2 177.6 400.9 533.2 645.0 672.0

Table 6. Wind speed persistence:July SYNTHETIC WINDS

ACTUAL WINDS LEVEI.S

S Et)UENCKS ABOVE

SEI)UENCES BEI.ON

SEOUENCES

AROVE

S EOUENCES BELOW

(m/s)

Z OF F MEAN Z OF MEAN ~QUENCE.~ iDnRATIIIN 5 EOUENCES I DURATION 2 4 6 8 In 12 14 16 18 2o 22 24

7.8 11.7 17.4 21.3 24.3 9.6 5.2 1.7 0.9 n.(I n.n

39.8 24.0 13.5 8.2 3.1 3.2 2.7 I.~ 1.5 O.0 ().0

O.&

o.o

7.3 11.2 16.7 20.6 24.(I 9.9 5.f, 2.1 1.3 0.4 0.4 ~).4

1.6 3.7 5.2 7. I I[I.Z 29.3 I 54.x ll4l.~, 1247.II 744.t, 744.r~ 144.,)

(13)

] ] i

7, OF MEAN Z OF r MEAN LSEOUENC'L'i IURATION SEQUENCES iDURATION 4.8 12.3 20.4 24.7 20.8 11.2 4.4 1.2 0.3 D.O (I.¢~

63.1 21.4 10.2 5.7 3.7 2.7 2.1 1.6 0./4 n. I (P.(t H.q)

4.4 11.8 19.8 24.2 20.5 11.3 4.7 1.5 O.h n.4 (i.4 n.4

).3 2.7 3.9 5.9 In.l 23.21 71.5 288.9 584. I 7O9.2 744.H 744.0

Generation of autocorrelated wind speeds for wind energy conversion system studies 20

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and integrated into any computer software dedicated to power system planning or operating. The use of synthetic data from this model for utility applications implies the assumption that wind and load are not significantly correlated. This assumption was verified for the Magdalen Islands data and this agrees with general expectation. The autoregressive model presented in this paper has been successfully used at H y d r o - Q u 6 b e c in a computer simulator that assesses the economic impact of wind turbines on power systems.

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REFERENCES

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5

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25 Hours

30

35

40

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Fig. 8. Comparison of actual and synthetic wind speeds for the 48 first hours of January 1976: D, actual wind speeds; @, synthetic speeds.

6. CONCLUSIONS Wind is a very unstable phenomenon that cannot be completely described by any probability distribution, however sophisticated, since the wind speed at any given hour is correlated with the wind speeds in the preceding hours. Therefore any m a t h e m a t i z a t i o n of the wind phenomenon t h a t intends to represent more than the event space of that random variable must integrate these autocorrelations in one way or another. It should be underlined that the approach presented may also be used to model data sampled more or less frequently than at each hour; an hourly model has been presented here since wind speed data is usually collected on an hourly basis. The methodology described proves flexible enough to model the main characteristics of wind speeds and sufficiently accurate for wind turbine energy applications. Two of the main advantages of the B o x - J e n k i n s approach is that it can be easily adapted to data from different sites and that it assumes no probability distribution for the data. W h e n an appropriate model is found for the wind speed, a forecasting or generating function can be p r o g r a m m e d with very little coding

1. R. H. Sherlock, Analyzing winds for frequency and duration. Meteor, Monogr., Am. Meteor. Soc. No. 4, 7279 (1951). 2. R. Corotis, A. Sigl and J. Klein, Probability models of wind velocity magnitude and persistence. Solar Energy 20,483 493 (1978). 3. C. G. Justus, W. R. Hargraves, A. Mikhail and D. Graver, Methods for estimation wind speed frequency distributions. Am. Meteor. Soc. 17, 350-353 (1978). 4. J. P. Hennessey, A comparison of the Weibull and Raleigh distribution for estimating wind power potential. Wind Engng 2, 156 164 (1978). 5. R. Corotis, Simulation of correlated wind speeds for sites and arrays. SUN2, Proc. Int. Solar Energy Soc. Silver Jubilee Cong. 3, 2257 2261 (1979). 6. D. Mage, Frequency distribution of hourly wind speed measurements. Atmos. Envir. 14, 367 374 (1980). 7. M. Stevens and P. Smulders, The estimation of the parameters of the Weibull wind speed distribution for wind energy utilization purposes. Wind Engng 3, 132145 (1979). 8. R. Corotis, H. Sigl and M. Cohen, Variance analysis of wind characteristics for energy conversion. J. Appl. Meteor. 16, 1149-1157 (1977). 9. K. Chow and R. Corotis, Simulation of hourly wind speed and array wind power. Solar Energy 26, 199-212 (1981). 10. P. Giorsetto and K. Utsurogi, Development of a new procedure for reliability modeling of wind turbine generators. IEEE PES Winter Meet. WM249 1, New York (1982). 11. W. V. Cliff, C. G. Justus and C. E. Elderkin, Simulation of the hourly wind speeds for randomly dispersed sites. U.S. Department of Energy. Contract No. EY-76-C06-1830 (1978). 12. G. E. P. Box and G. M. Jenkins, Time Series Analysis, Forecasting and Control. Holden Day, San Francisco (1970). 13. D.G. Luenberger, Introduction to Linear and Nonlinear Programming. Addison-Wesley, Reading, Massachusetts (1973).