1582 IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December 1999

Wind speed simulation in wind farms for steady-state security assessment of electrical power

systems A. E. Fei,j6o* J . Cidrb (IEEE member)* J. L. G. Dornelas**

* Departamento de Enxeiieria Elktrica, Universidade de Vigo E. T. S. E. I. M., Lagoas-Marcosende, Vigo, Spain

Tel.: +34 86 812600 Fax.: +34 86 812173 Email: afeijoo@uvigo.es ** Uni6n Fenosa, Avda. Arteixo, 171, A Coruha, Spain

Abstract-Two methods are proposed to asses8 the impact of multiple wind farms On a large electrical power system. The methods are based on the Monte Carlo simulation of the wind speed occurring in each wind farm, taking into account measurements of mean values and correlations. The flrst method is based on the wind speed distribution, assumed to be Rayleigh, and on t h e correlation matrix, obtained from previously known simultaneous wind speed series. The sec- ond method is based on the application of the simulation to the wind speed series, taking into account the previously ob- tained conditional probabilities of the wind speeds a s func- tions of other wind speeds in wind farms.

Ifegwmds- Wind energy, Rayleigh distributions, asyn- chronous machine, Monte Carlo simulation.

I. INTRODUCTION

Wind energy has become an increasing source of energy in recent, years and the future trend seems to continue this development [l]. As an example, in Galicia, a region lo- cated in the northwcst of Spain, about 2800 M W will be installed over the next ten years [2]. When the steady-state of the whole electrical power system must be studied, the real powers injected and the reactive powers consumed by these wind farms must be considered. This is not easy, because of the variability of the wind speed, and the diffi- culty of predicting tlie behavior of the wind simultaneously in several locations.

The methods proposed here consist of simulating the wind speed distributions in the locations where the wind farms are installed, taking into account the wind speed probabilities and their correlat,ions.

The wind speed probability distribution is generally con- sidered to follow the Weibull distribution [3]. In this paper we use the Rayleigh distribution, which is a special case of the Weibull distribution where the shape factor, I C , equals 2.

PE-144-EC-0-08-1998 A paper recommended and approved by the IEEE Energy Development and Power Generation Committee of the IEEE Power Engineering Society for publication in the IEEE Transactions on Energy Conversion. Manuscript submitled March 23, 1998; made available for printing August 14, 1998.

The use of the Rayleigh probability distribution is recom- mended by the International Electrotechnical Commission in its document IEC 61400 1 [4], relating to the design of wind turbine generator systems, for the classes I, 11, I11 and IV defined there.

Some information about the Rayleigh distribution can be found in appendix I.

The wind speeds are considered the same for all machines in a wind farm. A method for taking into account the influence of a machine on another machine situated behind the first one is presented in [5 ] .

11. SIMULATION OF WIND SPEEDS IN SEVERAL WIND

The calculation of the frequency distribution of the wind speed in a wind farm is not particularly difficult, if the Monte Carlo simulation method is employed, because, as can be read in appendix I, the Rayleigh distribution is eas- ily inverted.

The problem arises with the simulation of the wind speeds in several wind farms when they are simultaneously calculated, because a correlation between them may exist.

One way of solving the simulation of multivariate distri- butions with given correlations is explained in [6]. Accord- ing to this method, given a vector of independent variables z = (a, zz, . . . , zJT (where T denotes matrix or vector transposition), with means pz = (pal, f i z z , . . . , pLzn)T and covariance matrix:

FARMS. METHOD 1

a new vector 21 = (~1,212, . . . ,U,)!" of correlated variables can he obtained, with means f i = (py l ,pLarz , . . . , pyn)T and covariance matrix:

0885-8969/99/$10.00 0 1998 IEEE

1583

3. The values so obtained must be operated on as in (I), in order to obtain new values with the covariance matrix and the mean values desired.

If this method is applied to non-normal distributions, the following problems occur: 1. During the process of obtaining the correlated wind speeds from the independent ones, negative values appear. They will not be considered. 2. The shape of the frequency distributions have small dif- fcrences from the Rayleigh distributions.

Using seven wind sites, our simulations always resulted in fcwer than 5% negative values. The number of negative valucs seems to grow with the number of wind farms in thc simulation. Simulations with 24 wind farms in our elcct,rical grid have had fewer than 10% negative values. Dropping these negative values causes small errors in the vcctor of means and the covariance matrix. These errors can be ignored as will be seen in the example given later.

The second problem cannot be solved, and must also be assumed as an error of the procedure, which is not very important as will be seen.

. A good approximation is achieved. . No temporary sets of speeds are needed for the simula- tion Only the mean wind speeds are necessary.

Tbe advantages of this method are:

111. SIMULATION OF WIND SPEEDS IN SEVERAL WIND

The second proposed method is based on conditional prnbabilities. Thc joint distribution function of a set of dcpendent variables y = (yl, y2,. . . ,yn)T can be expressed through the conditional distributions [GI:

FARMS. METHOD 11

if the following procedure is applied:

where L is a lower triangular matrix such as:

R, = L L ~ (2)

The matrix L is not unique, and a good choice for it is

The relationship between the mean values and standard that obtainable through Cholesqui's technique [7].

deviations of the set of variables t and y are as follows:

E(Yj = L W + PY (3) n, = L C I , L ~ (4)

When the set of variables z has a means vector pLt = (O ,O, . , , , OjT and covariance matrix:

1 0 . . . 0

n z = [ 0 1 ':,; 0)

0 0 . . . 1

the result of the operations above is that the set of variables y has a means vector E(y) = p, and a covariance matrix

Moreover, when this procedure is applied to normally distributed variables, the resulting vector y is a set of nor- mally distributed variables. This property is not fulfilled by other distribution functions. Tbc problem of multivariate statistical simulation applied to some non-normal probabil- ity distribution functions has been studied in [SI, but there is not a specific treatment of the Weibull or the Rayleigh distributions. Also, in [Q] some cases of multivariate dis- tributions with Weibull properties have been studied, and a generalization of the multivariate Rayleigh distribution can he found in [lo]. In both cases, the mathematical com- plexity is not justified, and some simplifications must be made.

In our case, the method described above is applied to Fiayleigh distributions taking into account that the errors committed can be ignored, as will be justified. Our method consists of applying the preceeding concepts to the prob- lem of generating set,s of wind speeds with given Rayleigh distributions and correlation matrices, and can be summa- rized as follows: 1. Generate wind speeds by means of the Monte Carlo sim- ulation, according to the Rayleigh distribut,ion of each wind farm. 2. The mean value of the wind speed of each wind farm must be subtracted from the wind speed obtained, and the result of this operation must be divided by the standard deviation. With this operation, sets of uncorrelated stan- dardized variables are obtained.

n, = LLT.

( 5 ) F(Y) = R(vi)Fz(yz/yi = Y?). . .

0 0 F n ( u n l l / l = ~ l r . . . 1 ~ n - i =Yn-1)

wbcie F ~ ( y l j is thc distribution function of the variable y1, F~(u21y~ = 0:) is the distribution function of the variable y2, conditioned by the variable y1, and so on.

To sample the vector U , n random numbers T I , ~ 2 , . . . , vn must be generated, and the following equations must be solvcd:

Fi(y1) = r1 (0)

(7)

(8)

F2(Y211/1 = U 3 = 7.2

yn-i) = Tn 0 0

F n ( ~ n l ~ / i = VI , . . , 3~n-1

But, in our case the method is applied based on prob- abilities and not on distribution functions, although the conccpts to be used are the same. The only data that are needed are a set of simultaneous wind speeds in all the sim- ulated wind farms. The method consists of the following steps:

1584

1. Generate a random number to simulate the wind speed for a given hour a t the first wind farm. If there are n1 wind speed values for wind farm 1, then 0 5 T I 5 n1. 2. Select the wind speed value U,l in position T I from the speed vector of wind farm 1. This is the wind speed for the first wind farm. 3. Count the number of times that U71 appears in the data for the first wind farm. 4. Generate a random number r2 in the data set for wind farm 2, so that 0 5 1'2 5 U,I. 5. Select the wind speed value Ur2 in position r2 from the speed vector of wind farm 2. This is the wind speed for the second wind farm. 6. Count the number of times that UT2 appears in the data for the second wind farm, corresponding to the wind speed U,, in the first wind farm. 7. Repeat for each of the remaining wind farms. For wind farm i, the wind speed values for all wiud farms for all j in 1 5 j 5 i must he taken into account.

As can be seen, for the generation of the wind speed of the second wind farm, the wind speed of the first one is taken into account. For the generation of the wind speed of the third wind farm, the wind speeds of the first and the second wind farms are taken into account and so on.

The process must be repeated a number of times equal to the number of wind speeds desired for each wind farm.

As an example, in Fig. 1 an illustration of the proposed method can be seen. In the example, for the sake of sim- plicity, only four wiud farms are considered, and in each of them a set of only 20 wind speeds is taken into ac- count. Let us suppose first that the sets of the example are based on data taken in the wind farms. The process can be begun in each of them. Let us suppose then, that the'process is carricd out in the order 1, 2, 3, 4. For the first wind farm, the probabilities of wind speeds are

P(U1 = d) = $, P(U1 = e ) = & and P(Ul = f ) = 20 '

If the random number gives as a result a wind speed of a ms-' for the first wind farm, this will condition the pos- sible wind speeds in the second one. So, in this case, the probabilities for wind farm 2 are P(U2 = alU1 = a ) = 2 5

and P(U2 = blUl = a) = g, If the random number for this case gives as a result that the wind speed in wind farm 2 is b ms-', this will condition the probabilities of occur- rence of wind speeds in wind farm 3. For wind farm 3 these probabilities are P(U3 = a.lU1 = a, U2 = b ) = I 3 and P(U3 = ciUr = a,U2 = b ) = $; Let us suppose that the random generation gives that wind speed in the wind farm is c ms-'. Following the same process, finally the probabil- ities for the fourth wind farm are P(U4 = alU1 = a, UZ = b, U, = c) = 5 and P(U4 = dlUl = a, U2 = b, U, = c) = 4.

P(U1 = a ) = &, P(U1 = b) = 3 20 P(U1 = C) = 1,

1

The advantages of this method are: . A good accuracy is achieved in the mean speed values and in the correlation matrix.

6 7

TABLE 1 SCALE PARAMETERS, SPECIFIED MEAN VALUES AND CALCULATED

MEAN VALUES OB THE WIND SPEED DISTRIBUTIONS OF THE WIND

FARMS CONSIDRRED IN THE EXAMPLE

7.22 6.4 6.5 6.3 10.49 9.3 9.5 9.3

TABLE I1 CORRELATION MATRIX OBTAINED FROM THE TEMPORARY SETS

1 1 1 I 2 1 3 1 4 1 5 1 6 1 7 1 1 11 1.00 1 0.87 I 0.83 I 0.84 I 0.02 I -0.01 I -0.03 I

-0.01

1.00 0.91 0.89 0.06 0.04 0.02 -

IV. EXAMPLES

0.02 0.05 0.02 0.82 0.87 1.00 -

Examples of the proposed methods are presented here. Simulat,ions are carried out with seven wind farms, in or- der to prove that the errors committed when calculating the frequency of occurrence of each wind speed are small. The scale parameters, C, of the wind speed Rayleigh distri- butions of the seven wind farms are given in Table I. In the same table the specified mean wind speed (obtainable from the scale parameter, as can be seen in appendix I), and the mean wind speed obtained after the simulations by both methods, U?'' for method I and for method I1 are given. As has been explained, the differences between spec- ified and calculated values are due to the fact of removing negative values of the wind speeds.

In the example 50000 wind speeds for each wind farm were generated, and about 4.6% of the cases had at least me negativc value.

In Table I1 the specified correlation matrix for the exam- ple can be seen, and in Table I11 the calculated correlation matrix is given for the same example.

In Fig. 2 the Rayleigh distributions for the seven wind Farms simulated can be seen, and also the frequency dis- tributions obtained by simulating them with the proposed method.

Finally, in Table IV the correlation matrix can be seen, . No negative wind speeds are generated. so all speeds generated are valid.

wind obtained frorn the application of method 11, and the means obtained, U@", can be seen in Table I

1585

i 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 wind farm 1

wind farm 2

wind farm 3

wind farm 4

Fig. 1. Examalo of method I1 simulation proceis

Pig. 2. Wind speed Croquency distributions and Rsyioigh distribution density funclions for the wind farms simulated in the example

v. W I N D FARM MODEL BASED ON THE ASYNCHRONOUS MACHINE MODEL

In all the wind farms that will be installed in our re- gion the generators are asynchronous generators. This fact means it is useful to employ a wind farm model for the steady-state analysis based on the induction machine model., The model is a PQ model, where the reactive power depends on the active power.

In such a model, Ihe real power depends on the features of the turbine, and is found as a function of the wind speed:

p = J ( U ) (9)

A way to calculate t,he reactive power as a function of

the real powcr is proposed in [ll], through the following expression:

Q = -80 - QiP - QzP2 (10)

where thc constants Qo, QI and Qz are experimentally obtained.

It is not always possible to obtain these three constants experirncntally, based on thc steady-state model of the in- duction machine. If we consider the magnetizing reactance at the tmninnls of the machine, we can calculate the reac- tive power consumed as:

1586

2 3 4 5

TABLE I11 CALCULATED C 0 1 ~ R B L A T I O N MA'PRIX APPLYING METHOD 1

0.86 0.82 0.83 0.02

TABLE V DATA OCRRESPONDINC T O THE EIGHT WIND FARMS

0.88 0.85 1.00 0.04

1 1 1 I 2 1 3 / 4 1 5 1 6 1 7 1 I/ 1.00 I 0.86 I 0.82 I 0.83 1 0.02 I -0.01 I -0.03

0.05 0.08 0.04 1.00

wind farm 1) C I U I 1 1 8.46 I 7.5 I

6 7

1.00 0.90 0.88 0.05

-0.01 0.03 0.06 0.03 0.90 1.00 0.86 -0.03 0.01 0.04 0.01 0.81 0.86 1.00

0.90 1.00 0.85 0.08

0.03 0.06 0.03 0.90

0.01 0.04 0.01 0.81

7.5 7.5 7.5

6 6

7.5 6 -

TABLE IV CALCULATED CORllBLATlON MATRIX APPLYING METHOD 11 TABLE VI

ESTIMATED CORRELATION MATRIX F R O M THE BIGHT WIND FARMS -

2 0.87 1.00 0.91 0.89 0.09 0.08 0.06

- -

-

- 3

0.82 0.91 1.00 0.86 0.12 0.11 0.08

- __

-

- G

0.01 0.08 0.11 0.08 0.90 1.00 0.87

- __

-

- 7

-0.02 0.06 0.08 0.05 0.80 0.87 1.00

- -

-

- 2

0.2 1.0 0.8 0.2 0.0 0.0 0.0 0.0

- -

-

- 3

0.2 0.8 1.0 0.0 0.0 0.0 0.0 0.0

- -

-

- 4

0.0 0.2 0.0 1.0 0.0 0.0 0.9 0.0

- -

-

- 5

0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0

- -

-

- 6

0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0

xz3

-

0.0 0.0 0.0 0.0 0.0 0.0 0.9 0.0 0.0 0.0 0.0 0.0 1.0 0.0 I 0.0 1.0

(11) 1 v2 v 4 - 4 p = x =

Q = V 2 (- -&,) + E - ' 2x Xlfl influencc of wind farms on the steady-state security of the

electrical system. where V is the voltage, X, is the magnetizing reactance, ' B, is the snsceptance of the capacitors bank for rcactive power compensation' and is the Of the rotor stator leakage reactances.

As has been stated, the final model is a PQ one, and for this reason, the calculation of the reactive power as a function of the real power is made under the assumption that the voltage is 1 P.u., because the error is not impor- tant. However, if more precision is required, the reactive power can be recalculated in each iteration of the load flow analysis.

The model used hcre requires previous knowledge of the parameters of the machine. If they are not known, they can be estimated. A method for the estimation of the pa- rameters of an asynchronous machine is proposed in [12].

As a summary of the process for obtaining the model, the following two steps must be followed: 1. Calculate the real power using the relationship betwcen the mechanical power and the wind speed. The real power and the mechanical power are assumed to be equal. 2. Calculate the reactive power using the relationship be- tween the reactive power and the real power, using the known parameters of the machine.

In Fig. 3 a power network simulated with method I can be seen. The interest of it consists of knowing the probabil- ity of each possible power generated by all the wind farms, in order to know the conditions of working if only one of the interconnection lines

Method I was applied to predict the probability of occur- rence of all possible real powers in a part of the Galician electrical network, where eight wind farms are to be in- stalled. Method I1 is not used because no known sets of wind speeds are available.

The average wind speed, 0 and the Rayleigh shape fac- tor, C, in each wind farm are given in Table V, and the correlation matrix estimated is given in Table VI, accord- ing to the distance between wind farms and to previous data analyzed from the area.

The reactive powers can be calculated as suggested in (11).

be active.

VII. ~ ~ N d L u s r o N s Two methods have been proposed to calculate the prob-

ability ofoccurrence of wind speed in several wind farms si- multaneously. The following conclusions can be extracted: 1. Knowing the mean wind speed and the correlation of the wind speeds in several wind farms, a Monte Carlo sim- nlation can be carried out, obtaining combinations of wind speeds for all the wind farms, where the distributions of the wind speeds are Rayleigh distributions with very small

AppLrCAT1oN To THE OF STEADY-STATE SECURITY ASSESSMENT

The method of obtaining corrclated wind speeds having Rayleigh distributions is here applied to the analysis of the

1587

WINDPARM I ( IIMWI [2] J . Moral, “El plan e6lico estrategico de Galicia”, in Libro de poneneias de las Jornadas de energia eblica. Santiago de Com- postela. Oct. 1997, Instituto para la Diversificacih y Ahorro de la energfa (IDAE).

[3] L. L. Reris, Wind energy conversion systems, Prentico Hall, 1990.

[4] IEC 61400 1. Wind turbine generator systems. Pa r t 1: Sofety requirements, 1994.

[5] N. G. Mortensen, L. Landberg, I. Troen and E. L. Petersen, Wind atlos analysis and application program (WASP), 1993.

[GI J. Kleijnen and W. van Groendall, Simulation. A statistical perspectiue, Wiley and Sons, 1992.

[7] J . Cidris and E. Dias, Mdtodos numirims en el andlisis de sistemas lineales de gran dimensidn: tdcnicas de esplatacidn de matrices dispersas, T6rculo Artes Grificw, S. A. L., 1996. M. E. Johnson, Multivariate statistical simulation, Wiley and Sons, 1987. L. Lee, “Multivariate distributions having Weibull properties”, Journal of multivariate analysis , no. 9, pp. 267-277, 1979.

“A generalisation of the multivariate Rayleigh INm”C0“eCTIIIN LINB 1 distribution”, Sankhya: The indian Journal of statistics, vol.

[11] P. S~rensen, “Methods for calculation of the fiicker contributions from wind turbines”, Tech. Rep. Ris@-IL939(EN), Risvl National

[12] M. 11. Haque, “Estimation of three-phase induction motor pa- rameters”, Electric Power Systems Research, vol. 26, pp, 187-

WIND P A W 6 (4 MW)

WMDTARM 5 0 4 MWI

INC*RCII”ELT”N LlNP 1 [8]

[9]

[IO] D. 1%. Jensen,

- A, no. 32, pp, 193-208, 1970.

WINDFABMRCIMWI Laboratory, 1995.

Fig. 3. Electrical network simulated 193, 1993.

APPENDIX

I. THE RAYLEIGH DISTRIBUTION

The probability density function of the Rayleigh distri- bution can be writt,en as:

P ( U ) = a z e U -q:y 2 = ,V,-(R)~ C’ (12)

where U is the wind speed, a = a and C is the scale factor.

The cumulative probability function of the Rayleigh dis- tribution, which measures the probability that the wind speed is below a given wind speed U0 can be expressed as:

Jz

h--.------~ ~~

/I 2s I,, 7s ,,xi 12s 1x1 Rei,, p,,wor GCl‘bNlCd I / / MW

Fig. 4. Probability of generating real power

P(U < uO) = / u‘U -e-g ‘ ( v = l - e - z ( a ) 0 = 0 a2 (13)

” a errors. = 1 - e - ( + ) 2. One of the problems of the first method is that some negative speeds appear during the calculation, the number of which grows with the x~~~~~~ Of wind ing them, small errors appear in the average Value of the wind speeds and in the correlation matrix. 3. The second method is based on sets of known data, and

Carlo simulation. The final results are similar to the known sets, and not necessary to the Weibull or Rayleigh distri- butions.

Given a Raylei& distribution, its mean value and vari- ance can be calculated as a function of its scale parameter By

thc G~~~~ function, as:

(14) it reproduces possible situations by applying the Monte U(, = cr (i) = cT f i

VIII. ACKNOWLEDGEMENTS So the standard deviation of the Rayleigh distribution The financial support received from Uni6n Fenosa is

can he given as: gratefully acknowledged by the authors.

REFERENCES [I] K. Rehfeldt, ‘LWiiidenergienut~ung irn internationalen Vcrgle-

ich”, DEW1 Magazin, no. 11, pp. 24-29, 1997.

1588

The Rayleigh distribution is one of t,he distributions that can be easily reproduced using the Monte Carlo simulation method [8], because it can be inverted, as can be seen in the following equation:

BIOGRAPHIES Andr6s Elias Feij6o Lorenzo received his licenciate degree

in Electrical Engineering from the Universidade de Santi- ago de Compostela, Spain, in 1990. He is now with the Departamento de Emelieria Ele'ctrica (Electrical Engineer- ing Department) of the Universidade de Vigo (University of Vigo), Spain, and his current interest is wind energy.

Jose CidrLs Pidre received his Ph. D. degree in Electrical Engeneering from the Universidade de Santiago de Com- postela, Spain, in 1987. He is professor and head of the Departamento de EnxeEem'a Ele'ctrica of the Universidade de Vigo, Spain, and leads some investigation projects on wind energy, photovoltaics and planning.

Jose Luis Garcia Dornelas received his Ph. D. degree in Electrical Engeneering from the Universidad Polite'cnica de Madrid, Spain, in 1981. He is head of the Departamento de Planificacidn de nedes Eldetiicas (Electrical Network Planning Department) in Unidn Fenosa.