Stochastic simulation of hourly average wind speed sequences in Tangiers (Morocco)

Solar Energy Vol. 56, No. 3, pp. 301-314, 1996

0038092X(95)00103-4 Copyright 0 1996 Else&r Science Ltd

Printed in Great Britain. All rights reserved 003&092X/96 $15.00+0.00

STOCHASTIC SIMULATION OF HOURLY AVERAGE WIND SPEED SEQUENCES IN TANGIERS (MOROCCO)

H. NFAOUI,* J. BURET* and A. A. M. SAYIGH** *Laboratoire d’Energie Solaire, Department de Physique, Facultt des Sciences, B.P.1014, Rabat, Morocco

and **Hertfordshire University, 147 Hilmanton, Lower Earley, Reading RG6 4HN, U.K.

(Communicated by DAVID MILBORROW)

Abstract-Twelve years of hourly average wind speed data are used to build an autoregressive model (AR(2)) to simulate hourly average wind speed (HAWS). The model matches well the characteristics of the experimental values of the wind speed. Tests have been performed to validate the model. Comparisons have been made between generated and real series of data to check if the wind behaviour is reproductible. The model is then used to build uu a reference Year for Tangiers and may be used to forecast wind speed, with good results.

1. INTRODUCTION

The peak demand for power in Morroco during any typical day occurs in the afternoon. Since the wind speed in Tangiers also peaks in the afternoon, it is possible that wind energy conversion systems (WECS) could be used as alternative back-up supplies instead of conventional energy. These conventional plants require approx. a 15 min start-up time; and if WECS are to be used as alternative sources of back-up power to the national grid, a dynamic statistical model is necessary to forecast how much back- up power can be expected from WECS a few hours in advance. Since wind power is a function of wind speed, forecasts of power are generally derived from forecasts of speed.

Also, for an autonomous power supply con- sisting of a wind turbine and a single diesel generator, with an energy storage device, the major control decision is when to switch the diesel on and off. This control function could be improved by using wind speed forecasts.

Dynamic statistical analysis based on a stochastic model is used to simulate and to forecast wind speed while retaining the main statistical characteristics of the observed data (average wind speed, variance, autocorrelation coefficients, probability density, persistence, etc.) which are used to build the model.

The model developed is based on the hourly average wind speed (HAWS) values which were obtained from the National Meteorological Service at Tangiers International Airport. These data were collected during the period from January 1978 to December 1989 (12 yr) using methods recommended by the World

Meteorology Organization. In previous works (Nfaoui et al., 1990, 1991) it was shown that it is necessary to have at least 9 yr data to realize an accurate statistical study of hourly average wind speed (HAWS).

Although, certain anomalies were noticed in the results for the month of January, we studied the month’s measurements in detail hour by hour. Thus, we found some abnormally high values of wind speed in 1988, in which the wind speed increased abruptly from 0 to 16 m/s in the course of 1 h. So, we have eliminated the month of January 1988 from the data that we have used.

In order to predict the output of a wind generator system by using simulation methods, it is necessary to have large series of measurements for the site. The real data are variable. To be usable, this enormous volume of data must be reduced without losing any informa- tion. One way to achieve this is the statistical treatment of data. Another way consists of generating a typical year of synthetic data which represents accurately the actual statistics of multi-years. This is the usefulness of the stochastic model for generating one reference year data for Tangiers. This procedure allows one to characterise the wind speed properties of a locality with only few parameters and to provide the user with time saving calculation procedures.

2. LITERATURE REVIEW

The first statistical studies of wind speed as a discrete random variable began 40 yr ago (Blanchard and Desrochers, 1984). Over this

301

302 H. Nfaoui et al.

period, different distribution functions have (Blanchard and Desrochers, 1984; Chou and been suggested to represent wind speed includ- Cortis, 1981; Cortis et al., 1978; Conradse et al., ing those of Pearson, Raleigh and Weibull 1984). Several previous studies, for example,

Table 1. Monthly average wind speed, Weibull parameters (Nfaoui et al., 1990, 1991) and values of x determined from Weibull and Skewness statistic methods

- Wind speed Weibull Hybride Weibull x

Months Urn/s) k c k c WeFbull Hvbr:Weib SkewneP- >” .

J 5.70 1.59 6.55 1.68 F 5.94 1.63 7.06 1.74 M 5.50 1.59 6.65 1.72 A 5.80 1.63 6.96 1.85 M 5.70 1.63 7.09 1.84 J 5.52 1.50 6.90 1.63 J 6.47 1.62 7.95 1.59 A 5.56 1.62 7.54 1.68 s 6.37 1.76 7.60 1.75 0 6.10 1.67 7.28 1.76 N 6.15 1.85 7.01 1.97 D 5.55 1.166 6.65 1.75

V, k and c obtained from observed HAWS (1978-1989).

7.03 0.44 0.47 0.62 7.66 0.45 0.48 0.65 7.16 0.44 0.48 0.67 7.35 0.45 0.51 0.68 7.18 0.45 0.51 0.65 7.08 0.42 0.45 0.58 8.30 0.45 0.44 0.58 7.60 0.45 0.47 0.59 8.28 0.49 0.49 0.67 7.74 0.46 0.49 0.64 7.72 0.51 0.55 0.78 7.26 0.46 0.49 0.66

I’,‘,‘,‘,‘,‘, 14 13 April

v+ ??*** g 12

????

11 ??* ??**** ????

I ii

??**** 10 ?? ??’ 9 ??*****

?? ??*** 8

V[mk]7 k

/-y

Hours HQWS

-0 2 4 6 B 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 Hours Hours

Fig. 1. Mean value and standard deviation of V(hj,m,a), for a given hour h calculated for all the days in the 12 yr.

Average wind speed sequences 303

14 13 12 11 10 9 9 7 6 5 4 3 2 1 0 0 2 4 6 6 10 12 1416 16 20 22

Houla 0 2 4 6 8 10 12 14 16 18 20 22

14 13 12 11 10 9 8 7 6 5

ii OE.‘.‘.‘.‘-‘.‘.‘.‘.‘.‘.~~ 0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22

HOUrX Hour

Fig. 2. Daily variation of observed, transformed and standard deviation of transformed HAWS calculated for all the days of a given month.

Chou and Cortis (1981) and Blanchard and single month (December 1981) and 3 yr for only Desrochers (1984) have attempted to incorpo- 3 months (June, July, August, 1982-1984), rate autocorrelation into wind speed models respectively. However, the previous studies have using the methodology of time series analysis not used such a large series of hourly average developed by Box and Jenkins (1976) to repre- wind speeds as in the present study. The purpose sent the fluctuations of hourly wind speed data. of using several years’ data when compiling the However, none of these studies has covered a model is to increase the reliability of the esti- large number of significantly different locations mated parameters of the model. In particular, a and climatic conditions. Furthermore, some of wind speed model based on several years should the studies have neglected important properties be more representative of future speeds and of the data, such as the seasonal and daily more able to simulate synthetic series of HAWS. variations and nonGaussian shape of wind The first purpose of this study is to build a speed distributions. Thus, Blanchard has fitted stochastic model, AR(p), for different orders, the observed hourly wind speed data by using using a long series of measurements. The aim the stochastic model directly. Brown et al. (1984) of this paper is to show the influence of using and Daniel and Chen (1991) have taken into several years of wind speed data for a given account the disadvantages and the limitations month on the order of autoregressive and of the previous studies, but the data used to moving average processes (ARMA(p,q)) and its develop the model have been restricted to a ability to simulate the HAWS.

l,O

O,Q

‘V

0,7

0,6

0,5

h July 4 lJo tr2,h October

O,Q

0,8

0,7

0,6

0,s ~~~T~Q~~Q0_~~~~0~~00~~~~ -~~~nO~~Qo~~~~nO~~~o~~~~

s-s-~~~v-~~~~~~+J~~ 3-~~~v-~v-~v--r~c*l~~~

Hours Hours

Fig. 3. Comparison of probability density of transformed and standardized HAWS, V’*, and corresponding normal distribution.

3. CONSTRUCTION OF A STATIONARY VARIABLE TO OBTAIN GAUSSIAN

DISTRIBUTION

3.1. Introduction The modelling methodology described here,

consists of fitting ARMA process of various orders to HAWS data which have been, at first, transformed to make their distribution approximately Gaussian and standardized to remove diurnal nonstationarity. Seasonal nonstationarity is removed by fitting a separate model for each month.

3.2. Gaussian transformation Direct application of stochastic models to

HAWS series is not possible due to their non- Gaussian distribution. To resolve this problem, and with the aim of simulating HAWS, Daniel and Chen (1991) have proposed a transformation of HAWS using the Dubey method (1967).

In fact, Dubey showed that, for shape parameters close to 3.6, the Weibull distribution has a shape similar to the Gaussian one. This approach is reasonable because a Weibull random variable to the power of x is also a Weibull random variable. In particular, if wind speed V has a Weibull probability density function:

(1)

where k = shape parameter (dimensionless) c = scale parameter (m/s)

then I/‘” has a Weibull distribution with shape parameter k/x and scale parameter cX (Debuy, 1967). In order to find an appropriate power x for the transformation, it is only necessary to solve the equation k/x=3.6, or x= k/3.6.

The second approach to selecting a power


V January ??Data . V -

April m Data +Nonnal distribution A Normal distribution

0,5 July m Data 0,5 -

-Nomral distribution October m Dra -Normal distribution

%4

083

PD

082

O,l

0.0

Fig. 4. Correlation coefficients (second order) of the reduced variable V’*(hj,a,m) calculated at different hours h.

transformation to obtain an approximately Gaussian distribution is by ‘using the Skewness statistical method (Balouktsis et al., 1986):

S,(m)= i : i a=1 j=l h=l

X {(v;h,j,m,a)- ~“wY~m~/J~~ (2)

where h= the hour of the day j=the day of the month a=the year J=24 h

M = no. of days in a given month A = number of years of observation m = month number.

The Skewness statistical method requires the

transformation of the data using several values of x. S, takes a value of zero when the distribution is completely symmetric (Blanchard and Desrochers, 1984).

The k and c parameters are obtained from Weibull and Weibull hybrid density functions by an iterative method (Nfaoui et al., 1990, 1991). The values of x are determined from the Dubey equation and Skewness method (Dubey, 1967; Balouktsis et al., 1986).

Table 1 shows that the x values obtained from the statistical Skewness method are always higher than those obtained from the Weibull and Weibull hybrid. The maximum value of x occurs in November; this means that the distribution of HAWS is closest to a Gaussian distribution for this month.


January

I

W Data 0 Simulation

acf

098

0,8

094

Q2

3 4 5 Lag (hours)

8 7 8

0,6

I 8 1 2 3 4 5 6 7

Lag [ hours )

July

Fig. 5. Observed and

The model parameters are estimated, using x values obtained from Skewness statistics because this has been found to give the best results for all months.

3.3. Necessity of a stationary random variable To perform a modelling process for wind

speeds by using correlation coefficients requires the use of a second order stationary variable. Without this variable these coefficients lose all significance because they depend on the time origin. It is evident that V(hj,m,a) is not stationary even for the first order; this is illustrated in Fig. 1, which shows the daily variation of V(hj,m,a) taken over 12 yr for every month.

The presence of the night-time data is benefi- cial to a stochastic study of HAWS compared with hourly global solar radiation, but the prob-

4 5 6 7 I

8 Lq [ hours ]

October

simulated function vi.

5 6 7 Lq [ hours ]

li 8

lem of daily variation still exists. The daily nonstationarity can be removed by subtracting the hourly expected value from the data and then dividing by the hourly standard deviation to reduce the data to a normal process with a mean of 0 and variance of 1. The observed series in this study have seasonal statistical characteristics as it has been shown in a previous study (Nfaoui et al., 1990, 1991).

3.4. Elimination of the seasonal variation The issue of time scale should be set down

at the outset. Although previous time-series studies of wind speed have been performed on a seasonal (Brown et al., 1984; Daniel and Chen, 1984) or yearly basis (Blanchard and Desrochers, 1984), the choice of the month would be more representative. We choose to


Table 2. Summary of the model and parameter estimates

1. Model (a) V(t) = wind speed for hour t at anemometer height (b) u’*(r)= (~(t))‘.~s (c) v*(t)=(v’(t)- V)/a’ (d) u’*(t)=~tu’*(t-1)+~2 u’*(t-2)+w(t) where w(t) are uncorrelated and identically distributed Gaussian random variables with zero expected value and a,(2)=0.208 variance (white noise).

Table 2(a). Parameter estimates

/J(r) o’(r) t <LJ(t)>0.6s t (u(t))0.68

1 2.76 1 1.69 2 2.70 2 1.74 3 2.65 3 1.77 4 2.66 4 1.73 5 2.66 5 1.73 6 2.64 6 1.75 7 2.68 7 1.70 8 2.70 8 1.73 9 3.97 9 1.67

10 3.26 10 1.60 11 3.36 11 1.57 12 3.49 12 1.48 13 3.62 13 1.42 14 3.69 14 1.41 15 3.72 15 1.44 16 3.73 16 1.40 17 3.70 17 1.38 18 3.57 18 1.39 19 3.36 19 1.43 20 3.14 20 1.51 21 3.09 21 1.57 22 2.90 22 1.66 23 2.82 23 1.76 24 2.78 24 1.74

fp, =0.737 $,=0.114 0,(2)=0.196

segment the year into monthly periods for prac- tica! reasons. First, most wind system design calculations are performed on a monthly basis. Second, we prefer a time scale sufficiently short, due to the fact that in any particular location, the wind has a relatively homogeneous behaviour within a month. Such a procedure has also been used for the analysis of other meteorological variables such as solar radiation, cloud cover and temperatures (Lou and Cortis, 1985). Seasonal nonstationarity is removed by fitting a separate model for each month.

3.5. Elimination of the daily nonstationarity For a given month m (from 1 to 12), we have

about 360 days of data (12 months). We define for every hour h (from 1 to 24), the hourly wind speed mean in the hour h of month m by:

W,m) = -& i E W,m,j,a) o-l j-l

(3)

M = number of days of a given month A =number of years of observation.

Then, we define the variance for the month m at hour h by:

a2 (km) =A $ ,f$ CW,m,~+ W,m)12 a-l I-1

(4)

which represents for a given hour h of a given month m the dispersion for considered days of HAWS around the mean. Figure 1 shows the daily variation of HAWS for January (m= l), April (m = 4), July (m = 7) and October (m = lo), which represent all four seasons. The standard deviation a(h,m) at either side of the mean is represented by crosses (V(h,m)+a (h,m)). This shows that the dispersion of HAWS for a given hour h has the same amplitude around the mean value for all hours of the day. The same remark can be applied to other months.

From these series, V(h,m) and a(h,m), we define for each day j of a given month m, and for each hour h, the reduced variable by:

V*(hj,m,a) = Vhj,m,a) - v(h,m)

o(h,m) ’ (5)

This classical variable transformation pro- duces a centralized and normalized distribution (mean 0, variance 1).

The daily variation of the observed, transformed and standard deviation of the transformed HAWS are shown in Fig. 2. The variations of transformed wind speed V” and standard deviation of transformed data suggest that daily variations were approximately removed from the wind speed data; the ranges of values are relatively small. For example, if we calculate the mean HAWS for the first week of April (m=4), we find that it is close to that of its last week. The daily nonstationarity is nearly removed.

3.6. Construction of a stationary Gaussian variable

To obtain a centralized and normalized random variable (mean 0 and variance 1) with a Gaussian distribution, Brown et al. (1984) proposed the following formula:

I/‘* (hj,m,a) = V’(hj,m,a) - V’(h,m)

a’(h,m) (6)

where

where V’(hj,m,a)= P(hj,m,a).

SE 56-3-n


Table 2(b). Parameter estimates

p’(t) Hours J F M A M J J A S 0 N D

1 2.480 2 2.490 3 2.458 4 2.532 5 2.518 6 2.507 7 2.473 8 2.476 9 2.520

10 2.544 11 2.744 12 2.912 13 3.071 14 3.147 15 3.180 16 3.156 17 2.083 18 2.95 1 19 2.750 20 2.602 21 2.584 22 2.467 23 2.527 24 2.476

2.793 2.574 2.757 2.353 1.865 2.157 1.940 2.599 2.639 3.784 2.675 2.713 2.607 2.695 2.362 1.854 2.176 1.895 2.635 2.650 3.661 2.725 2.787 2.630 2.654 2.367 1.852 2.135 1.902 2.612 2.639 3.701 2.701 2.688 2.599 2.655 2.390 1.884 2.114 1.857 2.595 2.591 3.616 2.624 2.649 2.576 2.658 2.345 1.861 2.098 1.850 2.528 2.646 3.631 2.593 2.629 2.511 2.641 2.312 1.813 2.101 1.848 2.584 2.652 3.633 2.602 2.646 2.469 2.675 2.375 1.853 2.111 1.855 2.554 2.574 3.618 2.560 2.686 2.466 2.700 2.545 2.169 2.335 1.998 2.683 2.627 3.611 2.630 2.700 2.545 2.965 2.876 2.511 2.628 2.324 3.036 2.771 3.605 2.706 2.843 2.821 3.258 3.025 2.736 2.879 2.644 3.470 3.136 3.828 2.786 3.013 3.040 3.362 3.186 2.874 3.007 2.837 3.645 3.286 4.244 2.981 3.141 3.268 3.492 3.339 2.941 3.072 2.941 3.722 3.393 4.364 3.190 3.259 3.334 3.615 3.395 2.985 3.151 3.062 3.770 3.469 4.43 1 3.303 3.330 3.413 3.685 3.484 3.051 3.247 3.123 3.900 3.504 4.523 3.301 3.328 3.416 3.724 3.530 3.070 3.335 3.200 3.997 3.528 4.507 3.314 3.423 3.434 3.732 3.588 3.083 3.356 3.254 4.012 3.515 4.586 3.280 3.382 3.395 3.696 3.556 3.073 3.309 3.191 3.965 3.456 4.441 3.124 3.238 3.315 3.567 3.454 3.013 3.238 3.098 3.804 3.261 4.064 2.973 3.035 3.154 3.355 3.264 2.836 3.049 2.848 3.438 2.928 3.851 2.810 2.885 2.927 3.141 3.015 2.601 2.780 2.531 3.178 2.713 3.782 2.719 2.799 2.733 3.085 2.771 2.315 2.528 2.242 2.953 2.644 3.746 2.657 2.693 2.555 2.898 2.622 2.110 2.349 2.071 2.842 2.544 3.637 2.579 2.637 2.551 2.819 2.497 2.006 2.298 1.960 2.781 2.615 3.764 2.560 2.687 2.514 2.783 2.415 1.949 2.225 1.953 2.632 2.545 3.805 2.581

4t) Hours J F M A M J J A S 0 N D

1 1.514 1.652 1.784 1.691 2 1.483 1.697 1.724 1.744 3 1.520 1.686 1.735 1.767 4 1.462 1.684 1.715 1.734 5 1.445 1.721 1.715 1.726 6 1.442 1.683 1.704 1.750 7 1.430 1.718 1.681 1.704 8 1.396 1.656 1.698 1.729 9 1.324 1.657 1.719 1.669

10 1.443 1.689 1.762 1.597 11 1.472 1.734 1.665 1.568 12 1.490 1.653 1.590 1.475 13 1.397 1.555 1.518 1.416 14 1.325 1.527 1.501 1.411 15 1.305 1.483 1.531 1.438 16 1.326 1.471 1.560 1.397 17 1.369 1.493 1.604 1.384 18 1.386 1.540 1.649 1.394 19 1.356 1.528 1.592 1.431 20 1.471 1.643 1.653 1.505 21 1.495 1.675 1.721 1.571 22 1.544 1.732 1.772 1.663 23 1.493 1.792 1.780 1.762 24 1.495 1.722 1.762 1.741

1.641 1.601 1.626 1.577 1.570 1.565 1.524 1.519 1.458 1.354 1.278 1.227 1.136 1.166 1.166 1.146 1.176 1.217 1.276 1.362 1.515 1.528 1.609 1.625

41.47 1.586 1.561 1.974 1.649 2.098 1.702 91.42 1.567 1.547 1.957 1.626 2.112 1.686 51.39 1.545 1.506 1.914 1.611 2.034 1.651 41.37 1.524 1.522 1.892 1.643 2.066 1.671 21.34 1.504 1.506 1.918 1.559 2.088 1.703 81.36 1.534 1.511 1.884 1.539 2.118 1.719 11.39 1.527 1.515 1.899 1.548 2.109 1.659 51.35 1.548 1.521 1.898 1.588 2.106 1.609 91.17 1.467 1.497 1.936 1.640 2.236 1.545 81.05 1.287 1.370 1.691 1.581 2.274 1.650

9.990 1.235 1.234 1.640 1.447 2.200 1.696 1.019 1.229 1.198 1.567 1.379 2.180 1.626 1.026 1.208 1.174 1.540 1.229 2.090 1.560 1.012 1.176 1.154 1.541 1.324 2.096 1.596 1.001 1.180 1.138 1.536 1.389 2.098 1.573 1.014 1.176 1.112 1.556 1.410 2.081 1.542 1.031 1.182 1.129 1.529 1.398 2.103 1.597 1.101 1.212 1.155 1.578 1.431 2.197 1.571 1.072 1.253 1.262 1.691 1.528 2.215 1.625 1.204 1.404 1.365 1.804 1.608 2.196 1.640 1.320 1.519 1.482 1.931 1.646 2.292 1.654 1.379 1.568 1.557 2.004 1.647 2.231 1.703 1.406 1.623 1.581 1.997 1.643 2.135 1.708 1.443 1.626 1.577 2.020 1.660 2.145 1.697

V’(‘(h,m) and s’(h,m): mean and variance of trans- gested to estimate the autocorrelation functions formed HAWS in the hour h of the day of (acf) of order p for the hour h of a given month month m. m:

The V’(h,m) and a’(h,m) are supposed constant: V’(25,m)= V’( l,m), V’(26,m)= V’(2,m), etc. r&m) = i E V*(hj,m,a) V*(h + pj,m,a). a’(25,m) = a’( l,m), o’( 26,m) = a’( 2,m), etc. a=1 j=l

(7) 3.7. Stationarity consideration The standard deviation relating to that esti-

To prove V’*(hj,m,a) is really second order mation is the order of (A4_4)-“~5 (MA = 30 x 12 stationary, the following expression was sug- days). Figure 3 displays the correlation coeffi-


Months

January February March April May June July Augst September October November December

Table 3. Overfitting AR(2) model cient for p = 2 defining r,(m) by:

4, 6, o,(2) Q

0.737 0.111 0.186 45 0.780 0.076 0.163 47 0.734 0.126 0.174 51 0.734 0.115 0.198 42 0.757 0.076 0.201 60 0.759 0.077 0.186 54 0.780 0.097 0.161 51 0.751 0.066 0.214 61 0.753 0.099 0.164 51 0.765 0.074 0.190 63 0.748 0.087 0.191 63 0.741 0.132 0.180 38

Degrees of freedom

30 30 30 30 30 30 30 30 30 30 30 30

r,(m)= i r,(b) (8) h=l

r,(m): mean of r&m). If the process is stationary, r,, is the best

possible approximation, of correlation coeffi- cient of order p with our data. We notice that most of the r,(h,m) points are within the rp + (A4A)-“.5 lines (Fig. 3). This proves that the stationarity hypothesis does not contradict the test because the r&m) is an approximation of the correlation function with average limits f (MA)-0.5 (Blanchard and Desrocher, 1984).

Model rejected for Q > 51.

f January f April

097 0,7

0.5 0,s

033 OS3

O,l O,l

-0,l -0,l 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

Lag(hours) Lw [ hours ]

July o,g -

PWf October

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 L~JJ [ hours ] Lag [ hours ]

Fig. 6. Estimates of partial autocorrelation function vi of transformed and standardized data.


Table 4. Comparison of actual and synthetic series

Months Variance

0 r2 % error

on ii % error

on 0 % error

on rr

January

February

March

April

May

June

July

Augst

September

October

November

December

5.70 4.30 0.900 0.841 5.48 4.09 0.894 0.835 5.94 4.46 0.913 0.856 5.98 4.53 0.911 0.854 5.50 4.23 0.906 0.852 5.66 4.32 0.907 0.853 5.80 4.04 0.892 0.830 6.01 4.16 0.893 0.830 5.70 3.99 0.892 0.824 5.83 4.04 0.893 0.824 5.52 4.41 0.900 0.837 5.74 4.41 0.901 0.838 6.47 5.23 0.915 0.859 6.65 5.30 0.915 0.860 5.56 4.68 0.884 0.811 5.71 4.72 0.885 0.812 6.37 4.80 0.912 0.858 6.65 4.91 0.912 0.858 6.10 4.43 0.898 0.832 6.24 4.58 0.899 0.833 6.15 3.98 0.897 0.833 6.37 4.09 0.897 0.834 5.55 4.20 0.903 0.846 5.69 4.37 0.904 0.847

4

1

3

4

2

4

3

3

4

2

4

3

5

2

2

3

1

0

1

1

2

3

3

4

1

0

0

0

0

0

0

0

0

0

0

0

First line: actual data. Second line: synthetic data.

3.8. Normality consideration The application of the ARMA(p,q) model to

simulate the HAWS requires in addition to the stationary variable, a normal distribution variable. Figure 4 shows sample comparisons of the probability density function of the transformed and standardized variable for every month with the corresponding normal probability density. It can be noticed for every month that the Gaussian curve envelopes the major part of the histograms, indicating that the transformation is approximately Gaussian.

3.9. Monthly autocorrelation function For a given month m the acf for p= 1,2,3, . ..,

18 using formula (9) were calculated. The acf has also been calculated by an alternative method taking into account all the hours of the day (h= 1,2,3, . . . . 24) for a given month m in order to establish a series of successive values by replacing the theoretical autocorrelation by the one estimated by the Yule-Walker equations given by (Balouktsis et al., 1986):

1 c,(m) =

’ JMA - Ap

(V’* (h,m,a)-I/‘*) x (V’*(h+p,m,a)-_r*) (9)

where JM = total number of hours for a given month and

r,(m) = C,(m)/C,(m) reflect the dependence of wind speed at instant h on the wind speed at h+p.

Applying both methods to the month [formu- lae (8) and (lo)] gives similar characteristics of acf, which decays exponentially and slowly (Fig. 5). Such correlation until 8 seems to indi- cate an important stability of V’*(h,m,a) during the previous 8 h.

4. METHODOLOGY

4.1. Introduction A class of models which has been widely used

or suggested for the description of time series is the ARMA model. The model was built using the methodology for time series analysis developed by Box and Jenkins (1976), which allows dependant variables, to be studied and takes account of the nature of that dependance. It also allows the user to single out, from an entire class of models (ARMA), one that would best represent the original data.


0.10 - r 64

0.05 -

January *

Confidence region

0.10 r (a) April

1 Confidence region I

0,lO r (a) July

0.05 L c 4

0,lO . r (a)

0.05 -

October.

I Confkfenoe region

Fig. 7. Estimates of autocorrelation function of the residuals r&4.

4.2. Stochastic model presentation The ARMA model takes into account the

past values of the data and the random change

of the climate represented by the white-noise process w(t). The model order is defined by analysing the acf and the partial autocorrelation function (pacf) shape, which we will define subsequently. It is obvious that the moving average parameters will be equal to zero if the model is a pure autoregressive AR(p) process. An ARMA process of order p and q is expressed as (Box and Jenkins, 1976):

Z(t)= f qiZ(t-i)+ i OiW(t-i)+W(t) i=l i=l

(10)

where li = autoregressive parameters 8i = moving average parameters

Z(t) = standardized and transformed hourly wind speed for the hour t

w(t) = random noise normally distributed with zero mean, variance obtained from the observed data

p and q = p order of the autoregressive operator and q order of the moving average operator.

4.3. Model building process The identification step consists of the determi-

nation of the model parameters for each month. For a purely autoregressive model AR(p), the acf decreases exponentially and tails off while the pacf cuts off after p order. If both tail off, an ARMA(p,q) model is recommended. The white noise variance of the autoregressive AR(p) model is estimated by (Box and Jenkins, 1976):

02(p)=aZ(V*)(l-rr,~1-rr,~,- . . . -r,pp) (11)

v2( I/‘*): variance obtained from the transformed standardized observed data.

The partial autocorrelation functions ( pacf ) are obtained by the following formula (Box and Jenkins, 1976):

rk=v)1rk_1+v2rk-2+ . . . +pprkdp. (12)

The pacf pi are simply the autoregressive coefficients considering the dependence of wind speed at any given hour on the wind speeds in the preceding hours. The pi can be found from the previous system of equations using the Gauss resolution method (Box and Jenkins, 1976; Branger, 1977).

The stationarity of the model can be verified by the following conditions (Blanchard and Desrochers, 1984; Brown et al., 1984; Daniel


April PDF(%)

5

January 15

0 0 0 2 4 6 6 1012 14 16 16 2022 24 0 2 4 6 8 10 12 14 16 18 20 22 ;

Wind speed (m/s) Wind speed [mls]

-Simulatio

0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 speed Cm/s)

10 12 1416 18 2022 24 Wind Whd speed [m/s]

Fig. 8. Comparison of Weibull model and data-based probability density function of V.

and Chen, 1984):

The fitted model is adequate if the simulated data are in good agreement with observed data and the autocorrelation function T&I) of the residuals a(t)=o’*(t)-(p,u’*(t- l)+p,v’* (t-2) + w(t)) are uncorrelated and normally distributed with mean 0 and variance l/&VU--2) (Blanchard and Desrochers, 1984).

5. APPLICATION OF THE MODEL TO

GENERATEAREFERENCEYEARFORTHE

TANGIERS LOCATION

5.1. Introduction After the condensed outline of structure of

the stochastic model, model selection crititia are employed to identify the most appropriate

ARMA process, in order to estimate its parameters and to check its validity. To illustrate the whole model building process by showing how all these ideas may be brought together to built seasonal models, these models have been used to generate a typical year of the considered data, and may be used to forecast wind speed.

5.2. Parameters estimation of the current model The rp and pi are obtained from the standard-

ized and transformed data using 12 yr of HAWS (Tables 2 and 3, and Figs 5 and 6). Figure 6 shows the plot of rp versus lag(hs), it can be seen that the acf decreases exponentially and slowly. The pacf pI is positive and significantly high while q’z is also positive but smaller (Fig. 5). This fact and the shape of the acf (Fig. 6) suggested that an AR(p) model is appropriate since the acf tails off and pacf cuts off at two. The results obtained for each month are tabu-

Average wind speed sequences

Table 5. Results of five generations of 720 values of V for April

313

Mean Variance acf % error % error % error v u rl on V on ir on r,

Actual synoptic data 5.865 4.03 1 0.893

Series of 720 values Run number: 1 6.337 4.373 0.904 8 8 1

2 6.718 4.434 0.889 15 10 1 3 5.461 4.363 0.911 7 8 2 4 5.617 4.172 0.909 4 3 2 .

acf: autocorrelation function.

lated in Table 3. The estimates of vi and qz have the same behaviour for all months as shown in Table 3, hence an AR(2) process was selected to model HAWS which confirms previous studies (Blanchard and Desrochers, 1984; Brown et al., 1984; Daniel and Chen, 1984).

5.3. Acceptance of the AR(2) model The AR(2) model was used to generate a

series of 8640 values (12 months) of synthetic wind data for each month, each month has its specific parameters. The mean, the variance and the first eight acf were calculated; the results are listed in Table 4 together with the actual parameters. The acf of the simulated and the original series are shown in Fig. 6 for different months. It is evident that the two acfs were not very different.

If the fitted model is adequate, then the autocorrelations of the residuals r&r) should be uncorrelated and normally distributed with mean 0 and variance l/AMJ. The first 32 autocorrelations for the residuals are presented in Fig. 7 which also shows standard error limits +2 (MAJ)-‘.‘. If the assumption of normality is justified, then 95% of the r&z) should be within the +2 (MAJ)-“~5 boundary. This is verified for April and July for which only one of the 32 is outside the confidence region. The model is finally accepted if these residuals are found to be uncorrelated; a statistic test using the statistical Q is performed in order to accept or reject the model which is given by (Blanchard and Desrochers, 1984; Daniel and Chen, 1984):

Q=AMJ i r; (a) (14) k=l

where L = maximum lag considered. Assuming that the residuals are uncorrelated,

this statistic is x2 distributed with L-p degrees of freedom (p = 2). Table 4 shows that the model is accepted for 7 months, with a significance level of 1% (x2<51).

5.4. Model validation To check the validity of the AR(2) model in

relation to the statistical characteristics of observed series, we have compared the observed and simulated values for every month (monthly average, variance and the two first acfs). The results listed in Table 4 are satisfactory for all months (error does not exceed 4% for the mean, 5% for the variance and 1% for the first acf). These results lead to the conclusion that the main statistical characteristics are preserved by the AR( 2) model. The comparison of probability density function (PDF) of simulated series using the AR(2) model with observed series (Fig. 8) shows its ability to generate accurate PDF, particularly for wind speeds higher than 1 m/s. We expect that the AR(2) model will be more adequate for locations where the calm frequency (v=O) is feeble. When compared with the Weibull model, the AR(2) model enables the generation of a long series of data which has identical statistical characteristics to measured data (Fig. 8). In addition, the AR(2) model can generate synthetic time series data.

5.5. Determination of a reference meteorological year

In order to illustrate the method used to generate a reference month, we consider the month of April. Using the autoregressive AR(2) model selected for this month to generate a series of 720 values (30 x 24 values), we calculated the main characteristics of wind speed. The results for five separate generation runs of synthetic sequences are presented in Table 5. The analysis of this table shows that run 4 reproduces most satisfactorily the actual synoptic input parameters. In this way, we can generate one month of synthetic sequences that represent the actual statistical characteristics of 12 yr of data for April. By the same way for each month, we can generate a synthetic year of data which we call the reference year.

314 H. Nfa loui et al.

6. CONCLUSION

It is clear from the above analysis that the AR(2) autoregressive model is capable to simulate HAWS. The error percentage of monthly simulated average wind speed is not higher than 3% for any month. This result is better than the result obtained by Blanchard, the improvement is due to the fact that measurements were made over a longer period of time. The autoregressive AR( 2) model could be used to generate reference monthly data in order to establish a reference year by generating several series for each month and then selecting the series with statistical characteristics closest to the real series. This reference year can then be used to predict the performance of a wind generator in a given location (Gordon and Redy, 1988; Bossanyi, 1985; Bensaad, 1985; Boulay, 1990).

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Stochastic simulation of hourly average wind speed sequences in Tangiers (Morocco)