WIND ENERGY
Wind Energ.
2017; 20:465–477
Published online 4 August 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/we.2016
RESEARCH ARTICLE
Combining economic and fluid dynamic models to
determine the optimal spacing in very large wind farms
Richard J. A. M. Stevens
1,4
, Benjamin F. Hobbs
2
, Andrés Ramos
3
and Charles Meneveau
4
1
Department of Physics, Mesa+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede,
The Netherlands
2
Department of Geography and Environmental Engineering and the Environment, Energy, Sustainability and Health Institute, Johns
Hopkins University, Baltimore, Maryland 21218-2686, USA
3
ICAI School of Engineering, Institute for Research in Technology, Comillas Pontifical University, C/ Santa Cruz de Marcenado 26,
28015 Madrid, Spain
4
Department of Mechanical Engineering and Center for Environmental and Applied Fluid Mechanics, Johns Hopkins University,
Baltimore, Maryland 21218, USA
ABSTRACT
Wind turbine spacing is an important design parameter for wind farms. Placing turbines too close together reduces their
power extraction because of wake effects and increases maintenance costs because of unsteady loading. Conversely, placing
them further apart increases land and cabling costs, as well as electrical resistance losses. The asymptotic limit of very large
wind farms in which the flow conditions can be considered ‘fully developed’ provides a useful framework for studying
general trends in optimal layouts as a function of dimensionless cost parameters. Earlier analytical work by Meyers and
Meneveau (Wind Energy 15, 305–317 (2012)) revealed that in the limit of very large wind farms, the optimal turbine
spacing accounting for the turbine and land costs is significantly larger than the value found in typical existing wind farms.
Here, we generalize the analysis to include effects of cable and maintenance costs upon optimal wind turbine spacing in
very large wind farms under various economic criteria. For marginally profitable wind farms, minimum cost and maximum
profit turbine spacings coincide. Assuming linear-based and area-based costs that are representative of either offshore or
onshore sites we obtain for very large wind farms spacings that tend to be appreciably greater than occurring in actual farms
confirming earlier results but now including cabling costs. However, we show later that if wind farms are highly profitable
then optimization of the profit per unit area leads to tighter optimal spacings than would be implied by cost minimization.
In addition, we investigate the influence of the type of wind farm layout. © 2016 The Authors. Wind Energy Published by
John Wiley & Sons, Ltd.
KEYWORDS
wind farm; engineering economics; fluid dynamic models; coupled wake boundary layer model; optimal turbine spacing; wind farm
design; turbine wakes; renewable energy
Correspondence
Richard J. A. M. Stevens, Department of Physics, Mesa+ Institute and J. M. Burgers Centre for Fluid Dynamics, University of
Twente, 7500 AE Enschede, The Netherlands.
E-mail: r.j.a.m.stevens@utwente.nl
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which
permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no
modifications or adaptations are made.
Received 18 February 2016; Revised 24 June 2016; Accepted 6 July 2016
1. INTRODUCTION
Wind farms are becoming increasingly larger. For instance, the Alta and Roscoe wind farms in Texas have approximately
600 turbines. Therefore, it is important to better understand the influence of wake effects on the optimal turbine spacing in
wind farms with hundreds or thousands of turbines.
1
For smaller wind farms, the majority of the turbines can be placed such
that wake effects are rather limited. For the design of such wind farms, the industry uses site-specific, detailed optimization
© 2016 The Authors.
Wind Energy
Published by John Wiley & Sons, Ltd.
465
Combining economic and fluid dynamic wind farm models R. J. A. M. Stevens
et al
.
calculations for wind turbine placement based on wake type models.
2–5
Such calculations aim to place the turbines such
that wake effects are limited with respect to the main incoming wind direction at the site under consideration. There are
also academic studies that use wake models to optimize the placement of a limited number of turbines on a given land
area using Monte Carlo simulations,
6
genetic algorithms
7
or evolutionary algorithms.
8, 9
For an overview on this, we refer
to the review by Herbert–Acero et al.
10
The typical wind turbine spacing that is used in actual wind farms nowadays is
6 10D,whereD is the turbine diameter.
It is known that these wake type models do not explicitly capture the effect of the interaction between the atmosphere and
very large wind farms.
11–15
These effects are better described by top-down type models, which are based on momentum
analysis and horizontal averaging. These models give a vertical profile of the average velocity profile by assuming the
existence of two logarithmic velocity regions, one above the turbine hub-height and the other below.
16–23
The mean velocity
at hub-height is used to predict the wind farm performance as function of wind farm design parameters. The top-down
model approach allows one to analytically calculate the wind farm performance for very large wind farms. Meyers and
Meneveau
24
used this approach to analyze the optimal spacing in the limit of very large wind farms by accounting for the
turbine and land costs and found an optimal spacing of 12 15D, which is significantly larger than the value found in
actual wind farms. Later work by Stevens
23
showed that the predicted optimal spacing is much closer to the values found
in actual wind farms when the smaller number of turbines (and thus smaller contribution of wake effects) in these farms
are taken into account.
The purpose of this paper is to generalize the Meyers and Meneveau
24
analysis for very large farms by including effects
of cable and maintenance costs in addition to land and turbine costs upon optimal wind turbine spacing under various
economic criteria. Figure 1 shows in schematic fashion the general trade-offs that we analyze. The horizontal axis is the
spacing between turbines while the vertical axis expresses the levelized cost per MWh of energy production, defined as
the present worth (at an assumed interest rate) over the wind farm lifetime of the farm’s capital as well as operations
and maintenance (O&M) costs, divided by the present worth of energy (MWh) produced over the farm’s life. The cost
is normalized so that 1.0 is the hypothetical levelized cost of a single wind turbine whose output is unaffected by wakes
from other turbines and for which there are no capital costs beyond the turbine and its associated structure. A real wind
farm with multiple turbines will have a levelized cost greater than this level because there will be some wake effects
(decreasing the denominator) as well as additional costs for land acquisition (perhaps in the form of offshore leases), cables
between turbines and associated electrical resistance losses, and roads (in the case of onshore wind farms). The downward
sloping curve (turbine capital and O&M cost) shows how wakes affect the levelized turbine, structure and O&M costs; at
a small spacing, wake effects significantly shrink the MWh production (and thus the denominator of levelized costs). That
curve approaches the 1.0 asymptote for large spacings. Meanwhile, the upward sloping curve (quadratic and linear cost)
represents the levelized cost of land, cables and roads, which approaches zero at the origin and is convex (bending upwards)
if land costs are important, because they are a quadratic function of spacing. The sum of these two cost curves gives the
overall levelized cost of power production, which will be U-shaped and will have a minimum at the cost-optimal spacing.
The spacing that minimizes the cost of wind depends on shapes and levels of both the capital and O&M cost curve and the
quadratic/linear cost curve.
In this paper, we use the coupled wake boundary layer (CWBL) model of Stevens et al.
13, 14, 25
to account for wake
effects in the capital/O&M cost curve, and then combine it with an improved methodology for estimating the cost curve
for very large farms. Because the CWBL model is an analytical model, it allows a very fast evaluation of the wind farms’
performance. Important benefits of the CWBL model compared with the Calaf et al.
21
model are (i) that the CWBL model
Figure 1. Schematic of analysis of optimal spacing in a large wind farms, showing trade-off between wake losses (reflected in
higher capital and O&M costs for lower values of turbine spacing) and land, cable and other spacing-related expenditures (reflected
in increased ‘quadratic plus linear costs’ as a function of spacing).
Wind Energ.
2017; 20:465–477 © 2016 The Authors.
Wind Energy
Published by John Wiley & Sons, Ltd.
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DOI: 10.1002/we
R. J. A. M. Stevens
et al
. Combining economic and fluid dynamic wind farm models
is able to predict the effect of the turbine layout on the performance of very large wind farms and (ii) that the CWBL
model captures the entrance effects. Thus, the use of the CWBL model allows us to analyze the optimal spacing of wind
farms in more detail than the Calaf et al.
21
model. This CWBL approach can be used for wind farms of varying size. As
wake effects on the optimal spacing are most pronounced in very large fully developed wind farms, we focus on that case
here. Apriori, it is difficult to specify a very sharp definition of the ‘fully developed’ regime of large wind farms because
it depends on case-by-case arrangements. For the purpose of mean power optimization as carried out in this paper, we take
the view that fully developed means that the turbine power becomes nearly constant with downstream distance. Common
experience such as in Horns Rev (80 turbines) shows that at the end of the wind farm turbines is exposed to fully developed
wind conditions. Therefore, we expect fully developed wind conditions to become relevant for wind farms with several
hundred turbines.
The improved costing methodology used in this paper considers cost categories that are ‘quadratic’ (e.g., area-related
costs arising from leasing or occupying land) and ‘linear’ (e.g., the expense of inter-turbine cabling, electrical resistance
losses and roads) related to the inter-turbine spacing. Based on an order of magnitude estimation of the quadratic and lin-
ear costs, we argue in this paper that linear costs are significant and are especially important for offshore wind farms. In
addition, to account for the effect of both quadratic and linear cost components in this paper, we also present versions
of the model that address other issues not considered by Meyers and Meneveau.
24
One version accounts for the effect of
spacing-affected turbulence upon cost-optimal spacing, because closely spaced turbines are likely to experience increased
turbulence-related fatigue and damage. Another version recognizes that areas that are particularly profitable for wind devel-
opment can be limited, implying that the objective of design is not to minimize cost of production but to maximize profit
per unit area.
In Section 2, we start with the definition of the quadratic and linear cost components and the dimensionless cost param-
eters that are important for very large wind farms. In Section 3, we introduce the CWBL model
13, 25
that will be used to
estimate the power and turbulence intensity for different wind farm designs. In Section 4, the modeling approach intro-
duced by Meyers and Meneveau
24
is extended to include linear costs (e.g., the costs of cables, roads and resistance losses
that scale linearly with the distance between turbines) while in Section 5 the optimal turbine spacing is determined by opti-
mizing the profit per unit area of the wind farms instead of the normalized power per unit cost. In Section 6, the effect of
the maintenance costs, based on the predicted turbulence intensity by the CWBL model, is evaluated. Subsequently, we
will use the CWBL model to estimate the effect of the wind farms layout on the optimal turbine spacing in very large wind
farms in Section 7. Conclusions are given in Section 8.
2. DEFINITION OF DIMENSIONLESS PARAMETERS
In this paper, we will study the effect of different cost factors on the optimal turbine distance S D sD,whereD is the turbine
diameter, and s the dimensionless turbine distance that will be used in the remainder of this paper. For convenience, we
introduce the following dimensionless parameters to analyze the effect of the main economic influences on optimal turbine
spacing:
D
Cost
quadratic
Cost
turbine
=D
2
, ˇ D
Cost
linear
Cost
turbine
=D
, D
Rev
1
Cost
turbine
, D
Main
1
Cost
turbine
(1)
where all the cost factors have been normalized with respect to the turbine costs and the turbine diameter when appro-
priate. Cost
turbine
is defined as the present worth (at some discount rate) of the capital and O&M cost of a single turbine,
excluding all costs that depend on distance between turbines, such as cabling, land leasing and roads. Cost
quadratic
and
Cost
linear
are, respectively, the present worth of area and length dependent capital and other costs that increase with turbine
spacing. Quadratic costs include for example costs arising from leasing or occupying land while linear costs include items
such as inter-turbine cabling, electrical resistance losses and roads. ‘Rev
1
’ is the present worth of expected revenue of a sin-
gle wind turbine that does not experience wake effects. ‘Main
1
’ are the maintenance costs over lifetime, again for a single
turbine without wake effects, the estimate should indicate for which turbulence intensity this estimate was obtained, such
Ta b l e I . Reference estimates of reasonable dimensionless cost parame-
ters for wind farms, see details in the Appendix.
(quadratic) ˇ (linear) (revenue) (maintenance)
Onshore 1 10
3
5 10
3
1. 5 0 . 1
Offshore 2 10
5
1 10
2
1. 5 0 . 1
The cost parameters (quadratic), ˇ (linear), (revenue) and (mainte-
nance) are normalized with the turbine costs, see equation (1).
Wind Energ.
2017; 20:465–477 © 2016 The Authors.
Wind Energy
Published by John Wiley & Sons, Ltd.
DOI: 10.1002/we
467
Combining economic and fluid dynamic wind farm models R. J. A. M. Stevens
et al
.
that this can be accounted for in the calculations. Lifetime revenue is defined as being net of the expense of interconnecting
the farm to the main power grid, because that cost is not included in Cost
turbine
.
In Sections 4–7, we assess the optimal spacing and profitability for wide ranges of the linear (ˇ) and quadratic () turbine
cost parameters, in acknowledgement of the large uncertainty in their values for particular locations. In the Appendix, we
present some order of magnitude estimates for the dimensionless cost parameters for onshore and offshore wind farms,
see Table I, to direct the reader’s attention to more practically relevant regions of the parameter space. The parameter is
meant to include all sources of revenues, including tax incentives and renewable energy credits. The precise value is not
based on actual experience but upon an expectation that constructed wind farms will be profitable. Therefore, we assume
the same for both onshore and offshore facilities under the assumption that public policy and market conditions are such
that either are somewhat but not greatly profitable for development. Thus, although costs are higher for offshore facilities,
we anticipate that revenues will also be higher, so that the ratio of revenues to costs for financially viable wind farms will
somewhat but not greatly above 1.0. For instance, offshore farms are given more subsidies in the UK than onshore farms.
26
To assure the aforementioned assumption of wind farm profitability, we have selected D 1.5 for the onshore and offshore
reference cases.
3. INPUT PREDICTIONS FROM THE CWBL MODEL
In the CWBL model, the top-down approach by Calaf et al.
21
is used to calculate the performance of the wind turbines in
the fully developed region of the wind farm. In the CWBL model, the ratio of the mean velocity to the reference incoming
velocity at hub-height in the fully develop region is
h
ui.z
h
/
hu
0
i.z
h
/
D
ln
ı
H
=z
0,lo
ln
ı
H
=z
0,hi
ln
"
z
h
z
0,hi
1 C
D
2z
h
b
#
ln
z
h
z
0,lo
1
(2)
Here, ı
H
indicates the height of the atmospheric boundary layer in the fully developed region of the wind farm, b D
w
=.1 C
w
/,
w
28
p
C
T
=.8w
f
s
x
s
y
/, z
0,hi
denotes the roughness length of the wind farm, which is evaluated in
the model according to z
0,hi
D z
h
.1 C R
/
b
exp
Œq ClnŒ.z
h
=z
0,lo
/.1 R
/
b
/
2
1=2
,whereR
D D=.2z
h
/, q D
C
T
=.8w
f
s
x
s
y
2
/, hu
0
i.z
h
/ D u
= ln
z
h
=z
0,lo
and w
f
indicates the effective wake area coverage, which is obtained from
the two-way coupling with the wake model part of the CWBL model, and is equal to the ratio of wake area divided by
total area.
25
Because of the two-way coupling between the wake and the top-down model, w
f
depends on parameters such
as the streamwise distance between the turbines s
x
, the relative positioning of the turbines and the wake coefficient in the
fully developed region of the wind farm k
w,1
. As we focus on the optimal spacing for very large wind farms, we assume
that the average power output of the wind turbine is the same as the turbine power output in the fully developed region
of the wind farms. The power ratio P
1
=P
1
is given by the ratio of cubed mean velocity at hub-height with wind turbines
compared with the reference case without wind farms:
P
1
P
1
.s
x
, s
y
, layout, .../ D
h
ui.z
h
/
hu
0
i.z
h
/
3
(3)
Figure 2. The velocity field, obtained with the CWBL model, in the fully developed part of a very large wind farm with a spacing of
s
D 10.5 with an (a) aligned, (b) staggered and (c) full wake coverage limit layout. The black lines indicate the turbine positions. The
wake coverage area w
f
is defined as the percentage of the area in the fully developed regime of the wind farm where
u
< 0.95
u
0
.
25
Here, w
f
D 0.35 (aligned), w
f
D 0.60 (staggered) and w
f
D 1.00 (full wake coverage limit).
Wind Energ.
2017; 20:465–477 © 2016 The Authors.
Wind Energy
Published by John Wiley & Sons, Ltd.
468
DOI: 10.1002/we
R. J. A. M. Stevens
et al
. Combining economic and fluid dynamic wind farm models
In earlier work, we showed
13, 25
that the CWBL model gives improved predictions for the power output in the fully devel-
oped region compared with the top-down model introduced by Calaf et al.
21
or stand-alone wake models.
3
In addition,
the CWBL model is able to predict the difference between the performance of different wind farms geometries. Here, we
specifically focus on aligned, staggered and ‘full wake coverage limit’ layouts. Figure 2 shows a visualization of these dif-
ferent wind farm layouts. The full wake coverage limit is defined as a wind farm configuration for which the wake coverage
area w
f
D 1 in the CWBL model. In the full wake coverage limit layout, the performance of the wind farm is better than
for aligned or staggered wind farms, because the distance between directly upstream turbines is larger, by placing turbines
such that turbines on the next downstream row are just outside the wake. For s < 7, the staggered wind farm layout still
has a wake area coverage w
f
D 1, and therefore, the full wake coverage limit only outperforms the staggered layout for
s > 7. This effects is discussed in more detail in the work of Stevens et al.
27
We note that most calculations presented in
this paper, except in Section 7 where the effect of the wind farms layout is discussed, are based on the power estimates for
the full wake coverage limit layout.
4. OPTIMAL TURBINE SPACING OF VERY LARGE WIND FARMS WITH
CABLE COST
In order to investigate the robustness of the results from Meyers and Meneveau,
24
we include, in addition to the quadratic
(land) costs, a linear cost component defining the total cost per turbine as
Cost D Cost
turbine
C .sD/Cost
linear
C .sD/
2
Cost
quadratic
(4)
Following the approach by Meyers and Meneveau,
24
this leads to the following normalized power per unit cost
P
.s
x
, s
y
, layout, .../ D
P
1
Cost
D
P
1
Cost
turbine
C .sD/Cost
linear
C .sD/
2
Cost
quadratic
D
P
1
Cost
turbine
P
1
P
1
1
1 C ˇs C s
2
(5)
where P
1
D P
1
.s
x
, s
y
, layout, .../ is the average power production over the life of a typical turbine in a farm with
the assumed spacing and layout. P
1
is the average power production over the life of a turbine if it was lone-standing
without a wind farm. The ratio (P
1
=P
1
) thus contains the wind farm related power reductions because of wake and layout
effects. Costs are levelized to present worth. Thus, the ratio (P
1
=Cost
turbine
) is the reciprocal levelized cost of power for an
individual wind turbine, while P
.s
x
, s
y
, layout/ is the reciprocal levelized cost of power for a wind turbine in a wind farm.
First, in order to explore equation (5), we maximize it for the simple case of a very large fully developed wind farm.
We use equation (3) to determine the ratio P
1
=P
1
and replace it in equation (5) and find s that maximizes P
. Results are
shown in Figure 3(b). In agreement with the results of Meyers and Meneveau,
24
Figure 3(b) shows that without linear costs
(ˇ approaching zero) the optimal turbine spacing strongly depends on the quadratic costs. In addition, Figure 3(b) reveals
that with increasing linear costs, the predicted optimal spacing becomes smaller.
As can be seen, for offshore wind farms, it is the linear costs that determine the value of the optimal distance, while
for onshore farms, quadratic costs become equally or more important. This is indicated in Figure 3(b), where the reference
Figure 3. (a) The predicted power output in the fully developed region of a wind farms compared with a free standing wind turbine
(
P
1
=
P
1
) using equation (3) as function of the geometric mean turbine spacing
s
D
p
s
x
s
y
for
s
x
=
s
y
D 1. (b) Optimal spacing
as function of ‘quadratic costs’ and ‘linear costs’ ˇ , see equation (1). The circle and square indicates the reference offshore and
onshore case, respectively.
Wind Energ.
2017; 20:465–477 © 2016 The Authors.
Wind Energy
Published by John Wiley & Sons, Ltd.
DOI: 10.1002/we
469