short-cut desing of wind farms

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Energy Policy 29 (2001) 567}578

Short-cut design of wind farmsC.T. Kiranoudis*, N.G. Voros, Z.B. Maroulis

Department of Chemical Engineering, National Technical University of Athens, Heroon Polytexniou 9, Zografou Campus, Athens 15780, Greece

Received 10 February 2000

Abstract

The problem of designing wind parks has been addressed in terms of maximizing the economic bene"ts of such an investment. An

appropriate mathematical model for wind turbines was used, taking into account their construction characteristics and operational

performance. The regional wind "eld characteristics for a wide range of sites have been appropriately analyzed and a model involving

signi"cant physical parameters has been developed. The design problem was formulated as a mathematical programming problem,and solved using appropriate mathematical programming techniques. The optimization covered a wide range of site characteristics

and four types of commercially available wind turbines. The methodology introduced a short-cut design empirical equation for the

determination of the optimum number of wind turbines. The model is appropriate for regional planning purposes. Variation of unit

cost of wind farm area introduced an additional degree of freedom to the problem and more e$cient designs could be ob-

tained. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Regional planning; Wind turbines; Mathematical modelling; Optimization

1. Introduction

Energy derived from wind is receiving considerableattention due to its availability, low cost and environ-

ment friendly operation of such a technology. In most

recent applications, wind energy conversion systems are

considered for electricity generation. Many such machine

types, ranging in capacity from a few hundred watts to

a few megawatts, have been proposed and tested and

many are commercially available. However, the e!ect of

many design parameters on the energy output and eco-

nomic performance of such systems is not fully

documented (Golding, 1976, Hunt, 1981).

Early developments concentrated their e!orts in plac-

ing machines in open plains forming large arrays ofaerogenerators (i.e. wind farms), thus making convenient

and manageable power production units. In general, the

promotion of wind energy as an alternative resource, is

closely related to the design of reliable and cost e!ective

wind farms. In this sense, the design of a wind farm or

equivalently the determination of the type of machinery,

*Corresponding author. Tel.: #30-1-772-1503; fax: #30-1-772-

3228.

E-mail address: [email protected] (C.T. Kiranoudis).

number and layout of wind turbines in space, should

maximize the energy output as well as the lifetime of the

machines (Cavalo et al., 1993).The design of reliable and cost-e!ective wind farms

capable of large-scale electrical energy production is

a prerequisite for the e!ective use of wind power as an

alternative resource. In this sense, the design of a wind

farm or equivalently the determination of type, number

and layout of wind turbines should maximize the energy

output together with the lifetime of the machines (Mil-

borrow, 1980). In all cases, the design objective is closely

related to the total annual output of the wind farm

operation in power terms. Given the type of wind tur-

bines to be used, regional wind speed statistical data and

the area of the farm, we can estimate the total annualenergy output of the wind farm. In this case, the optimum

number of wind turbines to be installed has to be

determined under a certain economic environment. The

determination of the optimal farm characteristics must

be based on speci"c design objectives. In this case, the

design problem may be formulated as a mathematical

programming problem, involving an objective function

representing the investment e$ciency, which is expressed

by the pro"ts expected per unit of capital invested.

The construction and operation of various types of

wind turbines have been investigated to an extent that

0301-4215/01/$- see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 3 01 - 4 2 1 5 ( 0 0 ) 0 0 1 5 0 - 6


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Nomenclature

a, b, c,

a

, a

,

a

short-cut model parameters of Eqs.

(14)}(19)

A area covered by wind turbine rotor (m

)A$

wind farm area (m)

b

, b

,

b

, b

,

b


(14)}(19)

c

, c

,

c

capital cost coe$cients de"ned in Eq. (21)

c

, c

,

c

, c

,

c


(14)}(19)

c

power turbine coe$cient

c

nominal turbine power coe$cient

c#

unit cost of electricity produced ($/kWh)

c#*

conventional unit cost of electricity

($/kWh)

c$

cost of unit farm area ($/m)

c-.

operational cost coe$cient de"ned in Eq.

(16) ($/kW)

C!.

capital cost ($)

C-.

operational cost ($/h)

C2

total annual cost ($)

d distance of wind farm turbines (m)

D wind turbine diameter (m)

e percentage of the capital cost on an annualrate

E total energy recovered (kWh)

f Weibull cumulative distribution function

H height (m)

H0

reference height (m)

i wind farm turbine index

k Weibull function shape factor

M Weibull cumulative distribution function

N total number of wind farm turbines

P power generated by a wind turbine (kW)PI investment e$cieny

P

nominal turbine power (kW)

P* maximum power generated by a wind tur-

bine (kW)

P2

total turbine power output (kW)

P

nominal turbine power (kW)

s wind turbine power curve shape factor

S total annual pro"ts expected from the in-

vestment ($)

u wind speed (m/s)

u

Weibull function wind speed parameter

(m/s)

uG

onset speed of the ith turbine (m/s)

u

mean wind speed of a region (m/s)

u

nominal wind turbine speed (m/s)

u*

maximum wind turbine speed (m/s)

u0

reference wind speed (m/s)

Greek letters

di!usional turbulence coe$cient of Eq.(10)

gamma function

,

,

,

,

short-cut design model parameters de"nedin Eq. (27)

relative distance of wind farm turbines

wind farm e$ciency air density (kg/m)

several sets of operating experimental data are currently

available in the literature. Furthermore, wind speed stat-

istical data and wind power potential have been thor-

oughly investigated for a wide range of regions wherewind energy seems promising for exploitation. In addi-

tion, suitable mathematical models describing the opera-

tion of wind turbines have been developed and used for

the simulation of wind farms. Galanis and Christophides

(1990) presented a systematic investigation of the e!ects

of rated power, rated wind speed and tower height on the

annual energy production and the unit cost of energy

produced. Based on these results, a method for the selec-

tion of optimum wind energy conversion systems was

formulated. Voutsinas and Rados (1993) developed

a method for designing aerodynamically optimal wind

farms. The method consisted of positioning wind tur-

bines so as to maximize the energy they absorb or equiv-

alently minimize the loss of energy due to wake e!ects. In

this sense, the optimal design problem was formulated asa constraint minimization problem involving an objec-

tive that represented the loss of energy due to wake

e!ects. The wake e!ects of the farm were modeled by

means of an improved kinematic wake model.

Several commercial software products have been de-

veloped to analyze, design and optimise wind farms.

Most of them involve sophisticated algorithms for opti-

mising wind farms for increased energy or reduced cost of

energy, subject to environmental or physical constraints.

In addition, they calculate the energy yield of a farm

using integrated wind #ow modules and advanced wake

568 C.T. Kiranoudis et al. /Energy Policy 29 (2001) 567}578


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Fig. 1. Information #ow diagram for System Design and Energy

Planning.

models. They calculate long-term wind speed predictions

and use high-powered graphics capabilities in order to

visualize results (in some cases even 3D views are pro-

vided). Some very popular used by several consulting

"rms include WIndFarm by ReSoft, WinMap by Brower

and Company, WindFarmer by Ripe Software, WA1P byRISO National Laboratory, and WindOps by WindOps

Ltd. This philosophy, is oriented towards detailed design

of farms but it is cumbersome to apply for regional

planning reasons, where primary decisions have to be

taken in order to roughly decide upon penetration of

wind power technology on speci"c regions. The regional

planning procedures involve "rst stage design calcu-

lations that will estimate wind park feasibility for a vast

number of sites and therefore short-cut design proced-

ures are favored. Regional planning strategies are pre-

sented in Fig. 1.

Short-cut design is a technical procedure for express-ing, in a straightforward way, the optimal results of

a detailed design problem through emprical equations

involving the corresponding design variables. In this way,

all other model variables are directly computed through

the model equations. The parameters of the empirical

equations are evaluated by "tting the short-cut model

equations to the corresponding design problem optimal

results computed using the full process model. Short-cut

design is extremely important for preliminary selection of

alternative design scenaria for diversi"ed policies of in-

vestment at a regional level. In this way, short-cut evalu-

ations of optimal designs for certain sites is an indispens-

able tool for assessing regional planning strategies at

a national or international level. Moreover, it is extreme-

ly important for determining the way that alternative

energy sources could possibly penetrate the energy mar-

ket by an appropriate subsidy policy.

This work addresses the problem of wind farm short-

cut design in terms of maximizing the economic bene"tsof the investment. The mathematical model of wind tur-

bines was developed taking into consideration their per-

formance with respect to construction and operation.

The objective function to be maximized was the total

pro"ts expected from the plant operation. The wind "eld

characteristics of a site have been analyzed in terms of

signi"cant physical parameters and modeled appro-

priately. Optimization was carried out for a wide range of

site characteristics expressed by the corresponding model

parameters and four di!erent types of commercial wind

turbines. A short-cut design empirical equation was in-

troduced for determining the optimum number of windturbines. The model is appropriately for regional plann-

ing purposes. From the engineering point of view, such

an analysis will directly serve as an evaluation tool for

explicitly determining the pro"tability of wind power

exploitation in a certain region.

2. Mathematical modeling of wind turbines and farms

The useful power of a wind turbine is (Kiranoudis

et al., 1997)

P"CuA, (1)

where

A"D

4. (2)

The power coe$cient, C

, depends on wind speed and

actual turbine characteristics. Experimental data for

power coe$cient in the case of four commercial wind

turbines studied in this work (namely VESTAS, ENER-

CON, BONUS and FLODA; mentioned in increased

nominal power order) are given in Fig. 2, as a function of

wind speed perpendicular to the wind direction. Allcurves exhibit a maximum for a given wind speed value

representing the nominal performance of the turbine. An

expression for representing the power curve of wind

turbines is proposed (Kiranoudis et al., 1997):

C"C

exp

(lnu!lnu)

2(ln s) . (3)In this expression, the turbine characteristics are the

nominal power coe$cient, C

, the nominal wind speed,

u, and a parameter expressing the operating range of

C.T. Kiranoudis et al. /Energy Policy 29 (2001) 567}578 569


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Fig. 2. Wind turbine e$ciency curves (points are experimental data and lines are model predictions) for all types examined. (a) VESTAS,

(b) ENERCON, (c) BONUS, (d) FLODA.

wind speed, s. It must be noted that the nominal power of

the turbine is given by the following expression:

PP"

C

uA. (4)

Quite excellent "ts to actual data were detected when

Eq. (3) was used to real turbine data. The predictions of

the empirical equation proposed for the case of power

coe$cient experimental values, are also given in Fig. 2,

indicating the excellence of the "t and suggesting the

practical signi"cance of Eq. (3). The estimated turbine

parameters of Eq. (3) for all commercial wind turbines

studied, are given in Table 1. Eqs. (1) and (3) can now be

combined to express the actual power converted as

a function of wind speed. The corresponding turbinediameters are also given in Table 1. The experimentally

detected power values and model predictions for the

entire wind speed region and for all turbines employed

are given in Fig. 3. The model, and most experimental

data, detect an optimum for power at a certain wind

speed level. Using the above mentioned model for power

coe$cient, maximization with respect to wind speed sug-

gests a corresponding value for wind speed, at optimum

power (Kiranoudis et al., 1997)

uH"u

exp[3(lns)]. (5)

The maximum power values for each turbine em-

ployed are given by the following equation (Kiranoudiset al., 1997):

PH"c

u

exp[

(ln s)]A. (6)

Their corresponding values for all turbines employed

are also listed in Table 1.

When a record of the mean wind speeds for a speci"c

region are available, the frequency and wind duration

can be modeled accurately using mathematical functions

in predicting the output of wind turbines in various

locations. In this case, the Weibull distribution function

that estimates the probability that the wind speed ex-

ceeds a certain value, is widely used as a good match withexperimental data. The Weibull distribution has a cumu-

lative distribution function of the form (Bowden et al.,

1983)

M(u)"1!exp!u

u

I

(7)and a probability density function of the form (Bowden et

al., 1983)

f(u)"k

uu

u

I\exp!

u

u

I

. (8)



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Table 1

Wind turbine and wind park model parameters

Turbine VESTAS ENERCON BONUS FLODA

D (m) 28 32 37 42

P

(kW) 59 85 134 170

u

(m/s) 7.2 8.2 8.1 8.2

c 0.43 0.32 0.39 0.37s 1.7 1.7 1.7 1.7

u*

(m/s) 16.8 19.1 18.9 19.1

P* (kW) 211 302 475 602

a

43.4;10\ 25.0;10\ 29.8;10\ 27.7;10\

a

0.277 0.256 0.282 0.275

a

8.62;10\ 8.09;10\ 8.47;10\ 8.38;10\

b

4.02 4.09 4.04 4.05

b

1.21;10\ 1.54;10\ 1.53;10\ 1.62;10\

b

86.1 80.1 95.7 257

b

1.44;10\ 1.07;10\ 1.21;10\ 4.06;10\

c

0.846 0.824 0.818 0.817

c

1.84;10\ 1.38;10\ 1.42;10\ 1.31;10\

c

1.06 0.948 1.00 0.987

c 7.23;10\

0.102 7.78;10\

8.17;10\

Fig. 3. Wind turbine performance curves (symbols as in Fig. 2).

The Weibull parameters that characterise wind

speed distribution for a speci"ed site, are the shape

parameter k and the scale parameter u

. A com-

mon starting point for evaluating Weibull func-

tion parameters is the mean wind speed of the

region which is given by the following relation

(Bowden et al., 1983):

u"u

1#

1

k. (9)



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Fig. 4. Yearly wind speed duration curves (k"2). (1) u"5 m/s, (2)

u"7 m/s, (3) u

"9m/s.

The mean wind speed is the most commonly encoun-

tered variable for representing wind speed data for a spe-

ci"c region. The Weibull shape parameter k usually

varies between 1.8 and 2.2 for most regions. The yearly

wind speed duration curves expressing the time period

that wind speed lies within a speci"c range can be directly

computed by means of the cumulative Weibull distribu-

tion function. The e!ect of mean wind speed on yearlywind speed duration (8760h) is given in Fig. 4.

In most cases mean wind speed for a speci"c region is

measured at a standard height (usually 10 m). In order to

estimate the wind speed at di!erent heights so that the

corresponding values are corrected with respect to the

center of the wind turbine (where the total power output

is actually evaluated), the following power law has been

proposed (Voutsinas and Rados, 1993):

u

u0

"H

H0?, (10)

where the power index, , depends on the roughness ofthe terrain, the time of day, the wind stability and speed

level. In most cases a value of

is quite appropriate for

most regions.

The total annual power output of the wind turbine can

now be evaluated by means of the following relation:

P2"

P(u)f(u) du. (11)

Given the power curve of a speci"c turbine, the energy

per year generated by each individual turbine can be

computed. The ratio of the entire power generated by the

real farm to the one corresponding to all wind turbines

operating in the absence of wake e!ects is the wind farm

e$ciency. Thus, for a wind farm involving N wind tur-

bines placed in a prede"ned con"guration, the wind farm

e$ciency for a speci"c wind speed u and calculated onset

wind velocities (uG, i"1,2,N), is given by the following

equation (Lissaman, 1979):

"

,GJ

P(uG)

NP(u). (12)

Obviously, the wind farm e$ciency is a function of the

turbine type employed, the wind farm con"guration and

wind speed. Estimation of the overall e$ciency of a wind

farm is of crucial importance to a wind farm design

procedure since due to its explicit relation to total annual

power converted, it is considered a vital trade-o! be-

tween performance and cost. In general, the design objec-tives would seek the best wind farm con"guration, i.e.

number and layout of individual turbines, and the actual

type and size of each turbine, provided that wind statist-

ical data and region characteristics as well as economic

"gures are supplied. In the case where we seek for the

total optimal design of the wind farm, the procedure for

the determination of the layout of wind turbines must be

repeated within the optimization iterative loop, thus re-

quiring a problem of extreme di$culty to be solved.

A short-cut model for wind farm e$ciency is used for

the uniform turbine grid, i.e. the turbines are equally

displaced from each other. It is expressed as a function of

type of machinery used, wind speed and wind farm char-acteristics. The wind farm characteristics are in the case

of the uniform grid employed, the total number of tur-

bines, N, and their relative distance, d, that is related to

the turbine diameter and their absolute distance by

means of the following equation (Kiranoudis and

Maroulis, 1997):

"d

D. (13)

The short-cut model proposed is given by the follow-

ing equation (Kiranoudis and Maroulis, 1997) that esti-

mates the wind farm e$ciency as a function of wind

speed, the number of turbines and their relative distance:

"1au@ exp(!uA), (14)

where

a"a

exp(!a)[1!exp(!a

N)], (15)

b"b#(b

!b

)[1!exp(!b

)], (16)

c"c#(c

!c

)[1!exp(!c

)] (17)



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and

b"b

exp(b

N), (18)

c"c

exp(!c

N). (19)

Kiranoudis and Maroulis (1997) evaluated the 11

parameters of the proposed short-cut wind e$ciency

model for all types of turbines studied. These values arealso given in Table 1.

The annual energy obtained by the operation of the

wind farm is calculated by integrating Eq. (12) for the

entire year:

E"N

P(u) du. (20)

The installation cost of the farm is given by the follow-

ing equation as a function of the wind turbines' max-

imum power, their number plus the cost of the land

(Maroulis et al., 1996; Milborrow, 1995):

C!."c (PH

)AN

A#c$A$ . (21)

The annual operational cost of the plant is propor-

tional to the installed plant capacity and is given by the

following equation (Maroulis et al., 1996):

C-."c

-.NPH. (22)

The total annual cost of the plant is therefore cal-

culated by means of the following equation:

C2"eC

!.#C

-.. (23)

As a result, the unit cost of energy produced is the ratio

of total annual cost and annual energy recovered:

c#"

C2E

. (24)

The expected pro"ts from the operation of the wind

farm is therefore given below:

S"E(c#*!c

#). (25)

The investment e$ciency is expressed as the ratio of

expected pro"ts per invested capital:

PI"S

C!.

. (26)

Eqs. (1)}(26) constitute the mathematical model of theentire wind farm. In this case, the design objective is to

maximize the investment e$ciency from the operation of

such a plant. Given the type of wind turbine, the farm size

and the site wind "eld characteristics, there is only one

design variable to be computed by means of maximizing

the objective function; the number of wind turbines to be

installed. The optimization procedure throughout this

paper was carried out by means of the successive quad-

ratic programming algorithm implemented in the form of

the subroutine E04UCF/NAG. All runs were performed

on a SG Indy workstation under Unix.

3. Short-cut design of wind farms

On the basis of the above, the design strategy for wind

farms can now be clearly stated. Given the type of wind

turbine, the available farm area and the site wind "eld

characteristics (i.e. wind duration curve parameters)

the number of wind turbines to be installed must be

determined by means of optimizing appropriatetechnoeconomical criteria under speci"c operational and

environmental constraints.

As a consequence of the above, the determination of

the optimum plant con"guration must be based on speci-

"c design objectives. In practice, the representation of the

design problem for wind farms should focus to corre-

sponding mathematical "gures obtained through an ad-

equate mathematical model as previously formulated. In

all cases, the design procedure should involve an objec-

tive function representing the economic bene"ts from the

operation of such a plant or its e$ciency in terms of

energy availability towards regional demand. Certainalternative objective function types may be taken into

consideration regarding the bene"ts expected:

(i) Maximization of the investment ezciency: This case

suits to design problems confronted by individual

power-producing industries (either private or municipal)

that have invested or plan to invest in this "eld, in

countries where legislation permits so. In other words,

this objective refers to the direct economical bene"ts

expected from such an investment under a speci"c com-

petitive economic environment, thus determining the

feasibility of exploiting this type of renewable energy

source.

(ii) Maximization of the amount of energy annuallyproduced from the available wind potential: This case

suits to design problems usually confronted for regions

where no other sources of energy are technically exploit-

able, and the objective is to exploit the highest possible

energy potential of a region in order to cover the local

demand, assuming that the use of wind power is still

pro"table compared to the unit cost of electricty avail-

able in remote national regions due to increased trans-

poration cost.

Throughout this paper we choose the "rst possibility

for our objective function. Characteristic economical "g-

ures concerning capital and operational cost componentsfor a typical economic environment were taken into

consideration and are listed in Table 2. Between these

two cases, the former evaluates an optimum farm size

that is completely di!erent (smaller) than the latter. How-

ever, it can be shown that the optimal results of the "rst

objective that is an economically driven function co-

incide with the ones of the second objective that is a pure-

ly technical function (independent of economics) when

the unit cost of conventional electricity approaches in"n-

ity. In this case, the farm operates at the point of max-

imum energy recovery.



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Table 2

Design and cost data

c

($/kWh) 1000

c

0.67

c

0.75

c$

(k$/km) 1.7

c-.

($/kW) 135

c#* (c/kWh) 6.7e 0.1

Fig. 5. E!ect of number of wind park turbines on certain process

variables. (1) Investment e$ciency, (2) Energy recovered. (a)

c#*"6 c /kW h, (b) c

#*"8 c /kW h, (c) c

#*"14c/kWh, (d)

c#*"20c/k Wh.

Table 3

Short-cut model parameters

Turbine VESTAS ENERCON BONUS FLODA

191.84 171.01 131.45 127.23

2.42 1.82 1.46 1.53

!9041.8 !16760.7 !11258.5 !19475.3

1.97 2.19 2.06 2.33

131.83 169.38 158.00 153.09

In order to illustrate the above-mentioned situation,

we examine the case of a typical site involving a farm of

1 km area, a wind farm involving VESTAS wind tur-

bines and a wind "eld of annual mean speed of 8 m/s with

a Weibull function shape factor of value 2. The e!ect of

number of wind turbines on investment e$ciency and

total energy recovered is presented in Fig. 5. Obviously,

each case results in completely di!erent farm size and

economic "gures. For increasing unit cost of conven-tional electricity, optimum farm size determined by opti-

mizing the investment e$ciency increases to the one

determined by maximizing the energy recovery.

The optimization procedure for the determination of

the optimal number of wind turbines described earlier,

was concentrated on the solution of a speci"c design

problem involving a prede"ned turbine type, farm size

and site wind "eld characteristics. This procedure

can be extended to include a wide range of turbine

types, farm size and wind "eld particularities. When the

results of the optimization for each wind turbine and site

combination are systematically compiled and presented,

an empirical short-cut design equation can be evaluatedso that the design engineer can automatically determine

the optimal size of each plant and subsequently evaluate

its performance in terms of the recovered amount of

energy and the unit cost of power produced. A short-cut

design empirical equation of the following form is pro-

posed:

N"AA$#

u\A#

. (27)

It involves "ve parameters and expresses the optimal

number of wind farm turbines in terms of investment

e$ciency maximization. The determination of the intro-

duced short-cut model parameters was carried out by

"tting Eq. (27) to the optimal results of the full-design

problem for all wind turbines studied. The values of

model parameter for all turbines are given in Table 3. The

"tting of the short-cut empirical model to the optimal

number of wind farm turbines for all turbines studied are

given in Fig. 6. Visually, all "ts were extremely satisfac-

tory and Eq. (27) can be safely used for short-cut design

purposes in the case of wind farm design. The optimum

number of wind turbines increases with both farm area

and mean wind speed of the region. Also signi"cant

variation of the optimum number of wind turbines was

observed between all four types of wind turbines studied.



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Fig. 6. Fitting of the short-cut empirical equation (lines) on the number of wind park turbines (points). (1) VESTAS, (2) ENERCON, (3) BONUS, (4)

FLODA. (a) u"8 m/s, (b) u

"10 m/s, (c) u

"12 m/s, (d) u

"14m/s.

With increasing the size of the wind turbine, the optimum

number of turbines installed decreases.Fig. 6 expressing Eq. (27) is the essence of short-cut

design of wind farms. Given the type of wind turbine

used, the farm area and the site mean wind speed, the

engineer can automatically evaluate the optimum num-

ber of wind turbines to be installed, the corresponding

total amount of energy recovered and a reasonable es-

timation of the total plant cost and pro"ts expected at

a preliminary design level. At this stage of design, the

short-cut design equation for wind farms produced is

a tool of great signi"cance for feasibility studies on such

investments.

In order to determine the range of applicability of eachwind turbine, all wind turbines were directly compared

for a wide range of model parameters. The range of

application for all wind turbines are given in Fig. 7.

When all four wind turbines are compared, FLODA was

found to be the best. Among all three others, BONUS is

best for almost the entire range of wind speed values,

except for very small mean wind velocities and small to

moderate farm areas. In this case, VESTAS is preferrable,

while ENERCON is the worst of all. Between VESTAS

and ENERCON, VESTAS is more suitable for low to

moderate wind speed values, while ENERCON applies

for moderate to high mean wind velocities. Obviously,

larger wind turbines are superior to smaller for wind farmdesign. Small wind turbines apply only for lower mean

wind velocities.

The design of wind farms is greatly a!ected by the

mean wind speed of the speci"c region. The correspond-

ing wind speed distribution is also characterized by the

Weibull function parameter k which in most cases varies

between 1.8 and 2.2, as mentioned in the previous sec-

tions. The e!ect of this parameter on signi"cant wind

farm variables (optimum number of wind turbines, unit

cost of electricity and total energy recovered) is presented

in Fig. 8 for the typical (and practically observable for

most sites) case of mean wind speed of 8m/s and forVESTAS wind turbine. In this typical case, the deviations

for the complete range of k parameter variation is less

than 2% for the case of unit cost of electricity. Thus, the

overall e!ect of the Weibull function shape factor is

practically negligible, and clearly the only region variable

that should be taken into consideration for design pur-

poses is the mean wind speed value.

In all design cases mentioned above, the farm size was

taken to be constant dictated by availability reasoning.

In more realistic scenaria, we also study the e!ect of

external cost factors, such as the cost of the farm on the



10/12

Fig. 7. Regions of applicability of all wind turbines studied. (1) VES-

TAS-ENERCON, (2) VESTAS-ENERCON-BONUS, (3) VESTAS-

ENERCON-BONUS-FLODA.

Fig. 8. E!ect of Weibull constant k on process variables (u"8 m/s).

(1) unit cost of electricity, (2) number of turbines, (3) total energy

recovered, (a) k"1.8, (b) k"2, (c) k"2.2.optimum number of wind turbines to be installed and the

optimum farm area to be purchased (provided that it is

available). In this case, the farm size constitutes an addi-

tional design variable to the original problem of wind

farm design. This e!ect was studied for varying farm cost

and for VESTAS turbines operating for various mean

wind velocities. The results are graphically presented in

Fig. 9. As clearly indicated, the less the farm cost is, the

larger the optimum farm size to be purchased and the

more the optimum number of wind turbines to be instal-

led is.

In order to illustrate the e!ectiveness of the above-

mentioned procedure, the Greek island of Crete was

taken as a characteristic example. The individual sites

that have strong wind potential for the island are given in

Table 4, along with their characteristic available area and

mean wind speed. The application of the proposed model

resulted in the estimated number of turbines, energy

produced and unit cost of energy calculated. These values



11/12

Fig. 9. E!ect of park cost on process variables (VESTAS wind turbine).

(1) number of wind turbines, (2) wind park area. (a) u"6 m/s, (b)

u"8 m/s, (c) u

"10 m/s.

Fig. 10. Wind Park construction potential in Crete.

Table4

PotentialwindparksitesinCrete

Site

Park

area

(km)

Wind

speed

(m/s)

VESTAS

ENERCON

BONUS

FLODA

N

E (GWh/yr)

c#($/kWh)

N

E (GWh/yr)

c#($/kWh)

N

E (GWh/yr)

c#($/kWh)

N

E (GWh/yr)

c#($/kWh)

Perdikokor"

1.7

9.2

258

201.5

4.6

270

207.9

4.7

230

185.6

4.5

224

181.9

4.5

Kissamos

3.0

9.0

316

242.2

4.8

347

225.8

4.9

314

241.1

4.8

298

233.4

4.7

AnoMoulia

3.5

8.8

330

247.3

4.9

368

262.5

5.1

339

251.1

5.0

320

242.5

4.9

Kavenos

1.7

8.5

239

171.2

5.1

246

174.9

5.1

209

155.2

4.9

201

150.9

4.9

Rodopos

4.0

8.5

340

244.6

5.1

382

260.7

5.4

359

252.5

5.2

336

242.8

5.1

MoniToplou

1.1

8.0

183

121.6

5.4

176

118.3

5.4

142

99.5

5.2

137

96.6

5.2

Ko"nas

2.1

8.0

244

162.7

5.5

252

166.8

5.5

220

151.1

5.3

208

144.7

5.3

Achendrias

2.0

8.0

238

159.2

5.5

246

162.6

5.5

213

146.4

5.3

202

140.5

5.3

Prinias

0.8

7.6

142

88.7

5.8

127

80.8

5.7

97

64.5

5.5

92

61.7

5.5

Permabelas

1.4

7.4

179

107.9

6.0

169

103.4

6.0

140

88.6

5.8

130

83.1

5.8

Vrouhas

1.6

7.3

186

110.6

6.0

178

106.9

6.1

150

93.1

6.0

138

86.8

5.9

Vatos

1.7

7.2

187

109.1

6.3

179

105.4

6.2

153

92.6

6.1

139

85.7

6.1

Ravdouha

0.7

7.0

104

58.3

6.5

77

45.3

6.3

55

33.5

6.2

47

28.9

6.2

Erimoupolis

1.7

7.0

177

99.1

6.6

165

93.9

6.5

141

82.4

6.4

126

74.8

6.3

Achladia

3.0

7.0

240

134.1

6.7

249

137.5

6.7

231

130.3

6.6

207

119.6

6.5

MikronOros

0.8

7.0

113

63.5

6.5

88

51.3

6.3

65

39.3

6.2

56

34.3

6.2



12/12

are given in Table 3 for all turbines. Energy versus cost

correlation is depicted in Fig. 10. This Figure will act as

a strong compass to future investment plans in this

particular area. In addition, it serves as a valuable tool

for the state regional planner to appropriately guide

subsidies strategy in order to favor a speci"c site over

another one (thus selectively reduce energy produced

unit cost).

4. Conclusions

Regional planning requires short-cut design of wind

farms so that decisions related to the penetration of the

corresponding technology are taken on a uni"ed regional

rationale. Under this concept, the design of wind farms

can be properly analysed and addressed by means of

optimizing the expected bene"ts from such an investment

in the "eld of renewable energy exploitation. Optimiza-

tion can be carried out by developing the mathematicalmodel of wind turbines, taking into account their con-

struction characteristics and operational performance.

The model must also involve the regional characteristics

in terms of the yearly wind speed duration curve of

a speci"c site. The design problem can be formulated as

a mathematical programming one, and can be solved

using appropriate programming techniques. An empiri-

cal short-cut design equation describes optimal farm size

for a wide range of site characteristics and farm sizes and

four di!erent types of commercially available wind tur-

bines. In this case, the optimum number of wind turbines

installed, the amount of energy recovered and a reason-

able estimation of the plant cost can be automaticallydetermined. Generally, large wind turbines are preferred

for the most cases of wind farm design. Moreover, unit

cost of land area has a signi"cant impact on the design

problem, since its variation introduces an additional

degree of freedom to the problem and a more relaxed

optimization can be carried out.

Acknowledgements

This work was supported by the EU ALTENER

Project No.1030/93.

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