Energy for Sustainable Development 16 (2012) 471–475

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Energy for Sustainable Development

Optimising land use for wind farms

David Christie ⁎, Mark BradleyLews Castle College, University of the Highlands and Islands, Stornoway, Isle of Lewis, HS2 0XR, UK

⁎ Corresponding author.E-mail address: david.christie@uhi.ac.uk (D. Christie

0973-0826/$ – see front matter © 2012 International Enhttp://dx.doi.org/10.1016/j.esd.2012.07.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 17 November 2011Revised 20 July 2012Accepted 20 July 2012Available online 19 August 2012

Keywords:Wind farmsWakeLand use

The optimal scales and densities for wind turbine arrays are examined from the perspective of maximisingpower density, defined as power per unit area of land occupied. This is different from the usual aim ofminimising the cost of electricity production but could become increasingly important if available sites forwind farms become a limiting factor in their construction.A simple model is used to calculate the theoretical maximum power densities available from various config-urations of wind turbine array, taking into account the wake effect. The effects of array size, turbine separa-tion and perimeter set-back are investigated.It is observed that a wind farm designed to maximise power production per unit area of land could be verydifferent from one designed to maximise economic gain.

© 2012 International Energy Initiative. Published by Elsevier Inc. All rights reserved.

Introduction

In the last decades of the twentieth century commercial wind tur-bines doubled in size every four years (Stiesdal, 1999). Offshore tur-bines have continued this trend in the twenty-first century, althoughthere has been a levelling off in the size of onshore turbines (Too etal., 2009). The proliferation of larger turbines is driven by economicconsiderations. As the energy available from the wind is proportionalto the swept area of the turbine (Patel, 2005), increasing wind turbinesize has generally resulted in a decrease in total production cost perkilowatt hour of electricity produced (Thomsen, 2010).

Currently, minimising the cost of energy production is the princi-pal objective in wind farm design. Economic and related factors dic-tate the number and size of turbines in a development. Only afterhaving determined the general properties of the wind farm will thedesigner then seek to maximise the energy production for a givenland or sea area by adjusting the arrangement of the turbines. This isachieved using commercial software, which takes account of the to-pography and meteorology of a site, as well as the wake effect,where a turbine in the lee of other turbines experiences a reductionin available wind energy.

Efficiency is currently measured in terms of production cost, rath-er than land use. However, this situation may change as finding suit-able wind farm sites becomes a limiting factor. This is already beingobserved, for example in Japan (Kogaki et al., 2002) and Thailand(Bennui et al., 2007). This mirrors the situation with North Sea oil:as resources diminish, continuing demand results in continuing ex-traction, despite the rising costs. With increasing demand for land re-sources (Food and Agriculture Organization of the United Nations,

).

ergy Initiative. Published by Elsevi

1999), wind farm planners who have identified theoretically suitablesites must frequently face competition with other users. The processis also constrained by planning legislation and subjective opinionsabout the visual impact of turbines. These factors promote the needto use permissible sites efficiently and not just cheaply.

This article seeks to establish, in the most general terms, the prop-erties of a generic wind farm that maximises power yield per unitarea of land required. It cannot be assumed a priori that large individ-ual turbine size continues to be advantageous if maximising electric-ity production in a specified area is the main objective. It is possiblethat a larger number of small turbines would be able to deliver ahigher power density than fewer large turbines.

‘Real world’ onshore turbine arrangement varies greatly accordingto the topography of the land, and may be fine-tuned with optimisa-tion software. Prevailing winds will influence the design of both off-shore and onshore wind farms, and the atmospheric boundary layermust be considered for developments exceeding 10–20 km (Calaf etal., 2010). By contrast, we study a highly idealised situation, with reg-ular turbine spacing. By adopting such a simplified arrangement, wecan reduce a complex web of variable parameters to just three: thenumber of turbines, their separation, and the distance by whichthey must be “set back” from the perimeter of the wind farm. For var-ious combinations of these parameters, we estimate the total power,allowing for wake effects, per unit area footprint. Thus, we can estab-lish, in the most general terms the type of wind farm developmentthat yields the largest power density. The treatment is similar tothat of Phillips et al. (2010), where the separation and density of anidealised turbine grid were considered. However, the analysis hereis from the perspective of power density per unit area footprint ratherthan efficiency per turbine. Also, by restricting ourselves to the sim-plest wake model, we can explore a larger subset of the parameterspace.

er Inc. All rights reserved.

Fig. 1. Hexagonal turbine arrays with an edge length of n turbine separations. Theset-back distance and total area footprint, including “collar”, is shown for the n=4case.

472 D. Christie, M. Bradley / Energy for Sustainable Development 16 (2012) 471–475

Power density model

Arrangement of turbines

Our generic wind farm has a regular hexagonal arrangement ofturbines based on an equilateral triangular grid, similar to the config-uration 14×7 considered by Djerf and Mattson (2000).1 As shown inFig. 1, the turbine spacing is d, whilst the total calculated area foot-print includes a “collar” around the outer turbines, to allow them tobe set back a prescribed distance of p from the edge of the windfarm to allow for safety considerations (e.g. mechanical damage toturbines, ice shedding) as well as noise levels. The total area footprintA(n, d, p) and array size N(n) are given by2

A n;d;pð Þ ¼ 3ffiffiffi3

p

2n2d2 þ 6pndþ πp2; N nð Þ ¼ 3n nþ 1ð Þ þ 1: ð1Þ

Power yield

A simple model for power extraction from an ideal turbine was de-veloped by Betz (1926), and is described in wind energy handbook,for example Manwell et al. (2009) and Sathyajith (2006). For an ambi-ent wind velocity of u, directed into the turbine plane, the velocity im-mediately behind the turbine falls to γu. The power yield is given by

P ¼ π4ρr2Tu

3CP 1þ γð Þ 1−γ2� �

ð2Þ

where ρ is the density of the air and rT the turbine radius. Themaximumpower would be obtained for γ ¼ 1

3, giving

P≃1:14r2Tu3 ð3Þ

The coefficient in Eq. (3) is a theoretical maximum, and will belower for non-ideal turbines. However, taking a different value to ac-count for losses will scale all the results in the same way and will notaffect the qualitative comparisons. We attempt to develop the sim-plest model with as few variable parameters as possible and ignoreturbine power curves, individual design specifications, and intention-al operation below maximum rated power.

Wake effects

Immediately behind the turbine, the effective wind speed drops toγu. A region of reduced wind power, known as the wake, thenspreads out behind this, affecting neighbouring turbines. A simplemodel for such wakes, which is convenient to implement with fewfree variables, was obtained by Jensen (1983).

Assign to a turbine the Cartesian co-ordinates (xT1, yT1), with thex-axis in the direction of the wind. The wake is assumed to spread outlinearly in a frustum behind the turbine, which can be extrapolatedto a cone with vertex (xT1−x0, y0) in front of the turbine, and ahalf-angle φ, giving rise to an entrainment parameter α:

tanφ ¼ rTx0

≡α: ð4Þ

Momentum balance considerations give the wind speed withinthe wake cone as

vw ¼ u 1þ γ−1ð Þ r2T

r2w

" #; ð5Þ

1 For a given turbine separation, this is the densest theoretical arrangement. Real ar-rangements will vary with the individual characteristics of the site.

2 Note that n=0 for a solitary turbine.

where the cone radius rw=α(x−xT+x0). By inverting the equationof the wake cone, it is straightforward to show that the wind speedat a point (x, y) will be affected by any turbine situated in a frustrumupwind from it, whose positions (xTn, yTn) satisfy

y−α x−xTnþ x0

� �≤yTn

≤yþ α x−xTnþ x0

� �: ð6Þ

When multiple turbines lie upwind from the turbine of interest,the individual wakes must be superposed. Various methods areused for combining multiple wakes (VanLuvanee, 2006; Crespo etal., 1999; Vermeer et al., 2003; Lissaman, 1979; Katić et al., 1987;Voutsinas et al., 1990, 1993). Jensen's single turbine calculationused a volume flow balance to obtain a linear relation betweenspeed deficit and downwind distance. In the same spirit, one canshow that the wind speed deficit in an overlap area can be obtainedby summing the individual wakes arithmetically, using the formula

v ¼ uþ γ−1ð Þr2TXMn¼1

vTn

r2n: ð7Þ

Where the sum is over all the turbines Tn whose positions satisfyEq. (6), and vTn is the wind speed at each turbine.

This is straightforward to implement computationally for arrays ofturbines. Assign Cartesian co-ordinates to each turbine, with thex-axis in the wind direction. Consider each turbine in turn, in orderof increasing x-position. For the turbine in question, establish whichof the previous turbines' wakes will be in the upwind cone of influ-ence, using Eq. (6). Then, obtain the wind speed at the turbine inquestion using Eq. (7). Using Eq. (3) for the power in an ideal windturbine, the theoretical maximum power for the N turbines in thearray is given by

P ¼ 1:14r2TXNn¼1

v3Tn: ð8Þ

The structure of the overlapping wakes will vary with the wind di-rection, as well as turbine separation and array size. The power calcu-lation must therefore be repeated for each combination of theseparameters.

Implementation

For a given turbine configuration, the power density is calculatedby dividing the total array power incorporating wake effects, calculat-ed using the procedure in the section Wake effects, by the area foot-print as described in the section Arrangement of turbines. Weconsidered a sequence of hexagonal turbine grids (such as those inFig. 1) of various sizes and turbine separations. For each grid, thetotal power was calculated for wind directions increasing in incre-ments of 1∘ (by rotating the array so that the x-axis aligns with the

0.95 0.85

0.8

0.75

0.7

0.65

0.6

0.9

0.85

0.8R

elat

ive

Effi

cien

cy

Rel

ativ

e E

ffici

ency

0.750 100 200

Number of turbines in array300 5 6 7 8 9

Array spacing (turbine diameters)

Relative efficiency for different array sizes, 8D spacing Relative efficiencies for different turbine spacings

127 Turbines169 Turbines

Fig. 2. The plot on the left shows the relative efficiencies (per turbine) for hexagonal arrays of various sizes, with a spacing of 8 turbine diameters. The plot on the right shows therelative efficiency for various separations, for 127- and 169-turbine arrays.

6

5

4

3

2

1

Power densitiesVarious array sizes and separations

473D. Christie, M. Bradley / Energy for Sustainable Development 16 (2012) 471–475

wind direction). Due to the high degree of symmetry in our hexago-nal turbine array, we need only consider wind directions of up to30∘: with reflections and rotations, this basic unit can be repeated tocover the full 360∘. The full wake calculation must be repeated foreach array size, separation and wind direction, as the structure ofthe wake interactions differs in each case.

The entrainment angle φ was assumed constant. Following Jensen(1983), we select a value of φ that gives α=0.1 in Eq. (4).

We assume that the turbine separation and the set-back distance areboth expressed in terms of blade diameters. For a fixed α, a change in rTwill bring about an overall change in scale, but will not affect the struc-ture of the overlapping wakes. The area footprint and the theoreticalmaximum power (given by Eq. (8)) are both quadratic in rT. If we ne-glect the variation ofwind speedwithhub height (andhence, indirectly,turbine size), the power per unit area will then be independent of theturbine radius, and will only depend on array size, separation andset-back. This simplifies the analysis considerably.

The wind speed is taken to be 9:32ms−1, which is the averagewind speed at hub height for the Horshader Community Wind Devel-opment on the Isle of Lewis (Anderson, 2009). This is a necessarily ar-bitrary choice. However, in our simple model, the wakes comprisefractional falls in wind speed. In our idealised expression (8) for the-oretical maximum power, a change in ambient wind speed will onlyaffect overall scaling, and will not alter the comparisons betweenarrays.

Guidelines on set-back distances vary. In the section Prescribedset‐back of 5 turbine diameters, we take a relatively conservativevalue of 5 turbine diameters as the set-back distance from the outerturbines to the boundary of the wind farm, matching Clay County(2009). In the section Variable set-back distance, we examine the ef-fect on calculated power density by varying this parameter.

We considered several different turbine separations. The smallestseparation was 5 turbine diameters: below this threshold value, tur-bulent and other collective phenomena begin to have a detrimentaleffect on the array performance (Seifert, 2003). The largest separationconsidered is 9 turbine diameters.3

Preliminary example

Although the analysis will consider power per unit area, we in-clude an example where a different measure of efficiency is used.This enables comparison with existing treatments, in particular(Phillips et al., 2010), where power outputs of regular square arrays

3 For an array to have a higher power density than turbines sited individually, its ar-ea footprint per turbine cannot exceed that of a single turbine. Thus, we demand that

A n;d;pð ÞN nð Þ bA 0; d; pð Þ. Using Eq. (1), this criterion becomes db2

ffiffi3

p3 −1

n þ 1n2 þ

ffiffiffi3

pπ

2nþ 1n

� �12

" #p.

This upper bound increases with increasing n, but will never exceed ~1.9p. For a set-back of 5 diameters, this means a maximum separation of 9 diameters.

of turbines were calculated with four widely used methods, includingthe Jensen/Park model we have adopted. Two slices of the parameterspace were considered by Phillips et al. (2010). Firstly, the separationwas held at 8 turbine diameters, and the array size was varied. Sec-ondly, the size of array was held at 144 turbines, and the separationvaried. Efficiency was defined as power per turbine, divided by thepower of a solitary turbine without wakes. The behaviour of themodels was broadly similar, albeit with some differences in detail.

For comparison, Fig. 2 plots our wake calculation for similar sets ofturbine arrays. The plots are of very similar shape to those in Phillipset al. (2010), but the efficiencies calculated (per turbine) are some-what lower. This is because a triangular array is a denser arrangementof turbines than a square array of the same separation so that thewakes will be larger. Different values of the entrainment constanthave also been used.

Prescribed set-back of 5 turbine diameters

Here, ten different sized hexagonal arrays were considered. For eacharray, the power calculation was performed for turbine separations of5, 6, 7, 8 and 9 turbine diameters, and the full 30° range of uniquewind directions was considered in increments of 1∘. For Fig. 3, meanpower densities were obtained by taking a mean over the 30 differentwind directions. These were plotted against array size (on the x-axisof the plot) and turbine separation (on the y-axis). The power densityis seen to decrease with increasing turbine separation. The reductionin wake loss for widely separated turbines is hence more than offsetby the increase in area. The smallest spacing, five turbine diameters,

07 19 37 61 91 127 169 217 271 331

Number of turbines in array

9 8 7 6 5

Separation

(turbine diameters)

Fig. 3. The mean power for various array sizes (x-axis) and separation distances(y-axis).

8

7

6

5

Setback Distance v Power Density for Various Array Sizes

3 4 5Setback Distance (turb. diameters)

6 7

Fig. 5. Power density plotted against set‐back distance for various sizes of array, withseparation 5 turbine diameters. The array sizes range from 1 to 127 turbines. For thelowest set‐back, the smaller the array, the greater the power density. For largest set‐backs, this is reversed.

474 D. Christie, M. Bradley / Energy for Sustainable Development 16 (2012) 471–475

yields the greatest power density. For a given turbine separation, thepower density increases with array size, then tails off, so that there isa definite optimum number of turbines in each case, although the over-all variation is small by this measure. The optimal arrangement overallis a 61 turbine wind farm with a separation of 5 turbine diameters.The turbine separations show strikingly little variation in power densitywith array size. In Fig. 2, the usualmeasure of efficiency per turbine for an8 diameter separation dropped off significantly for larger arrays. How-ever, the reduction in area footprint per turbine in large arrays compen-sates for this, so that the power density for the 8 diameter separation isrelatively unchanged.

In Fig. 4, the spread of power densities over the 30∘ range of winddirections is plotted. The x-axis shows the array size; different turbineseparations are represented by different colours/shades of grey. Thelarger spacings give lower power densities due to their increasedarea footprints, but they have the benefit of considerably lower vari-ability of the variability of power yield with wind direction.

Variable set-back distance

In this section, the effect of a variable set-back distance on powerdensity is tracked for various wind farm configurations.

In Fig. 5, the power density (averaged over wind direction) of eacharray size is plotted against set-back distance for a 5 diameter turbineseparation. It can be seen straight away that if the set-back is 5 turbinediameters, then the 61 turbine array is marginally the highest-yieldingarrangement. This matches the results of the section Prescribed set‐back of 5 turbine diameters. As one might expect, smaller set‐backs fa-vour smaller arrays, but Fig. 5 enables us to find the crossover points.Thus, for example, if a set-back of 4 turbine diameters is prescribed,then a 19 turbine array is the most efficient use of land.

Fig. 6 gives the results for arrays with spacings of 8 turbine diam-eters. As the array separations increase, the power densities fall, andthe lines cross each other at larger set-back distances.

Discussion

This article investigates the scale and density of wind turbinearray required to maximise the theoretical power yield per unitarea of land. The results indicate that a pivotal element in this pro-cess is the ratio between the perimeter set-back of a wind farm andthe spacing between individual turbines within it. Set-back, like tur-bine spacing, is not standardised on land and varies with local build-ing codes, noise policy and potential ice throw (RERL, 2009). At sea,factors include the proximity of shipping routes, areas used as an-chorages, fishing grounds, and the level of traffic in the area (UKMCA).

If the set-back distance is less than a half of the distance betweenindividual turbines in an array, a greater power density will be

7Spread of power densities with wind direction

6

5

4

3

2

1

07 19 37 61 91 127 169 217 271 331

Number of turbines in array

Pow

er D

ensi

ty(W

atts

/squ

are

met

re)

5 6 7 8 9

Fig. 4. The maximum and minimum power densities for various array sizes and sepa-ration distances.

achieved by using a single turbine, and arrays with fewer turbineswill give a higher power density than larger arrays. Where a small pe-rimeter size is acceptable, the smallest number of individually largeturbines will maximise power density. As set-backs increase, largerarrays are increasingly favoured.

Morgan and Bossanyi (1998) consider 5 turbine diameters as aset-back suitably safe from “significant risk”. With this set-back, a61 turbine array with 5 diameter spacing maximises power density.For our approximate analysis, this remains the same whatever thescale of the farm, so for a finite farm area, individual turbine sizewould ideally be scaled up or down to produce 61 turbines at a 5 di-ameter spacing. Arrays with 31 and 91 turbines come a close second:the results give an order-of-magnitude indication of the optimal arraysize. This disagrees with Roy (1996) who argues that large turbinesobtain a higher energy output per unit area of land. Increasing the dis-tance between a fixed number of turbines would reduce the wake ef-fect but not enough to compensate for the additional land arearequired. Conversely, for wider separations, more numerous arrayslead to increased wake effect, with individual turbines performingmore poorly, but a more efficient use of space means that thepower density barely varies with array size.

Other factors will influence wind farm design. Fewer large tur-bines mean that any one being taken “off line” has a greater signifi-cance on the total energy production (Patel, 2005). Rogers et al.)consider the issues of maintenance and repair to be critical issues inthe design of wind power systems. However, before decisions oncompromises can be made the theoretical optimum situation needsto be understood.

Wind farms are eventually limited by size, and caution should beexercised when modelling larger wind farms, where the approximatewake model begins to break down (Schlez and Neubert, 2009).

3.4

3.3

3.2

3.1

3

2.9

2.8

Setback Distance v Power Density for Various Array Sizes

Setback Distance (turb. diameters)5 5.5 6 6.5 7

Fig. 6. Power density plotted against set‐back distance for various sizes of array, withseparation 8 turbine diameters.

475D. Christie, M. Bradley / Energy for Sustainable Development 16 (2012) 471–475

Collective effects such as resonances may also play an increasing rolefor large, regular turbine arrays, and the size of a wind farm is ulti-mately constrained by the boundary layer that can build up above itand significantly impact on the power production of the downstreamturbines (Calaf et al., 2010). Intelligent interactive wind farms canalso sacrifice the performance of individual turbines to enhance theoverall performance of the array (Abdelkafi and Krichen, 2011).

Our analysis also neglects the variation in wind speed withaltitude. Turbines with larger blades tend to be taller, enablingthem to reach the greater wind speeds encountered at higher alti-tudes (Mackay, 2009). The authors tentatively suggest that talltowers with small blades would allow for both the optimum turbinespacing and the advantages of higher wind speeds. Such a farm of“sunflower” turbines would be very different in appearance fromcurrent designs.

Our idealized treatment does not take into account topographical,political or economic constraints on wind farm design. The simplewake model also neglects ground effects, as well as external effectsfrom neighbouring farms. Wake turbulence, and especially partialwake turbulence (Seifert, 2003), has a large effect on a turbine's lifespan, with fatigue damage being a major cost consideration (Eecen etal., 2007). A uniformly variable and equally strong 360∘ wind is alsoassumed: prevailing wind patterns would in reality alter the optimumlayout. An applied design to maximise area use efficiency would un-doubtedly need to includemore criteria than the model presented con-siders. The interplay between rotor diameter, turbine density and setback distancewould however remain fundamental factors in the designprocess, and our simple model produced plots in Fig. 2 that broadlyagree with those obtained in Phillips et al. (2010) using more sophisti-cated techniques. It should also be noted that the simplicity or other-wise of any model selected does not alter the suggested change inemphasis from maximising economic gain to maximising power pro-duction per unit area occupied. It is reasonable to conclude that awind farm designed to maximise power production per unit area ofland could look very different from current installations.

Acknowledgements

This research was carried out with support from the UHI StrategicDevelopment Body, the European Regional Development Fund, High-lands and Islands Enterprise and Comhairle nan Eilean Siar. The au-thors are grateful to Neil Finlayson, Eddie Graham and Arne Vöglerfor useful comments.

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