Optimal turbine spacing in fully developed wind-farm
boundary layers
Johan Meyers
1
and Charles Meneveau
2
1
Department of Mechanical Engineering, Katholieke Universiteit Leuven,
Celestijnenlaan 300A — bus 2421, B3001 Leuven, Belgium
2
Department of Mechanical Engineering & Center for Environmental and
Applied Fluid Mechanics, Johns Hopkins University,
3400 North Charles Street, Baltimore MD 21218, USA
February 11, 2011
Abstract
As wind farms become larger, the asymptotic limit of the “fully developed”, or “infinite”,
wind farm has been receiving increased interest. This limit is relevant for wind farms on flat
terrain whose length exceeds the height of the atmospheric boundary layer by over an order of
magnitude. Recent computational studies based on Large Eddy Simulation have identified vari-
ous mean velocity equilibrium layers, and have led to parameterizations of the effective roughness
height that allow predicting the wind velocity at hub-height as function of parameters such as wind
turbine spacing and loading factors. In the current paper, we employ this as a tool to make predic-
tions of optimal wind turbine spacing as function of these parameters, as well as in terms of the
ratio of turbine costs to land-surface costs. For realistic cost ratios, we find that the optimal av-
erage turbine spacing may be considerably higher than conventionally used in current wind-farm
implementations.
Keywords: wind farm, wind energy, optimal wind turbine spacing, large-eddy simulation
1. INTRODUCTION
Recently, wind energy has received renewed interest. This originates in part from large funding pro-
grams by American and European governments, and comes from the realization that wind energy will
be an important contributor in the production of affordable and clean energy in the next decades. In
various scenarios,
1, 2
a contribution of wind energy to the overall electricity production up to 20%
is aimed at by 2030. To realize these targets, larger wind farms (both on- and off-shore), cover-
ing increasingly larger surface areas are required. When large-scale wind-farm implementations are
considered, the total drag induced by all turbines in the farm may change the equilibrium in the atmo-
spheric surface layer. In particular, with a characteristic height of the ABL of about 1 km, wind farms
with horizontal extents exceeding 10–20 km may therefore approach the asymptotic limit of “infi-
nite” wind farms, and the boundary layer flow may approach a new fully developed regime, which
depends on the additional surface drag induced by the wind farm. In the current study, we focus
1
Preprint version. The final version is published in Wind Energy 15, 305-317 (2012)
DOI: 10.1002/we.469
on this asymptotic “infinite” wind-farm regime, and investigate the optimal wind-turbine spacing in
these wind farms to either optimize the ratio of total power output per land surface, or the ratio of
total power output per unit of total cost that also includes cost of turbines. Depending on the ratio
between total costs per turbine and total costs per land surface, in the case of “infinite” wind farms,
we find that the optimal average turbine spacing may be considerably higher then conventionally used
in current wind-farm implementations.
Design and optimization of single wind turbines is well explored nowadays, often using blade-
element–momentum theory, and Glauert’s theory for rotor aerodynamics.
3, 4
Also effects of turbine
wake aerodynamics have received much attention.
5
Studies of the interaction of large wind farms and
the atmospheric boundary layer (ABL) are far less prevalent. In this area, pioneering work was per-
formed by Frandsen,
6
who formulated a model for the surface roughness induced by “infinite” wind
turbine arrays. More recently, the subject gained renewed interest in the context of off-shore wind-
farm under performance.
7
Very recently, studies employed large-eddy simulations to study wind-
farm–ABL interactions,
8, 9
focusing on the ‘infinite’ wind-farm limit. Moreover, in Ref. 8, Frandsen’s
model for the induced wind-farm surface roughness was refined, to include effects of turbine-wake
mixing.
When turbine spacing is considered in a more conventional approach, minimum wind-turbine
spacing in wind farms is mainly governed by the desire to limit wake-induced fatigue loads in turbines
located downstream of a prior row of turbines.
5
However, large wind farms increase the effective sur-
face roughness experienced by the ABL,
6, 8
such that the effective wind velocity at turbine-hub height
decreases compared to an unloaded ABL. Hence, increasing the installed power per land surface area
(i.e. decreasing the average wind-turbine spacing) has an inverse effect on the total extracted power
per turbine. Depending on the cost per turbine, and the cost of land used for wind farms, this leads to
an optimization problem for wind-turbine spacing in wind farms, where the optimal spacing is given
by economical constraints. In the current work the refined effective roughness model of Ref. 8 is used
as the basis to elaborate a model for overall wind-farm power output per land surface, taking fully
developed wind-farm–ABL interactions into account. A detailed discussion is presented on optimal
turbine spacing, and its dependence on economical parameter, and operating regimes.
Wind-turbine operation is often classified into three regions: region I–III.
4, 10
The first region is
at very low wind speeds where aerodynamic forces cannot overcome the turbine’s internal friction
losses. At very high wind speeds (Region III), the power output of turbines is restricted by load-
ing constraints on its mechanical structures and by economical constraints on the size of the power
generator. In this region, turbine power is controlled at a constant level, independent of wind speed,
either by stalling the turbine blades, or by feathering the turbine.
4
In region II, power output is not
restricted, and wind turbines work close to their aerodynamical optimal operating conditions. In the
current work, we focus in large part on optimization of turbine-spacing in region II, where turbine
thrust and power coefficients are close to optimal. At the end of Section 3, region III operation and
its influence on optimal turbine spacing in wind farms, is discussed. It will be argued that feathering
may have an impact on the optimal turbine spacing in the equilibrium wind-farm ABL, while stalling
the turbine keeps the optimal spacing at the region-II optimum.
The paper is further organized as follows. First, in Section 2, the model for wind-farm optimiza-
tion is elaborated. In Section 3, optimization results are presented and discussed. Finally, conclusions
are presented in Section 4.
2. MODEL FOR WIND-FARM OPTIMIZATION
First, some definitions and conventions for wind-turbine thrust and power, which will be further used
in the current study, are introduced in §2.1. Next, in §2.2, standard relations for the atmospheric
2
boundary layer are briefly reviewed. Subsequently, the induced surface-roughness model for wind
farms
8
is discussed in §2.3. Finally, in §2.4 the wind-farm optimization problem is defined in terms
of normalized farm power.
2.1. Definitions and conventions
In conventional wind-turbine momentum theory, the thrust of a single wind-turbine on the surrounding
flow is expressed as
F
T
= −
1
2
C
T
ρU
2
∞
A, (1)
with C
T
the thrust coefficient, U
∞
the upstream undisturbed flow velocity at hub height, and A =
πD
2
/4 the turbine-rotor area (with D the rotor diameter). However, for large wind-turbine arrays
with significant interactions among wind turbines and wakes, this reference velocity U
∞
is not readily
known and would require arbitrary decisions about what upstream distance to use when specifying the
velocity. Moreover, such a reference velocity would depend on farm parameters such as the average
turbine spacing, and turbine loading. Instead, for wind farms, it is useful to base the relations for
thrust on the prevailing axial velocity at the rotor-disk position, U
d
, such that
F
T
= −
1
2
C
′
T
ρU
2
d
A. (2)
Note that the value of C
′
T
is straightforwardly related to the lift and drag coefficients of the turbine
blades (see e.g. Ref. 9 for an elaboration), and much less sensitive to farm parameters such as average
turbine spacing. Moreover, in large-eddy simulations of wind farms,
8
U
d
is readily available during
the simulation, such that Eq. (2) can be directly employed as a force model.
For a lone-standing turbine, it is possible to relate C
′
T
to C
T
by using classic actuator disk theory.
This allows us to express
U
d
= U
∞
(1 − a), C
′
T
=
C
T
(1 − a)
2
, (3)
with a the axial induction factor.
9
For the Betz limit
4
(i.e., C
T
= 8/9, and a = 1/3), we obtain
C
′
T
= 2. Using typical values C
T
= 0.75, and a = 1/4 (which have been used before for modeling
wind turbines)
11
leads to C
′
T
= 4/3. Obviously, for wind farms, Eq. (3) is not valid, though the typical
values for C
′
T
remain applicable.
For wind-turbine farms, it is further useful to express the thrust in relation to the average land
surface area S per turbine (S = ℓ
x
ℓ
y
, with ℓ
x
, ℓ
y
the average turbine-spacing in stream-wise and
span-wise directions), leading to
F
T
=
1
2
c
′
ft
ρU
2
d
S, (4)
with a friction coefficient c
′
ft
based on the horizontal surface rather than frontal area. Further,
c
′
ft
=
πC
′
T
4s
x
s
y
=
πC
′
T
4s
2
, (5)
with s
x
= ℓ
x
/D, s
y
= ℓ
y
/D, and s =
√
s
x
s
y
.
The power extracted on average by wind turbines from the atmospheric boundary layer corre-
sponds to
P =
1
2
C
′
T
ρU
3
d
A =
1
2
c
′
ft
ρU
3
d
S. (6)
This is not equivalent to the power P
ax
on the turbine axis. The latter relates to the torque and rotational
velocity of the turbine. The drag forces on the turbine blades increase thrust, but reduce torque. From
3
an energetic point of view, the drag forces lead to losses, corresponding to a conversion of mean-flow
energy in the atmospheric boundary layer into turbulent motion and heat. Using the power coefficient
C
P
, and C
′
P
(respectively with respect to U
∞
, and U
d
), the power on the turbine axis corresponds to
P
ax
= P
C
′
P
C
′
T
= P
C
P
C
T
(1 − a)
(7)
Using actuator disk theory, it is straightforward to find that C
P
= C
′
P
(1 − a)
3
. For the Betz limit (i.e.,
C
P
= 16/27, and a = 1/3), C
′
P
= 2.0.
For wind turbines, typical optimal values may be C
P
≈ 0.34 and a ≈ 1/4, such that C
′
P
≈ 0.8, and
P
ax
≈ 0.6P. In reality, the ratio C
′
P
/C
′
T
depends on the turbine working region. In region II, C
′
P
/C
′
T
is
close to optimal, with high values for C
′
T
, and C
′
P
. Consequently, in this operating region, optimization
to P or P
ax
is roughly equivalent. In region III, the turbine’s power output is controlled to be constant.
Depending on the control mechanism, this may lead to a large decrease in C
′
P
/C
′
T
. Consequences of
region III operation on the optimization results in the current work are addressed, separately, at the
end of Section 3. Until then, we assume P ∼ P
ax
, and formulate the wind-farm–ABL optimization
problem in terms of P.
2.2. Geostrophic wind and ABL relations
In the current subsection, we briefly review classical relations for the atmospheric boundary layer, as,
e.g., well documented in Ref. 12.
In the atmospheric boundary layer (ABL), the driving force is the geostrophic wind, of velocity
magnitude G, on top of the ABL, which is given by geostrophic balance condition without the effects
of friction. Inside the boundary layer, a balance exists between pressure forces, Reynolds stresses,
and Coriolis forces induced by the Earth’s rotation. Since the velocity in the ABL decreases towards
the surface, Coriolis forces also decrease, which causes the velocity to turn away from the geostrophic
wind direction at lower altitudes, often referred to as the Ekman spiral. Conventionally, a reference
frame is selected which is aligned with the wind speed near the surface (in the inner layer of the
boundary layer). In this case, the geostrophic wind G is defined with two components, i.e. U
G
in
stream-wise, and V
G
in span-wise direction, such that G = (U
2
G
+ V
2
G
)
1/2
, and γ = arctan(−V
G
/U
G
)
the angle between the geostrophic wind direction, and the wind direction near the surface. Classical
similarity theory then leads to
12
U
G
u
∗
=
1
κ
ln
u
∗
f z
0
−C, (8)
V
G
u
∗
= −A (9)
with κ = 0.4 the Von Kármán constant, and where C ≈ 4.5, and A ≈ 11.25 are found to be good
values.
6
Further, z
0
is the surface roughness. In the context of wind-farms, this is related to total
roughness induced by the ground surface and the wind turbines on the ABL. Likewise, u
∗
is the
friction velocity, which is related to the total friction exerted by the ground and wind turbines on the
boundary layer. Further details on z
0
, and u
∗
, and their relation to the wind-farm parametrization, etc.,
are provided in §2.3. Finally,
f = 2Ω sin ϕ (10)
is the Coriolis parameter. For Ω = 2π/(24 ×3600 s) = 7.27 ×10
−5
1/s, and, e.g., at 40 degree latitude,
we get f = 9.34 × 10
−5
1/s.
4
10
−3
10
−2
10
−1
0.4
0.5
0.6
0.7
0.8
0.9
1
z
0
/z
h
u/G
u
h
/G
V
G
/G
U
G
/G
Figure 1: Relation between geostrophic wind and wind speed at turbine hub height as function of the
surface roughness in the ABL (cf. Eqs.(8,9,11) with f = 9.34 ×10
−5
1/s, and z
h
= 100m). (—): u
h
/G;
(−−): U
G
/G; and (−·): V
G
/G. (N): Ro
h
= 1000; (•): Ro
h
= 2000; and (H): Ro
h
= 3000;
Combining Eq. (8), and (9), leads to
G
u
∗
=
A
2
+
1
κ
ln
u
∗
G
Ro
−C
2
, (11)
where the dimensionless group Ro = G/( f z
0
) has the form of a Rossby number, expressing a ratio
between inertia and Coriolis forces. In the current work, we are mainly interested in the reaction of
the ABL to changes in the surface roughness induced by wind turbines. Therefore, we introduce an
alternative Rossby number, using the turbine hub-height as reference length scale, such that
Ro
h
=
G
f z
h
= Ro
z
0
z
h
, (12)
and we will evaluate the effect of variations in z
0
/z
h
, while keeping Ro
h
constant. A representative
reference value for Ro
h
may, e.g., be estimated using f = 9.34 ×10
−5
1/s, G = 20m/s, and z
h
= 100m,
leading to Ro
h
≈ 2140.
Using expression (11), it is useful to investigate the relation between the geostrophic velocity G,
and u
h
, the mean streamwise velocity at turbine hub-height, which we estimate here using Monin-
Obukhov similarity under neutral stratification conditions (the log-law for rough walls). One can
write u
h
≈ u
∗
/κ ln(z
h
/z
0
), with z
h
the turbine-hub height. To this end, G/u
∗
is solved numerically from
Eq. (11), using MATLAB’s fsolve function. Alternatively, fits to the inverse function may be em-
ployed, as, e.g., proposed in Ref. 7 and further explored in the Appendix, where such an approximate
expression is given explicitly (since it involves errors on the order of 7% for G/u
∗
, here we continue
to use the numerical solution). In Figure 1, u
h
/G is displayed, together with the separate geostrophic
components U
G
/G, and V
G
/G as function of the surface roughness z
0
(with z
0
covering a range be-
tween 0.1m and 10m – as may be encountered in large wind farms
8
– normalized by z
h
= 100). In
the figure, three different values of Ro
h
are displayed, i.e. Ro
h
= 1000, Ro
h
= 2000, and Ro
h
= 3000.
5
