The iteration method for tower height matching in wind farm design

The iteration method for tower height matching in wind farm design

K. Chen, M.X. Song, X. Zhang n

Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing100084, PR China

a r t i c l e i n f o

Article history:Received 18 December 2013Received in revised form16 June 2014Accepted 17 June 2014

Keywords:Tower height matchingWind turbine positioning optimizationGreedy algorithmIteration methodWind farm

a b s t r a c t

This paper studies the tower height matching problem in wind turbine positioning optimization. Variousmodels are introduced, including the power law wind speed model with height in the wind farm, thelinear wake flow model for flat terrain, the particle wake flow model for complex terrain and the powercurve model with power control mechanisms. The greedy algorithm is employed to solve the windturbine positioning optimization at a specified tower height. The optimization objective is to maximizethe Turbine-Site Matching Index (TSMI), which includes both the production and the cost of wind farm.Assuming that the optimized layout for each tower height is the same, an iteration method is developedto obtain the approximated optimal height. The convergence of the proposed iteration method isdiscussed through the mathematical analysis. The proposed iteration method is validated through thenumerical cases over both flat terrain and complex terrain. The results indicate that the proposedmethod can obtain better optimized height in shorter computational time than previous studies.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays, energy shortage problem has become one of theserious problems all over the world. In order to mitigate thisproblem, people start to pay attention to all kinds of renewableenergy resources, including solar energy, geothermal, biomass,tide and wind energy. Among these renewable energies, windenergy is an important alternative energy due to the advantages ofclean and rich resources (Chen and Zhang, 2007). In China, thereare a large amount of wind energy resources. The potential windpower at 10 m height on the land and on the sea are 253 GW and750 GW, respectively (Tong and Dong, 2012). In the past decadeyears, wind energy is developing rapidly in China. The totalinstalled capacity of wind turbines reached 75.3 GW up to 2012,which was the largest in the world (Song, 2012).

Wind energy is extracted by wind turbine in wind farm.The power output of the wind turbine increases as the windspeed increases. Meanwhile, the wind turbine will generate awake region downstream due to the extraction of the wind powerand the disturbance of the wind rotor. In the wake region, thewind speed is reduced and the turbulence is increased. Therefore,the wind turbine positions should be designed to reduce the wakeeffect and increase the total power output of the wind farm.

Much investigation has been done on wind turbine positioningoptimization (WTPO). Researchers introduced many optimizationalgorithms to solve the problem. Genetic algorithm was the firstalgorithm introduced to solve WTPO by Mosetti et al. (1994). Thisalgorithm simulates the biological evolution process. Bases on thepopulation, the algorithm searches the optimized solution throughthe selection, crossover and mutation operators. In Mosetti's study,binary coding method was used combined with the linear wakemodel and 3-order power curve model. The target was to max-imize the production per unit cost. The effectiveness of geneticalgorithm on WTPO was validated by three numerical cases. Basedon Mosetti's study, others have used larger population and moregenerations (Grady et al., 2005), and more realistic models (Moraet al., 2007; Kusiak and Song, 2010) to improve the optimizedresults. Wan et al. (2009) used real coding genetic algorithm tooptimize the wind turbine positions with the target of maximizingthe total power output with the number of wind turbines fixed,obtaining better results than the ones by binary coding geneticalgorithm. Another type of optimization algorithms used in WTPOis greedy algorithm. Greedy algorithm is based on a single turbinelayout and the turbines are placed in the positions one by one thatmake the objective value maximum in each step. Compared togenetic algorithm, greedy algorithm requires less computationand the optimized result does not have randomness. Ozturk andNorman (2004) combined greedy algorithm with the adding,removing and moving operators to optimize the wind turbinepositioning problem. Based on the submodular property in theoptimization problem with linear wake model, Zhang et al. (2011)

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Journal of Wind Engineeringand Industrial Aerodynamics

http://dx.doi.org/10.1016/j.jweia.2014.06.0170167-6105/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: [email protected] (X. Zhang).

J. Wind Eng. Ind. Aerodyn. 132 (2014) 37–48

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used the lazy algorithm to reduce the computational time ofthe algorithm. Song et al. (2012) developed a particle wake modelthat can be used over complex terrain. Based on this particlewake model, the greedy algorithm with repeated adjustment wasdeveloped to optimize wind turbine positioning problem overcomplex terrain (Song et al., 2013). Besides, some other optimiza-tion methods were also introduced in WTPO, including simulatedannealing method (Rivas et al., 2009), Monte Carlo method(Marmidis et al., 2008) and particle swarm optimization method(Wan et al., 2010).

The wind speed of the wind farm and the cost of the windturbines will increase with the tower height. Therefore, the towerheight of the turbines should match the potential site to achievemaximum power output per unit cost. In the literature, the towerheight matching problem has been considered to further improvethe turbine layout based on WTPO (Chen et al., 2013a). TheTurbine-Site Matching Index (TSMI) was introduced as the objec-tive function, including the production and the cost of the turbinelayout. The greedy algorithm with repeat adjustment was intro-duced to solve WTPO. The optimal height of wind turbine can beobtained through the enumeration method. That is, apply WTPOat each optional tower height. Then the height with the maximumobjective value is the optimal tower height. However, it requires alarge amount of computational time, especially when the turbinehas a large range of optional tower height. In previous study, thefitting method was developed to obtain the optimized height inless computation. When using the fitting method, the normalizedpower output (L) is defined. The optional height range of the windturbine is divided into several parts with the same interval and thesplitting points are obtained. Then apply WTPO at each splittingpoint and reproduce the whole L curve using the L values at thesepoints through polynomial fitting. Finally, calculate the extremepoints of TSMI using the L curve and obtain the optimized point.The extreme points with the maximum objective value is theoptimized height. Three numerical cases were used to test theperformance of the fitting method. The results indicated that thefitting method can obtain the approximated optimal height infewer less times of applying WTPO than the enumeration method(Chen et al., 2013a). However, it needs at least 5-order fitting toobtain the optimized results with the error less than 5% for themulti-direction wind situations. In this paper, the tower heightmatching for WTPO is studied. Assuming that the optimized layoutfor each optional tower height is the same, an iteration method isdeveloped to obtain the optimized tower height. The convergenceof the iteration method is discussed through the mathematicalanalysis. The effectiveness of the proposed method is validated bythe numerical cases over both flat terrain and complex terrain.

The optimized results by three methods are compared for eachcase, including

� Enumeration method: Apply WTPO at each optional height andtake the height with maximum objective value as the optimalone. The result of this method is treated as the optimal result ofthe tower height matching problem.

� Fitting method: Obtain the optimized height through fitting theL curve, developed in previous study (Chen et al., 2013a).

� Iteration method: Assuming that the optimized layout for eachoptional tower height is the same, the optimized tower heightis obtained by an iteration process, developed in present study.

The remainder of the paper is organized as follows. Section 2presents the models introduced in WTPO. Section 3 introduces theoptimization methodology. Section 4 presents the iterationmethod for tower height matching problem. Section 5 discussesthe numerical results of the test cases. Section 6 presents theconclusions.

2. Models

2.1. Linear wake model

In this paper, the linear wake model used in the study ofMosetti et al. (1994) is employed to calculate the wind turbinewake effect for the wind farm on flat terrain. The wake model isconsidered to be a conical area, as shown in Fig. 1. The velocityinside the wake region is calculated by the following algebraicexpression:

u¼ u0 1� 2a

1þα xr1

� �2264

375 ð1Þ

where u0 is the local wind speed without placing the turbine, x isthe distance downstream the turbine rotor, r1 is the downstreamrotor radius, a is the axial induction factor and α is the entrain-ment constant, which are expressed as follows:

a¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffi1�CT

p2

ð2Þ

r1 ¼ r

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�a1�2a

rð3Þ

α¼ 0:5lnðh=z0Þ

ð4Þ

where CT is the trust coefficient, r is the radius of the wind rotor,h is the tower height of the wind turbine, and z0 is the surfaceroughness.

The size of the wake region is described by the wake influencedradius R, which is the radius of the wake region at a specifiedsection in the crosswind direction, expressed as

R¼ αxþr ð5ÞConsidering multiple wake interference effect, the velocity of theith turbine is calculated by Gonzalez et al. (2010)

ui ¼ u0i 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑N

j ¼ 1

Aij

πr2i1� uij

u0j

� �2" #vuut

0@

1A ð6Þ

where u0i and u0j are the local velocities at the ith and the jthturbines' positions without placing the turbines. They are equal tothe inlet speed of wind farm over flat terrain. uij is the wind speedat the wind rotor of ithe turbine in the wake region of the jthturbine, N is the number of wind turbines, ri is the rotor radius of

x

u0

uR=αx+r

u0

r

Fig. 1. Schematic of linear wake model (Chen et al., 2013a).

K. Chen et al. / J. Wind Eng. Ind. Aerodyn. 132 (2014) 37–4838

the ith turbine, and Aij is the rotor area inside the jth turbine wake,calculated by

Aij ¼

πr2i ; dijrrwj�ri12r2i ðθ1� sin θ1Þþ

12r2wjðθ2� sin θ2Þ; rwj�riodijorwjþri

0; dijZrwjþri

8>>><>>>:

ð7Þwhere

θ19∠AOjB¼ 2 arccosr2wjþd2ij�r2i

2rwjdijð8Þ

θ29∠AOiB¼ 2 arccosr2i þd2ij�r2wj

2ridijð9Þ

where rwj is the wake influenced radius of the jth turbine by Eq.(5), dij is the crosswind distance between the ith and the jth windturbines, and θ1 and θ2 are shown in Fig. 2. As the near wake flowis very complicated due to the disturbance of wind rotor, the linearwake model cannot reproduce the near wake flow. This model isonly applicable to the far wake flow, which is usually four times ofthe wind rotor diameter away from the turbine hub in down-stream. On the other hand, this wake model does not consider theeffect of the increase in turbulence intensity on the power output.However, the linear wake model is a good approximation to wakeflow over flat terrain. Furthermore, the wake flow is calculatedusing the algebraic expression, which can obtain the wake velocityin short time. Therefore, the linear wake model is commonly usedcombining with the optimization algorithms to solve the turbinepositioning problem over flat terrain.

2.2. Particle wake model

For complex terrain, the shape of the wake flow will change,so the linear wake model cannot be applied. Song et al. (2012)proposed the particle wake model that can be applied overcomplex terrain. Previous study has shown that particle wakeflow can be combined with the optimization algorithm to optimizethe wind turbine positioning problem (Song et al., 2013). In theparticle wake model, the turbine wake flow is calculated byparticle simulation and the velocity deficit in the wake region isrepresented by the particle concentration. The steps of the particlewake model are shown as follows (Song et al., 2012):

1. The velocity of the field should be obtained through theComputational Fluid Dynamics (CFD) method. The obtainedflow field is named the pre-calculated flow field.

2.

The particles are generated within the area of the wind rotorand moved based on the pre-calculated flow field.

3. In each time step of particle simulation, the convective effect,the diffusive effect and the attenuation effect are considered.The convective displacement and the diffusive displacementsatisfying Gaussian distribution are added to each particle.The total displacement is shown as

Δx¼ ð1þσrÞuΔt ð10Þwhere u is the local velocity interpolated through the pre-calculated flow field and Δt is the span of a time step, σ is thediffusive coefficient of the particle wake model, and r is aGaussian distributed random number. During the particlesimulation, the particles are disappeared according to aattenuation coefficient γ.

4. A cube with the side length as same as the wind rotor diameteris used to calculate the particle concentration. The particleconcentration of each time step is summed and averaged. Afterhundreds of time steps of simulation, the particle distributionbecomes stable. The averaged number of particles inside thecube during the particle simulation is normalized to therelative particle concentration, denoted by c.

5. The wake influenced velocity is calculated by the followingtransformation expression:

u0 ¼ uð1�βcÞ ð11Þwhere u0 is the wake influenced velocity, u is the velocity of thepre-calculated flow field, β is the transformation constant and cis the relative particle concentration.

In the particle wake model, σ, β and γ are three parameters tobe determined, which depend on the characteristics of the windturbine. The particle model still does not consider the effect of theincrease in turbulence intensity on the power output. However,this model contains the influence of the terrain through theparticle simulation based on pre-calculated flow field. Therefore,it can calculate the wake flow over complex terrain.

2.3. Total power output of wind farm

The power output of a wind turbine is expressed by the powercurve model, which gives the power output for each wind speed.The turbine power curve is characterized by the characteristicspeeds of the wind turbine, expressed as (Albadi and El-Saadany,2010)

PeðuÞ ¼ Pr �0; uouc or uZuf

PascðuÞ; ucruour

1; urruouf

8><>: ð12Þ

oj oi

wjr

ir

ijdoj

oi

A

1

2

B

wjr

irijd

Fig. 2. Schematic of wake area (Chen et al., 2013a): (a) dijrrwj�ri and (b) rwj�riodijorwjþri.

K. Chen et al. / J. Wind Eng. Ind. Aerodyn. 132 (2014) 37–48 39

where uc, ur and uf are the cut-in speed, the rated speed and thecut-out speed of the turbine, respectively. Pe is the actual poweroutput of the wind turbine, Pr is the rated power output and Pasc isPe as the percentage of Pr. In the present study, Pasc is calculated bythe cubic model, expressed as (Albadi and El-Saadany, 2010)

PascðuÞ ¼ u3

u3r

ð13Þ

The 2-parameter Weibull distribution is commonly used to modelthe wind speed characteristics, shown as (Burton et al., 2001)

f ðuÞ ¼ kc

uc

� �ðk�1Þe�ðu=cÞk ð14Þ

where k is the shape parameter and c is the scale parameter. Afterdiscretized, the discrete wind speeds ui and the corresponding piare obtained.

The total power output is the sum of the outputs of all theturbines, expressed as

Ptot ¼ ∑M

i ¼ 1½piPlayoutðuiÞ� ¼ ∑

M

i ¼ 1pi ∑

N

j ¼ 1PeðuijÞ

!" #ð15Þ

where Ptot is the total power output of the wind farm, M is thenumber of the wind cases, pi is the probability of the ith wind caseand PlayoutðuiÞ is the power output of the turbine layout with theincoming velocity ui. N is the number of the wind turbines and Pe

is the power output calculated by Eq. (12).Normally, wind speed will increase with the height from the

ground which is commonly modeled by power law or logarithmiclaw (Burton et al., 2001). In the present study, the power lawmodel is used, shown as

u¼ urefhhref

� �αu

ð16Þ

where uref is the wind speed at the reference height href , namedthe reference wind speed. αu is the wind shear coefficient. Eq. (16)is substituted into Eq. (15), shown as

Ptot ¼ ∑M

i ¼ 1pi ∑

N

j ¼ 1PeðHðhÞðuijÞref Þ

!" #ð17Þ

The wind speed increases with the tower height, increasing thetotal power output of the wind farm.

2.4. Turbine-site matching index

When constructing wind farm, the profit is always desirable foroptimality, through increasing the power output and decreasingthe cost. In this paper, the Turbine-Site Matching Index (TSMI)is introduced as the objective of the tower height matchingproblem, the same as the previous study (Chen et al., 2013a).TSMI represents the production per unit cost of the wind farm,expressed as

TSMI¼ Ptot=Pr

ICCð18Þ

where the numerator of TSMI is the total power output of windfarm normalized by Pr. The denominator is the normalized costICC, modeled by Albadi and El-Saadany (2010)

ICCðhÞ ¼ ICCðhÞICC80 m

¼ aC � hþbC ð19Þ

where ICC80 m is the ICC value at the height of 80 m, which is thecharacteristic ICC value. In present study, aC and bC are chosen as1:1875� 10�3 and 0.905 (Albadi and El-Saadany, 2010). As thetower height increases, Ptot and ICC both increase. There exists aoptimal height for the largest TSMI value.

A potential factor (PF) is defined to describe the total heighteffect on TSMI, shown as

PF¼ TSMImax�TSMImin

TSMIminð20Þ

where TSMImax and TSMImin are the maximum TSMI and theminimum TSMI, respectively. PF describes the maximum potentialimprovement of TSMI after optimization. The larger PF is, thelarger the possibility of potential improvement is.

3. Wind turbine positioning optimization

3.1. Greedy algorithm with repeated adjustment

For a specified tower height, tower height matching problembecomes a traditional WTPO problem, which can be solved by theoptimization methods in the literature. In order to compare withthe results of previous study, the greedy algorithm with repeatedadjustment (Chen et al., 2013a) is introduced to solve the WTPOproblem. The algorithm consists of two stages: locating stage andrelocating stage. Locating stage starts from the empty wind farm.The steps of locating stage are listed below.

1. The wind farm is meshed by square grids and all the grids arenumbered.

2. Add a turbine at the grid with the maximum evaluation value.3. Try each empty grid with a wind turbine and calculate the

evaluation value.4. Add a turbine at the grid with the maximum evaluation value

and record the order of the turbines adding in the wind farm.When there are some grids having the same maximumevaluation value, it will choose the grids with small numberto place the wind turbine.

5. If the specified number of turbines is reached, locating stage iscompleted. Otherwise, return to step 4 and continue theprocedure.

When locating stage is finished, a turbine layout with thespecified number of wind turbines is obtained. However, thestrategy of turbine placement in locating stage cannot guaranteeobtaining the optimal solution, so relocating stage is used forfurther optimization of the turbine layout. The steps of relocatingstage are shown as follows:

1. Remove one turbine according to the order of the turbines inthe adding process.

2. Try each empty grid with a turbine and calculate theevaluation value.

3. Add a turbine back to the layout at the grid with the maximumevaluation value.

4. A cycle is defined when all the turbines go through a relocatingprocedure. If the layout does not change during the wholecycle, relocating stage is completed. Otherwise, return to Step1 and continue the procedure.

When relocating stage is finished, the final optimized layout isobtained. In the algorithm, the evaluation function is chosen as theobjective function (TSMI), which is desirable for optimum.

3.2. Penalization of the objective

In the wind farm, the wind turbines should be placed farenough in case of damaging others when falling down. This is thesafe distance condition. In order to guarantee this safe distancecondition satisfied, the objective value is penalized and the


evaluation function is shown as

E¼0; min

i;j

Dij

Rij

� �oλ

TSMI; mini;j

Dij

Rij

� �Zλ

8>>><>>>:

ð21Þ

where λ is the adjust coefficient, Rij is the sum of the heights oftwo turbines, including the hub height and the rotor radius, and Dij

is the horizontal distance between two turbines. Through penali-zation, the wind turbines tend to locate at the positions satisfyingthe distance condition ðDijZλRijÞ. The distances among the tur-bines can be adjusted through changing the value of λ. λ should begreater than 1 to guarantee the safe distance condition.

4. Iteration method for tower height matching

When the power curve of the wind turbine satisfies the cubicmodel for all wind speeds, previous study has shown that theWTPO is the same optimization problem at each optional heightwhen ignoring the height dependency in wake effect and con-sidering the distances among turbines constant (Chen et al.,2013a). Therefore, the optimal wind turbine layout for each towerheight is the same. In this situation, through applying one WTPOat a specified height, the optimized layout for this height can beobtained. Translate the optimized layout to other tower heightsand evaluate the TSMI values to reproduce the TSMI curve. Thenthe height with the maximum TSMI value is the optimal height forthe tower height matching problem.

For the real situation, the optimal wind turbine layout for eachheight is different due to the power control mechanism and theheight effect of the models. However, as an assumption, theoptimal layout for each height is still considered to be the same.Then an iteration method for tower height matching can bedeveloped. The detail steps of the method are shown below.

1. Choose an initial standard height, denoted by h0.2. Apply WTPO at h0, obtaining the relevant optimized turbine

layout.3. Translate the optimized layout to other heights and reproduce

the TSMI curve through evaluate the layouts.4. Take the height with the maximum TSMI value as the opti-

mized height, denoted by hite. If hite equals to h0 or the maxi-mum number of iteration step is reached, stop the procedure.Otherwise, let h0 equal to hite, return to Step 2 and continue theprocedure.

5. When the iteration procedure is finished, the optimized heightfor each iteration step is obtained. Choose the height with thelargest TSMI value among these optimized heights as the finaloptimized height.

As the iteration method cannot guarantee obtaining a conver-gence optimized height, a maximum number of iteration steps isselected. When the maximum number is reached, the iterationprocedure is stopped. The approximated optimal height in thetower height matching problem can be obtained through theiteration method. Generally, the optimal layout depends on theoptimization method. Thus, the optimal layouts for similar towerheights may be quite different, but the objective values of theoptimal layout for similar tower heights are very close to eachother. This is the characteristic that makes the iteration methodeffective. The convergence of the proposed iteration method isdiscussed in Appendix A.

5. Numerical study

In this section, the proposed iteration method is validated bynumerical cases. Section 5.1 considers three typical cases over flatterrain in previous study, including the single-directional windcase with a single speed, the multi-directional wind casewith uniform speeds and the multi-directional wind case withvariable speeds. The proposed method is compared to the fittingmethods and the enumeration method through these three cases.In Section 5.2, Weibull distribution is introduced to model thewind speed characteristic and a cases with Weibull distributionover flat terrain is considered. Section 5.3 considers a multi-directional wind case over complex terrain to further test theiteration method.

5.1. Three typical cases

In this section, three typical cases over flat terrain are used totest the proposed iteration method and compare to the previousstudy (Chen et al., 2013a). The reference wind speed distributionsof the cases are shown as follows.

� Case 1: Single wind direction with a single wind speed.� Case 2: Multi-directional (16 directions with intervals of 22.51)

wind with uniform speeds.� Case 3: Multi-directional (16 directions with intervals of 22.51)

wind with variable speeds.

The parameters of the numerical study are listed in Table 1.The calculation domain is chosen as a square area with the sizeof 2000 m �2000 m. The ground roughness of the wind farm is0.3 m and the wind shear coefficient is 1/7 (Chen et al., 2013a). Thewind from west to east is defined as 01 and the one from south tonorth is defined as 901. The linear wake model is used to calculatethe wind turbine wake flow. The number of wind turbines to beplaced in the domain is 30 and the adjust coefficient λ in Eq. (21) isset as 1.05. Table 2 lists the properties of the wind turbine used.The optional tower height of the turbine is in the range ½40;140�.The power curve model is expressed as Eq. (12). Greedy algorithmwith repeated adjustment is used to solve the wind turbinepositioning problem with a specified tower height. The grids for

Table 1The parameters of the numerical study (Chen et al.,2013a).

Parameter Value

Domain 2000 m �2000 mGround roughness 0.3 mWind shear coefficient 1/7Grids for optimization 30 �30Number of wind turbines 30Adjust coefficient 1.05

Table 2Wind turbine properties (Chen et al., 2013a).

Property Value

Rotor diameter D (m) 40Trust coefficient CT 0.88Tower height (m) 40–140Cut-in speed (m/s) 2Cut-out speed (m/s) 18Rated speed (m/s) 12.8Rated power (kW) 630


optimization is set as 30�30. The initial tower height of theiteration method is chosen as 40 m and the maximum iterationsteps is 4. Note that the distances among the wind turbines may be

less than 160 m (four times of the wind rotor diameter) when thetower height is less than 57 m for λ being 1.05. In this situation,the linear wake model cannot be applied. Therefore, when the

Fig. 3. TSMI curves of Case 1: (a) uref ¼ 9 m=s; (b) uref ¼ 9:4 m=s; (c) uref ¼ 9:6 m=s; and (d) uref ¼ 10 m=s.

Table 3Comparison of results for Case 1.

uref (m/s) PF (%) Enumeration Fitting method (m) Iteration method (m)

hopt (m) hfit;3 hfit;4 hite;1 hite;2 hite;3 hite;4

9 44.9 128 131 135 118 137 128 1289.4 32.4 98 100 98 101 101 – –

9.6 26.1 88 75 83 89 89 – –

10 14.8 63 61 64 66 66 – –

Averaged error – 1.51% 1.19% 2.22% 1.13% 0.72% 0.72%Nopt 101 5 6 2 3 4 5

0 500 1000 1500 20000

500

1000

1500

2000

0 500 1000 1500 20000

500

1000

1500

2000

Fig. 4. Optimized wind turbine layouts of Case 1 ðuref ¼ 10 m=sÞ: (a) h¼63 m and (b) h¼66 m.


tower height is less than 57 m, the distances among wind turbinesare set to be at least 160 m to guarantee the validity of the linearwake model.

The tower height matching can be solved by the enumerationmethod, which applies optimizations in the interval of 1 m foreach height. The height with the largest objective value is theoptimal height. The optimized results by the fitting method inprevious study (Chen et al., 2013a) are also presented for

comparison. The relative error is defined to evaluate the optimizedresult, shown as

Errorðf iÞ ¼jf optimized� f optimaljmaxiðf iÞ�miniðf iÞ

ð22Þ

where f optimized is the optimized objective value using the iterationmethod or the fitting method and f optimal is the optimal objectivevalue using the enumeration method. The denominator of therelative error is the discrepancy of the best objective value and theworst objective value. The relative error ranges from 0 to 1. Whenthe relative error equals to 0, it obtains the best optimized result.When the relative error equals to 1, it obtains the worst optimizedresult. The smaller the relative error is, the better the optimizedresult is.

5.1.1. Case 1In Case 1, the single wind direction with one single speed is

considered and various values of uref are studied. Fig. 3 shows theTSMI curve with the height for each uref , which contains theresults of the enumeration method and the proposed iterationmethod. In Fig. 3, hn represents that the standard height of the nthiteration step. The results show that the TSMI curves of variousstandard heights are close to the accurate TSMI curve. Especiallyaround hn, the errors of the curves are small. As the iteration stepsincreases, hn gets closer and closer to the optimal height of theaccurate TSMI curve.

Table 3 shows the optimized results of the three methodsfor Case 1. hopt is the optimal tower height by the enumerationmethod, which is treated as the optimal height. hfit;n represents theoptimized height of n-order fitting using the fitting method. hite;n

represents the optimized height of the nth iteration step by theproposed iteration method. Nopt is the times of applying WTPOand is used to evaluate the total computation of the methods. Itcan be seen that the iteration method can obtain the convergenceoptimized results in 4 steps for various uref . hite;2 for various valuesof uref are close to hopt. The average error of objective value is only1.13%, which is similar to the ones by 4-order fitting (1.19%) for thefitting method. Furthermore, the iteration method only needs toapply WTPO three times to obtain hite;2, which is less than theenumeration method and the fitting method. As the number ofiteration steps increases, hite;n is closer to hopt and the averageerror of objective value decreases. For the 4th iteration step, theaverage error of objective value is only 0.72%. Generally, theproposed iteration method can obtain better optimized height inless computational time than the fitting method. For the selectedvalues of uref , the maximum PF is 44.9% and the minimum is 14.8%.Therefore, the height effect should be taken into consideration formore power output and less cost. The value of PF increases as thewind speed decreases. For a wind farm with lower wind speed, ithas more potential improvement in the tower height matchingproblem.

Fig. 4 shows the optimized layouts for the enumeration methodand the iteration method for the situation of uref ¼ 10 m=s. It canbe seen that the objective values are close, which is 1.019 for 63 m

Fig. 5. The wind rose of Case 2 (Chen et al., 2013a).

Fig. 6. TSMI curves of Case 2.


uref (m/s) PF Enumeration Fitting method (m) Iteration method (m)


10 18.0% Optimized height 74 78 77 72 73 74 72Relative error – 2.60% 5.38% 3.33% 6.54% 0% 3.33%Nopt 101 6 7 2 3 4 5


and is 1.017 for 66 m. The relative error calculated by Eq. (22) isonly 0.2%.

5.1.2. Case 2In Case 2, 16-directional wind with single speed of 10 m/s is

studied. The wind rose of Case 2 is presented in Fig. 5. Fig. 6 showsthe TSMI curves and Table 4 lists the optimized results by the threemethods. For the fitting method, the best optimized height is 78 mand the relative error is 2.60%. The error of the TSMI curves aroundthe optimal tower height between the enumeration methodand the iteration method decreases as the iteration step increases.As hite;4 equals to hite;1, the iteration method does not obtain aconvergence optimized height in this situation. The best heightamong hite;n of each step is the final optimized height, which is theone of the third step 74 m. For Case 2, the iteration method findsthe optimal height. Furthermore, it needs to apply fewer less

WTPOs for the iteration method to obtain the optimized heightthan other methods. Fig. 7 shows the optimized wind turbinelayout of Case 2.

5.1.3. Case 3In Case 3, multi-directional wind with variable speeds is

considered. The velocities are in the range from 9 m/s to 11 m/s.The wind rose of Case 3 is shown in Fig. 8. Fig. 9 shows the TSMIcurves of Case 3. Similar to the above cases, the error of the TSMIcurves around the optimal height between the iteration methodand the enumeration method decreases as the iteration stepincreases. Table 5 lists the optimized results by the three methods.For the fitting method, the best optimized height is 96 m and theerror is 0.48%. While for the iteration method, the optimizedheight is 93 m when the iteration is convergence, which is veryclose to the optimal one 92 m. However, though the optimizedheight for the iteration method is closer to hopt than the one of thefitting method, the error of the best TSMI value for the iterationmethod is 1.67%, a little larger than the one for the fitting method.Due to the influences of the grids and the safe distances among theturbines, there are few extreme points around the maximum pointin the TSMI curves, which can be seen in Fig. 9. Therefore, theresult by the iteration method may converge to a certain localoptimum. Through increasing the number of the grids in WTPO,the TSMI curves may be more smooth and the result by theiteration method may converge to the global optimum. Again, thevalue of Nopt for the iteration method is less than other methods.Fig. 10 shows the optimized layouts for the enumeration methodand the iteration method.

5.2. Case 4: a case with realistic models

In this section, a case with realistic models is considered tofurther test the proposed iteration method. Wind condition with16 wind directions is considered and each direction satisfy a

0 500 1000 1500 20000

500

1000

1500

2000

Fig. 7. Optimized wind turbine layout of Case 2.

Fig. 8. The wind rose of Case 3 (Chen et al., 2013a). Fig. 9. TSMI curves of Case 3.


uref (m/s) PF Enumeration Fitting method (m) Iteration method (m)


9–11 9.8% Optimized height 92 79 96 87 91 93 93Relative error – 5.13% 0.48% 4.47% 2.24% 1.67% 1.67%Nopt 101 6 8 2 3 4 5


Weibull distribution. The parameters of the Weibull distributionsare generated randomly, as shown in Fig. 11. The wind turbine inthe study of Chen et al. (2013b) is used, which has a tower heightranging from 46 m to 78 m. The parameters of the turbines arelisted in Table 6. The power curve of the turbine are shown as

PðuÞ ¼0; uo2:5 or uZ280:3084� u3; 2:5ruo13:0185680; 13:0185ruo28

8><>: ð23Þ

The cost model in the study of Chen et al. (2013b) can be expressedas below

Cost¼ C1þC2h ð24Þwhere C1 is turbine cost and C2 is tower cost per meter. Theobjective is chosen as

Objective¼ Ptot

∑Ni Costi

ð25Þ

where N is the number of the wind turbine placed in thewind farm.

The domain of the wind farm is chosen as a square area withthe size of 1000 m �1000 m, meshed by 20 �20. The groundroughness of the wind farm is 0.3 m. The number of wind turbinesto be placed in the domain is 10. The distance among the windturbine is fixed as 200 m, which is five times of the wind rotordiameter. The tower height matching is solved by the enumerationmethod and the proposed iteration method. For the enumerationmethod, the height with the best objective value is treated as theoptimal height. The relative error is calculated by Eq. (22). For Case4, wind speeds satisfy various Weibull distributions for various

wind directions. The initial standard height is chosen as 46 m.Table 7 shows the optimized results. Through the enumerationmethod, the maximum objective value corresponds to the towerheight of 55 m and the minimum one corresponds to 78 m. The PFvalue is 1.2%. It only needs to apply twice of optimizations toobtain the convergence optimized height for iteration method. Theoptimized height is 55 m, equals to the optimal one. Fig. 12 showsthe optimized layouts for the tower heights of 55 m and 78 m.When the tower height is large, the increase of the cost overcomesthe increase of the power output. Thus, the objective valuedecreases. It can be seen that the iteration method has goodperformance for Case 4.

5.3. Case 5: a case over complex terrain

In this section, a case over complex terrain is used to validatethe proposed iteration method. A typical complex terrain with thesize of 2000 m �2000 m is introduced and the contour of altitudeis shown in Fig. 13. Consider the same wind cases and the samewind turbine in Case 3. The number of wind turbines to be placedin the site is 30. The flow fields of the complex terrain of variouswind directions are obtained through CFD calculation. The particlewake flow model is introduced to calculate the turbine wake flow.For the wind turbine in Case 3, the parameters β, σ and γ of theparticle wake model are valued as 0.65, 0.3 and 0.005, respectively(Song et al., 2012). The greedy algorithm is used to optimize thewind turbine positioning problem for a specified tower height.Since the particle wake model contains randomness and needs


Parameter Enumeration Iteration method

Optimized height (m) 55 55Relative error – 0%Nopt 33 2

0 500 1000 1500 20000

500

1000

1500

2000

0 500 1000 1500 20000

500

1000

1500

2000

Fig. 10. Optimized wind turbine layouts of Case 3: (a) h¼92 m and (b) h¼93 m.

0 100 200 300 4001

2

3

k

0 100 200 300 4005

10

15

c(m/s)

0 100 200 300 4000

0.1

0.2

Direction(degrees)

p

Fig. 11. Wind data for Case 4.

Table 6Parameters of the wind turbine in Case 4.

Parameter Value

Diameter (m) 40Rated power output (kW) 680Rated speed (m/s) 13.0158Cut-in speed (m/s) 2.5Cut-out speed (m/s) 28Thrust coefficient 0.8888Tower height (m) 46–78Turbine cost ð€Þ 593,867Tower cost ð€=mÞ 1500


more computational time to obtain the wake flow result, only thelocating stage of greedy algorithm is considered. The number ofoptimization grids is chosen as 50�50. Other parameters are thesame as Case 3.

Fig. 14 shows the comparison of the TSMI curves of the flatterrain and the complex terrain for the wind cases of Case 3. Forcomplex terrain, the wind speed for a specified height in the windfarm is not uniform. Large wind speeds are obtained at some partsof low height due to the disturbance of the terrain. When theheight is low, the TSMI value of the complex terrain is larger thanthe one of the flat terrain. When the height is large, the poweroutputs and the TSMI values of both terrains are similar. Therefore,the optimal tower height of the complex terrain is lower and therelevant TSMI value is larger than the ones of the flat terrain.PF value is 6% for Case 5. As the minimum TSMI value of complexterrain increases, PF value of complex terrain decreases comparedto flat terrain.

Table 8 shows the optimized results of the enumerationmethod and the iteration method for Case 5. It can be seen thatthe optimized tower height for the 4th iteration steps is 82 m,close to the optimal one 76 m by the enumeration method. Therelative error of the TSMI value is only 0.08%. Fig. 15 shows theoptimized layouts of 76 m and 82 m. In the wind farm, the locationwith higher altitude usually has higher wind speed, so some windturbines are placed on the top of the hill. Besides, most of the windturbines are located at the edge of the wind farm to enlarge thedistances from others, reducing the influence of the wake effect.

It can be concluded from Case 5 that the iteration method can alsobe applied to complex terrain.

6. Conclusions

In this paper, the tower height matching for wind turbinepositioning optimization is studied. The wind speed in wind farmis modeled by the power law with 1/7 order to the height. Thewake flow is calculated by the linear wake model over flat terrainand by the particle wake model over complex terrain. The powercurve of the wind turbine is characterized by a simplified 3-orderpower law with power control mechanism. The wind turbine

0 200 400 600 800 10000

200

400

600

800

1000

0 200 400 600 800 10000

200

400

600

800

1000


Fig. 13. The contour of altitude of the complex terrain.

Fig. 14. The comparison of TSMI curves of Cases 3 and 5.

Table 8Optimized results for Case 5.

uref

(m/s)PF Enumeration Iteration method (m)

hopt (m) hite;1 hite;2 hite;3 hite;4

9–11 6.0% Optimized height 76 75 77 82 83Relative error – 0.33% 0.33% 0.08% 0.34%Nopt 101 2 3 4 5


positioning optimization at a specified height is solved by greedyalgorithm, with the target of maximizing the Turbine-Site Match-ing Index (TSMI). Assuming that the optimized layout for eachtower height is the same, the iteration method is developed toobtain the approximated optimal height. The convergence of theproposed iteration method is discussed through the mathematicalanalysis. The numerical cases over flat terrain and complex terrainare introduced to test the proposed method. The results show thatthe TSMI curve generated by a selected standard height is close tothe accurate TSMI curve, especially around the selected height. Formost wind situations, the proposed iteration method can obtainthe optimized tower height in less computational time and theerrors of the results are less than the one in previous studies. As aconclusion, the iteration method is an effective way to solve towerheight matching problem in wind farm design.

Acknowledgments

This research is supported by the International Scientific andTechnological Cooperation Program of China (No. 2011DFG13020),the China Postdoctoral Science Foundation (2013M530043) andthe National High-Tech R&D Program (863 Program) of China (No.2007AA05Z426).

Appendix A. Convergence of the iteration method

Denote the accurate TSMI curve and the approximated TSMIcurve as T(h) and TaðhÞ, respectively, which are expressed as

TðhÞ ¼ PðhÞCðhÞ; TaðhÞ ¼

PaðhÞCðhÞ ; PaðhÞ ¼ PðhÞ�pðhÞ ðA:1Þ

where P(h) and Pa(h) are the accurate power output and theapproximated power output through translating the optimizedlayout of the standard height, respectively. C(h) is the costfunction, expressed by Eq. (19). p(h) is the error between PaðhÞand P(h). Denote the maxima of T(h) and TaðhÞ as hm and ham,respectively. The standard height of the optimized turbine layoutfor PaðhÞ is h0. We have

Tðh0Þ ¼ Taðh0Þ; Pðh0Þ ¼ Paðh0Þ; pðh0Þ ¼ 0

T 0ðhmÞ ¼ P0ðhmÞCðhmÞ�PðhmÞC0ðhmÞ½CðhmÞ�2

¼ 0; T 0aðhamÞ ¼ 0 ðA:2Þ

In order to prove the convergence of the iteration method, thefollowing assumptions are introduced.

Assumption 1. All the functions in Eq. (A.1) are at least second-order differentiable, that is

TðhÞ; TaðhÞ; PðhÞ; PaðhÞ; pðhÞ; CðhÞAC2 ðA:3Þ

Assumption 2. T(h) is a strictly concave function. Therefore,around hm, it obtains

T 0ðh0Þ40; when h0ohmT 0ðh0Þo0; when h04hm

(ðA:4Þ

Assumption 3. p(h) is a strictly convex function and h0 is theminima of the p(h), that is

pðhÞZ0; p″ðhÞZ0; hASp0ðh0Þ ¼ 0; p″ðh0ÞZ0 ðA:5Þwhere

S :½h0;hm�; when h0rhm

½hm;h0�; when h04hm

(

Assumption 4. Let FðhÞ ¼ pðhÞC0ðhÞ�p0ðhÞCðhÞ, which monotoni-cally decreases in S, that is

F 0ðhÞ ¼ pðhÞC″ðhÞ�p″ðhÞCðhÞr0; hAS ðA:6ÞBased on these four assumptions, the convergence of the

iteration method can be proved as follows:(1) When h0ohm, then

T 0ðh0Þ ¼P0ðh0ÞCðh0Þ�Pðh0ÞC0ðh0Þ

½Cðh0Þ�240 ðA:7Þ

Consider Eq. (A.2) and Assumption 3, it obtains

T 0aðh0Þ ¼

½P0ðh0Þ�p0ðh0Þ�Cðh0Þ�½Pðh0Þ�pðh0Þ�C0ðh0Þ½Cðh0Þ�2

¼ P0ðh0ÞCðh0Þ�Pðh0ÞC0ðh0Þ½Cðh0Þ�2

¼ T 0ðh0Þ40 ðA:8Þ

T 0aðhmÞ ¼

½P0ðhmÞ�p0ðhmÞ�CðhmÞ�½PðhmÞ�pðhmÞ�C 0ðhmÞ½CðhmÞ�2

¼ pðhmÞC0ðhmÞ�p0ðhmÞCðhmÞ½CðhmÞ�2

¼ FðhmÞ½CðhmÞ�2

ðA:9Þ

According to Assumption 4, F 0ðhÞr0. As h0ohm, then

T 0aðhmÞ ¼

FðhmÞ½CðhmÞ�2

r Fðh0Þ½CðhmÞ�2

¼ 0 ðA:10Þ

According to the analysis above, we have

T 0aðh0Þ40; T 0

aðhmÞr0 ðA:11Þ



Therefore, it obtains

h0ohamrhm; when h0ohm ðA:12Þ(2) When h04hm, then

Tðh0Þ ¼P0ðh0ÞCðh0Þ�Pðh0ÞC 0ðh0Þ

½Cðh0Þ�2o0 ðA:13Þ

Consider Eq. (A.2) and Assumption 3, it obtains

T 0aðh0Þ ¼



¼ T 0ðh0Þo0 ðA:14Þ

T 0aðhmÞ ¼

½P0ðhmÞ�p0ðhmÞ�CðhmÞ�½PðhmÞ�pðhmÞ�C0ðhmÞ½CðhmÞ�2

¼ pðhmÞC0ðhmÞ�p0ðhmÞCðhmÞ½CðhmÞ�2

ðA:15Þ

According to Assumption 4, F 0ðhÞr0. As h04hm, then

Ta0ðhmÞ ¼

FðhmÞ½CðhmÞ�2

ZFðh0Þ

½CðhmÞ�2¼ 0 ðA:16Þ

According to the analysis above, we have

T 0aðhmÞZ0; T 0

aðh0Þo0 ðA:17ÞTherefore, it obtains

hmrhamoh0; when h04hm ðA:18Þ(3) When h0 ¼ hm, then

T 0aðh0Þ ¼



¼ T 0ðh0Þ ¼ T 0ðhmÞ ¼ 0 ðA:19Þ

Therefore, ham ¼ h0 ¼ hm.According to the analysis above, based on the four assumptions,

we have

h0ohamrhm; when h0ohmham ¼ hm; when h0 ¼ hm

hmrhamoh0; when h04hm

8><>: ðA:20Þ

Therefore, from the process of the iteration, a sequence can beconstructed as follows:

ham;kþ1 ¼φðham;kÞ; ham;1 ¼ h0 ðA:21Þwhich satisfies

ham;koham;kþ1rhm; when h0ohm

ham;k ¼ ham;kþ1 ¼ hm; when h0 ¼ hmhmrham;kþ1oham;k; when h04hm

8><>: ðA:22Þ

where φ represents the iteration process. It can be seen that fham;kgis bounded and monotonous. According to calculus theory, fham;kgis convergence to a real number which is within the range ½h0;hm�or ½hm;h0�. However, it can only guarantee the convergence, butcannot guarantee that it can converge to hm. Nevertheless, when

h0 is chosen to be close to hm, the convergence result can be closeto hm.

The convergence is based on the four assumptions. Theoreti-cally, the tower height and the positions of turbines are contin-uous, so Assumption 1 is satisfied. A large amount of test casesshow that T(h) is a strictly concave function for most situations.That is, Assumption 2 is satisfied for most situations. In the presentstudy, C″ðhÞ ¼ 0 and CðhÞ40. According to Assumption 3, p″ðhÞZ0.Therefore, F 0ðhÞ ¼ �p″ðhÞCðhÞr0. Assumption 4 is satisfied whenAssumption 3 is satisfied. In reality, p(h) is not a strictly convexfunction and Assumption 3 is not satisfied for all the situations.However, after performing a large amount of wind cases, it isfound that the final height of the iteration method can converge tothe optimal one or at least around the optimal one for mostsituations.

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The iteration method for tower height matching in wind farm design