Journal of Wind Engineering and Industrial Aerodynamics 51 (1994) 105-116 105 Elsevier

Optimization of wind turbine positioning in large windfarms by means of a genetic algorithm

G. Moset t i , C. Po lon i and B. Div iacco

Dipartirnento di Energetica, Universit~ degli Studi di Trieste, Via Valerio 10, 34100 Trieste, Italy

(Received October 21, 1992; accepted in revised form May 17, 1993)

S u m m a r y

In this paper a novel approach to the optimization of large windfarms is presented. The wind turbine distribution at a given site is optimized in order to extract the maximum energy for the minimum installation costs. The optimization is made by associating a windfarm simulation model based on wake superposition with a genetic search code. The purpose of the paper is to prove the feasibility of the method by analyzing the results obtained in some simple applications. As a test case, a square site subdivided into 100 square cells as possible turbine locations has been taken, and the optimization is applied to the number and position of the turbines for three wind cases: single direction, constant intensity with variable direction, and variable intensity with variable direction.

1. I n t r o d u c t i o n

Wind tu rb ines for p roduc ing e lec t r ica l ene rgy n o w a d a y s are m a t u r e in t echnology . W h e r e l a rge wind r e sources exist, wind ene rgy conve r t e r s a re even becoming economica l ly compe t i t i ve c o m p a r e d to e lec t r i ca l power p roduced by o the r conven t iona l means . However , for m u l t i - m e g a w a t t p roduc t ion , a l a rge n u m b e r of wind tu rb ines m u s t be ins ta l led, and the efficiency of the windfa rm is h igh ly inf luenced by the i r pos i t ioning. The p re sen t work discusses a nove l a p p r o a c h to the wind t u rb i ne pos i t ion ing problem. The me thod adop ted is based on the pr inc ip les of gene t ics and n a t u r a l evolut ion , t h a t m a k e s the op t imiza t ion p rocedu re i ndependen t of local op t ima or func t ion gradien ts .

The op t imiza t ion p rob lem of the wind tu rb ine d i s t r ibu t ion a t a g iven si te mus t cons ider the fo l lowing s t a t ement s : - the wind tu rb ines inf luence each other ,

Correspondence to: Prof. C. Poloni, Dipartimento di Energetica, Universitfi degli Studi di Trieste, Via Valerio 10, 34100 Trieste, Italy.

0167-6105/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0167-6105(93)E0033-U

106 G. Mosetti et al./Optimization of wind turbine positioning

- the wind var ies in d i rec t ion and intensi ty , - t h e wind farm should ex t rac t the maximum energy for the minimum ins ta l la t ion cost. In order to obta in the "bes t so lu t ion" it is necessary: - to know the boundar ies of the problem (i.e. the ex ten t of the ter ra in , the wind dis t r ibut ion, the type of turb ines avai lable and the i r cost etc.), - to be able to model the behav ior of a wind farm conf igurat ion, - to be able to eva lua te its "goodness" , - to be able to conven ien t ly r each an opt imum by explor ing a small number of possible configurat ions. For these purposes two dis t inct a lgor i thms have been prepared, one for the windfarm per formance eva lua t ion and one for the opt imizat ion procedure . The windfarm model l ing has been made using the RISO approach repor ted in Refs. [1,2], while the opt imizat ion p rocedure is based on a genet ic a lgor i thm [3].

In the present work the to ta l ex t rac ted power and the inves tment cost have been optimized assuming a simple re la t ion be tween the cost of a single tu rb ine and the number of machines installed. A more sophis t ica ted windfarm model wi th a la rger number of object ives and cons t ra in t s as well as a more real is t ic economical model could be implemented in order to obta in more prac t ica l opt imizat ion results.

2 . W i n d f a r m m o d e l l i n g

The purpose of this work is to demons t ra te the appl icabi l i ty of the method and, therefore , some simple assumptions have been made concern ing the windfarm modelling. The model used is s imilar to the wake decay model developed by N.O. Jensen [1,2]. Assuming t ha t the momentum is conserved in the wake, the wind speed u downs t ream the tu rb ine is:

(1)

where a is the axial induc t ion factor, x is the dis tance downs t ream the turbine , rl is the down s t ream ro to r radius and ~ is the en t r a in m en t cons tan t (see Fig. 1).

The tu rb ine t h rus t coefficient and the down s t ream ro to r radius are l inked to the axial induct ion factor, a, and the ro to r radius, RR, t h rough the Betz relat ions:

/ 1 - a CT = 4a(1 -- a), r l = rr _ ] :---~- • (2)

~] l - - z a

G. Mosetti et al./Optirnization of wind turbine positioning 107

~dary

Mi I ~ --I Mi+l

Fig. 1. Schematic multiple wake model.

The entrainment constant is given by empirical means [2] as:

0.5 (3)

where Zo is the surface roughness of the site and z is the hub height of the wind turbine. Assuming that the kinetic energy deficit of a mixed wake is equal to the sum of the energy deficits, the resulting velocity downstream of n turbines can be calculated using the following expression:

1 - - - - . (4) \ Uo/ i= 1 Uo/

The thrust coefficient and the electrical power produced by a single turbine are given as a function of the local wind speed. The wind characteristics are given by the three parameters direction, intensity, and probability (i.e. time/year that a particular wind exists), while the turbine characteristics considered are rotor diameter, hub height, and thrust coefficient and electrical power extracted as a function of the wind speed.

In order to optimize the windfarm from the economical point of view, it is necessary to model the cost. In the present example, a simple model is used that considers only the number of turbines as relevant variable to determine the cost/year of the windfarm.

We assume that the non-dimensionalized cost/year of a single turbine is 1 and that a maximum cost reduction of 1/3 can be obtained for each turbine if a large number of machines are installed. We then assume that the total

108 G. Mosetti et al./Optimization of wind turbine positioning

cost/year of the whole windfarm can be expressed by the following relation:

costtot = Nt ( 2 + ½ e - o.oo 174N~), (5)

where Nt is the number of turbine installed. The objective of the optimization is to produce the highest amount of energy

at the minimum cost and can be seen as the minimization of the following function:

1 costtot Objective = ~ - w 1 + --:---- w2, (6)

- - t o t "/')tot

where Ptot is the total energy produced in one year, Nt is the number of turbines, wl and w2 are arbitrarily chosen weights, and costtot is the cost/year of the whole windfarm. In the following examples wl has been kept small, since we are looking more for the lowest cost per energy produced.

3. Optimization algorithm

Typical optimization problems are usually solved by means of a "hill climbing procedure" based mainly on local gradients of a stated "cost func- tion". However, a typical drawback of this approach is the risk of finding only a local optimum, and the impossibility of finding a global optimum starting from the same configuration, if the configuration space is not simply connec- ted. The windfarm positioning problem is a typical discrete problem that, similar to the travelling salesman problem, is impossible to solve exactly, and where a simple gradient-based method cannot easily be applied. A grid of 10 x 10 possible windturbine locations, even if for each grid point consideration is restricted only to the two possibilities of having or not having a turbine at each location, means 21°° possibilities to explore, which far exceeds the capa- bility of any existing computer. A genetic search algorithm, developed first by Holland [4], can be applied to this case in a straightforward way. In what follows only a brief summary of the method will be given, while a more extensive discussion on the subject can be found in Ref. [5].

The basic idea of the method is the one of the biological evolution through simple transformation of a coded configuration. As in the natural process of reproduction, the genetic information stored in a chromosomal string of two individuals is used to create the genetic code of a new individual. Evolution and adaptation of the specie are guaranteed because the best individuals have the highest probability of surviving and reproducing.

The basic process that can occur in the construction of a new chromosomal string are random mutation of a gene, an exchange of genetic information between the reproducing parents and an inversion of the chromosomal string. In the following we will consider only the first two, namely mutation and crossover. The choice of the individuals subjected to the reproduction process, namely crossover and mutation, is randomly made, assigning a probability to

G. Mosetti et al./Optimization of wind turbine positioning 109

be extracted to each individual proport ional to their fitness. The following correspondence of terms can be made:

gene = design parameter, individual = design configuration, generat ion = evolution stage of the design, fitness function = design quality, social success = optimality of the design.

It must be noted that in simulating a biological evolution of a specie the optimization process will start searching from multiple points of the configura- t ion space, while t radi t ional gradient based optimization or even adaptive random search, essentially start from single point of the design space.

3.1. The genetic algorithm applied to windfarms The target of the evolution is to find the best windfarm configuration on

a given terrain. For simplicity we can consider as design variable only the position of a number of turbines of a given type. Let us subdivide the available ter ra in into cells where a turbine could be installed. In this manner a string representat ion of a windfarm can be easily found: 1 if in the relative cell a turbine exists, 0 if it does not.

A windfarm configuration in a terrain divided into 10 cells will then be represented by a binary number between A=0000000000 and B = 1111111111. A mutat ion will switch one bit, while a crossover between two parents will perform the following transformation: Parent 1 = 0010110011 Parent 2 = 1110001010

Child 1 = 0010001011 Child 2 = 1110110010 In the example, the crossover locations that are exchanged between the two

parents to create the children are from the fourth to the eighth bit. The "goodness" of each individual is defined by the fitness function that will be maximized, and that can easily be defined as the inverse of the objective function previously defined.

Star t ing from a given population the fitness of each individual of the popula- tion must be evaluated through the windfarm model previously described and on the basis of the fitness value the next generat ion must be constructed. The new population is obtained through crossover and mutat ion among the fittest individuals and at random locations. Both operations will occur with a certain probabili ty 0 .6<P¢<0.9 for a crossover and 0.01<Pro<0.1 for a mutation. A crossover has a higher probability because it is mainly responsible for the "local evolut ion" of the population while a mutat ion rarely occurs as it is responsible for the random introduct ion of new characters into the population. If only crossovers are applied, the population will soon get "steril ized" and probably converge to a local optimum, while if only mutat ions are applied, the algorithm becomes a sort of random search.

110 G. Mosetti et al./Optimization of wind turbine positioning

It must be noted that the method in the end looks for a better population and not directly for the best individual. Therefore, the best solution could even be found at the early stages of the evolution. However, as the mean fitness of the population will increase, the fittest individual will probably be part of the most evolved population.

4. Re su l t s

In what follows some numerical experiments will be il lustrated in order to demonstrate the capabilities of the method, and not for the solution of a practi- cal optimization problem, and, therefore, some crude assumptions about wind distribution and turbine characterist ics will be made. The optimization of a windfarm on a square shaped terrain subdivided into 100 possible turbine location will be considered under three significant conditions: the case of a single wind direction and intensity, the case of a constant intensity and 360 ° variable direction, and the case of a more realistic wind distribution. The possible turbine location will be the center of a square cell having a side size equal to five turbine diameters. A 5D cell guarantee the accuracy of the wake decay model that at a smaller distance could be inappropriate. The total size of the windfarm site is therefore 50D × 50D. All optimization procedures will start from random configurations even though for practical cases, a better initialization of the genetic search could sensibly reduce the computation time required.

Only one wind turbine type has been considered having a h = 6 0 m hub height, a 40 m diameter, a thrust coefficient equal to Ct=0.88, constant for the wind speeds considered, and a power curve as shown in Fig. 2. The site roughness is Zo = 0.3 m.

700

600

500

400

30O

2OO

100

0 , ~ , = .

0 2 4 6 8 10 12 Wind Speed (M/s)

4 16 18

Fig. 2. Power curve for the turbine considered.

G. Mosetti et al./ Optimization of wind turbine positioning 111

4.1. The case of a single w ind direction In this simple case the optimum can be intuitively understood. The objective

function to be maximized will consider the inverse of the investment cost per kW installed (that will reach a practical minimum with about 30 machines), and the maximum power extracted that would reach its maximum when 100 machines are installed. Intuitively, the optimal solution will have less than 30 turbines at the largest reciprocal distance, in order to minimize the single turbine performance deterioration due to the presence of other machines. Figure 3 shows the behavior of the minimum, maximum and average of the fitness function as a function of the population for a given population size. It can be noted that the difference between maximum and minimum fitness decreases as the convergence proceeds. Figure 4 shows that population sizes of 200 and 300 converge to approximately the same maximum fitness value while a 100 population size has not enough variability and therefore converges to a lower fitness.

A random configuration with 40 to 50 machines would typically have a wind farm efficiency of 0.50, defined as the ratio between the total energy extracted by the windfarm having Nt turbines and Nt times the energy extracted by an isolated turbine with the same undisturbed wind, while an optimized configura- tion would reach an efficiency of 0.95 with about 25 machines. As can be expected for a single wind direction, the optimized configuration presents two to three turbines per column at distances larger than 15D as shown in Fig. 5a.

4.2. The case of multiple w ind direction with constant intensity In this case a preferable orientation does not exist. The efficiency of each

turbine is mainly determined by the distance from any other turbine. The

. . . . F i t m i n .

. . . . F i t a v e . Pc ~u la t ion s i z e 2 0 0

- - F i t m a x . 0 . 8 , i

4 i i . ~ , , ~ ' ~ ' ~ I i , i r

0 . 7 ~= i _,~-.-~ ~ - " -'-~ " - " ~ ~ ' .A=I ~ s - , t

¢n t ~'i • / " t,~,'~ ri i m f ~ . ~ i Q Y~ L,~ L e- ® 0 . 6 . [ ] ., i ~ - ~ * ~= i

L L ........... ~.,~ .............. :~¢~.~ ~ .~, ~ ~ . . . . . . . . . . . .

, L ! ~ . . .~ i 0 . 4 ~ ,]--.-..,- i , '

i i i i [ 0 1 0 0 2 0 0 3 0 0 4 0 0

G e n e r a t i o n

Fig. 3. F i tness curves for a g iven populat ion s ize for the s ing le d irect ion case.

112 G. Mosetti et al.] Optimization of wind turbine positioning

09

f : LL

0 . 8

0 . 7 4

0 . 6 8

0 . 6 2

0 . 5 6

0 . 5

. . . . F i t m a x . 3 0 0

. . . . F i t m a x . 1 0 0 - - F i t m a x . 2 0 0 M a x . F i t n e s s

*~..,... ~ ....... ! ........ 4 ................... i

~ : ) ~ [ I [ ) . . ; ".i.~...:~ ( . .) g . . . . . . . . i

1 0 0 2 0 0 3 0 0 4 0 0

G e n e r a t i o n

Fig. 4. Maximum fitness curve for var ious populat ion sizes for the single direct ion case

• • • •

• • •

• •

a)

• • •

• • •

• • • •

• • •

• • •

m • •

• • • •

b)

Fig. 5. Positions of the turbines for (a) a single direction, and (b) mult iple directions.

G. Mosetti et al./ Optimization of wind turbine positioning 113

optimal configuration, as confirmed in Fig. 5b, will therefore have the most of the turbines along the edges of the domain. In this case an efficiency of 0.35 for a random configuration having 19 turbines.

In Fig. 6, the behavior of the fitness function is reported for different population sizes. It can be seen that even in this case, a population of 200 individuals with 350 iterations can be sufficient to reach a well optimized configuration even starting from a random distribution.

4.3. The case of multiple wind direction and intensity In this case the optimal solution cannot be intuitively found as wind speed,

direction and frequency play a fundamental role in the mutual influence between different turbines. For the wind distribution, summarized in Fig. 7, three intensities and 36 directions (every 10 °) are assumed, while the frequency is given as a fraction of time unit for which a certain wind exists. In this case only a calculation with 200 configurations times 400 iterations has been made. Figure 8 shows the maximum, minimum and average fitness in the population as a function of the number of iterations, and Fig. 9 shows the number of turbines for the best configuration found in each iteration. As can be seen, the optimum number is between 15 and 18 turbines. Figure 10 shows the positions of the turbines for the multiple direction and multiple intensities case.

The results obtained for all the optimized parameters are summarized in Table 1, where the parameters for a random configuration having 42 turbines installed and for an optimized configuration having 15 turbines installed shown in Fig. 9 are compared. The wind farm efficiency, which is dependent on the wind orientation only and not on the integral of the produced energy, goes from 0.34 to 0.84, the energy produced is decreased by 29% but the installation cost is reduced by 50%.

LL

2 4

2 2 . 4

2 0 . 8

1 9 . 2

1 7 . 6

1 6

- - - Fit max. 1 0 0

- - - F i t m a x . 2 0 0

- - F i t . m a x . 3 0 0

i ~ ~'~ .;;,~'¢ ; •

i,W:, i,,,." ¢ ~ ' ! ,i. ,

I 7=' Ji J

. . . . . . . . . . . . . . i i

M a x F i t n e s s ! .

.~, :~.?.;~,r~, ",,~," ~.f ,.

, - -n , r . , i ,1

i. i

200 300 Genera t ion

0 1 0 0 4 0 0

Fig. 6. Maximum fitness curve for various population sizes for the multiple direction case.

114 G. Mosetti et al./Optimization of wind turbine positioning

[] 8 m/s [] 12 m/s

[ ] 17 m/s W ind d i s t r i b u t i o n

o-e 3

(D c -

" O ¢- 1

0 0 7 0 140 2 1 0 2 8 0 3 5 0

Ang le (o)

Fig. 7. Wind d i s t r ibu t ion for the va r i ab le in t ens i ty and va r i ab l e d i rec t ion case.

c -

U_

28

25

22

19

16

13

10

....... Fit min. - - -F i t ave.

i F i t max. Population size 200 a I i I i i i i

.............. L.o.. A ............. L : , I : . : : : : : : . . ; ; £ ; ' C

..... : . . . . . . . . . . . . . . . . . . . i ...................

:~ , i ; : . , f ~ .............. i " i i ' ! i i i i

..~ + -i ~ ! !.

0 100 200 300 400

Generat ion

Fig. 8. Max imum fitness curve for va r ious popu la t ion sizes for the mul t ip le d i rec t ion and mul t ip le in t ens i t i e s case.

5. C o m p u t a t i o n t ime

The computation time required for the optimization is mainly determined by the windfarm simulation module. More than 90% of the cpu time is spent in the routines that evaluate the fitness of the configuration. The computation time is, therefore, almost proportional to the number of calls to the windfarm

G. Mosetti et al./Optimization of wind turbine positioning 115

4 2

m 3 4 ¢D

~ 2 6 o

n

E z 1 8

10

Population size 200

1 0 0 2 0 0 3 0 0 4 0 0

Generation

Fig. 9. Number of tu rb ines as the convergence proceeds.

• • •

Fig. 10. Posi t ions of the tu rb ines for the mult iple d i rec t ion and mult iple in tens i t ies case.

Table 1

Optimized pa ramete r s compared wi th a non optimized conf igurat ion for the th ree cases considered

Efficiency Pt= (kWyear) cos t /kWyear Nt

Random 0.50 13025 2.57 x 10- 3 50 Optimized 0.95 12375 1.57 × 10-3 25

Random 0.35 9117 3.68 x 10- 3 50 Optimized 0.88 8711 1.84 x 10- 3 19

Random 0.34 4767 7.04 × 10- 3 50 Optimized 0.84 3695 3.61 x 10-3 15

116 G. Mosetti et al./Optimization of wind turbine positioning

simulation routines. The time needed for the optimization is about 1.25E-3 s per configuration and wind considered on a CRAY XMP 14 which means that for the last case considered in this paper about 16000 s of cpu have been used.

However, no attempt has been made yet to optimize the software and, therefore, a better performance could be expected by increasing the efficiency of the windfarm evaluation routines.

6. Conclusion

A novel approach to the optimization of large wind farms has been presented. The feasibility of the approach has been demonstrated and could be applied to more sophisticated windfarm simulation models. In the present work only a single turbine type has been considered with simple assumptions, but the application of the method to more realistic problems would be straightforward if a realistic "cost" function together with real turbine data are used.

All the optimizations have been performed start ing from fully random config- urations. If some "rule of thumb" is used to generate the first population, the computational time required to reach an optimized configuration would prob- ably be highly reduced.

Acknowledgement

This Research was made with M.U.R.S.T. 40% funding.

References

[1] I. Katic, J. Hojstrup and N.O. Jensen, A simple model for cluster efficiency, EWEC '86 Rome, Italy, 7-9 October 1986.

[2] S. Frandsen, On the wind speed reduction in the center of large clusters of wind turbines, EWEC '91 Amsterdam, The Netherlands.

[3] P. Hajela, Genetic search - an approach to the nonconvex optimization problem, AIAA J. 28(7) (1990).

[4] J.H. Holland, Adaptation in natural and artificial systems (Univ. of Michigan Press, Ann Arbour, 1975).

[5] L. Davis and M. Steenstrup, Genetic search and simulated annealing (Lawrence Davis and Morgan Kauffman, Los Altos, 1987).