1
Optimal control of energy extraction in
wind-farm boundary layers
Jay P. Goit and Johan Meyers †
Department of Mechanical Engineering, KU Leuven,
Celestijnenlaan 300A, B3001 Leuven, Belgium
(Received 21 May 2014; Revised 28 November 2014; Accepted 27 January 2015)
In very large wind farms the vertical interaction with the atmospheric boundary layer
plays an important role, i.e. the total energy extraction is governed by the vertical trans-
port of kinetic energy from higher regions in the boundary layer towards the turbine
level. In the current study, we investigate optimal control of wind-farm boundary lay-
ers, considering the individual wind turbines as flow actuators, whose energy extraction
can be dynamically regulated in time so as to optimally influence the flow field and the
vertical energy transport. To this end, we use Large-Eddy Simulations (LES) of a fully-
developed pressure-driven wind-farm boundary layer in a receding-horizon optimal con-
trol framework. For the optimization of the wind-turbine controls, a conjugate-gradient
optimization method is used in combination with adjoint LESs for the determination of
the gradients of the cost functional. In a first control study, wind-farm energy extraction
is optimized in an aligned wind farm. Results are accumulated over one hour of operation.
We find that the energy extraction is increased by 16% compared to the uncontrolled
reference. This is directly related to an increase of the vertical fluxes of energy towards
the wind turbines, and vertical shear stresses increase considerably. A further analysis,
decomposing total stresses in dispersive and Reynolds stresses, shows that the dispersive
stresses increase drastically, and that the Reynolds stresses decrease on average, but in-
crease in the wake region, leading to better wake recovery. We further observe that also
turbulent dissipation levels in the boundary layer increase, and overall the outer layer of
the boundary layer enters into a transient decelerating regime, while the inner layer and
the turbine region attain a new statistically steady equilibrium within approximately one
wind-farm through-flow time. Two additional optimal-control cases study penalization
of turbulent dissipation. For the current wind-farm geometry, it is found that the ratio
between wind-farm energy extraction and turbulent boundary-layer dissipation remains
roughly around 70%, but can be slightly increased with a few percent by penalizing
the dissipation in the optimization objective. For a pressure-driven boundary layer in
equilibrium, we estimate that such a shift can lead to an increase in wind-farm energy
extraction of 6%.
1. Introduction
In large wind farms, the effect of turbine wakes, and the accumulated local energy
extraction from the atmospheric boundary layer leads to a reduction in farm efficiency,
with power generated by turbines in a farm being lower than that of a lone-standing
turbine by up to 50% (see, e.g., Barthelmie et al. 2010; Hansen et al. 2011 for detailed
measurements in the Horn’s Rev wind farm). In very large wind farms or ‘deep arrays’,
the interaction of the wind farms with the planetary boundary layer plays a dominant
† Email address for correspondence: johan.meyers@kuleuven.be
Preprint; published in Journal of Fluid Mechanics (2015), vol. 768, pp.5-50
doi: 10.1017/jfm.2015.70
2 J. P. Goit and J. Meyers
role in this efficiency loss. For such cases, Calaf, Meneveau & Meyers (2010), and Cal
et al. (2010) demonstrated that the wind-farm energy extraction is dominated by the
vertical turbulent transport of kinetic energy from higher regions in the boundary layer
towards the turbine level. Later this was further corroborated in a series of studies, both
relying on simulations (see, e.g., Lu & Port´e-Agel 2011; Yang et al. 2012; Meyers &
Meneveau 2013; Abkar & Port´e-Agel 2013; Andersen et al. 2013), as well as on wind-
tunnel experiments (cf. Markfort et al. 2012; Lebron et al. 2012; Newman et al. 2013).
In the current study, we investigate the use of optimal control techniques combined
with Large-Eddy Simulations (LES) of wind-farm boundary layer interactions for the
increase of total energy extraction in very large ‘infinite’ wind farms. We consider the
individual wind turbines as flow actuators, whose energy extraction can be dynamically
regulated in time so as to optimally influence the flow field and the vertical turbulent
energy transport, maximizing the wind farm power. Note that in contrast to control of
conventional boundary layers with the aim of drag reduction (see, e.g., Bewley et al. 2001;
Kim 2003), wind-farm boundary-layer control does not benefit from large reductions
of turbulence levels or relaminarization of the boundary layer, as the turbulent fluxes
provide an important physical mechanism for transport of energy towards the turbines
and the wake regions.
In practice, modern wind turbines are controlled by the generator torque, and by a
blade-pitch controller, which controls the aerodynamic torque by changing the angle of
attack of the turbine blades. Above rated wind speed, these controls are used to limit
the aerodynamic power extraction to the maximum generator power. Increasing power
extraction in this operating regime is not relevant, as it is limited by design constraints.
Below rated wind speed, generator-torque control is used to keep the turbine at an op-
timal tip-speed ratio, while the blade pitch is kept at its optimal design value. For a
lone-standing turbine, such an approach is aerodynamically optimal, and maximizes en-
ergy extraction (Burton et al. 2001; Manwell et al. 2002; Pao & Johnson 2009). However,
for wind turbines in a wind farm, this is not necessarily the case.
A lot of studies have considered optimization of wind-farm performance, many of them
focussing on optimization of farm lay-out both in small farms (e.g., Kaminsky et al.
1987; Kusiak & Song 2010; Chowdhury et al. 2012), and in large arrays (e.g., Newman
1976; Meyers & Meneveau 2012; Stevens 2014). Also farm control has received consid-
erable attention, focussing on various aspects of wind-farm operation such as reduction
of structural loads, power regulation and grid support, or increasing energy extraction
(Spruce 1993; Hansen et al. 2006; Johnson & Thomas 2009; Soleimanzadeh et al. 2012;
Fleming et al. 2013). However, as far as wind-farm–flow interactions are included in
these studies, they are all based on fast heuristic models: e.g., models for wake inter-
action and merging such as presented by Lissaman (1979), Jensen (1983) or Rathmann
et al. (2007) (see also Sanderse et al. 2011, for a review), or models for boundary-layer
response in large farms (e.g. Frandsen 1992; Calaf et al. 2010; Meneveau 2012). In the
current work, we consider the optimal control of wind farms using large-eddy simulations
of the wind-farm boundary layer as the state model, allowing for a detailed optimiza-
tion of the dynamic interaction of the farm’s turbines with the boundary layer and its
large-scale three-dimensional turbulent structures.
To date, the combination of optimal control techniques with time-resolved turbulent
flow simulations such as DNS (Direct Numerical Simulation) or LES has b een mainly
used for drag reduction in boundary layers (e.g. Bewley et al. 2001; Chang & Collis 1999,
among others), noise control in jets (Wei & Freund 2006), control of wakes (Li et al.
2003), or optimal nonlinear growth of mixing layers (Delport et al. 2009, 2011; Badreddine
et al. 2014). All of these cases are PDE-constrained optimization problems with a large
Optimal control of wind-farm boundary layers 3
number of degrees of freedom in control space, and a huge number in the state space.
For instance, in the current study, the number of degrees of freedom in control space is
approximately twenty thousand, while the space-time state space has about 1.5 billion
degrees of freedom. In such a case, the only viable optimization approach is gradient-
based optimization in combination with an adjoint-based gradient metho d. In the current
work, we follow the approach by Bewley et al. (2001), and combine the non-linear Polak-
Ribi`ere conjugate-gradient variant, with the Brent line search algorithm (Press et al. 1996;
Luenberger 2005; Nocedal & Wright 2006). Moreover, similar to Bewley et al. (2001),
we use optimal model-predictive control (or optimal receding horizon control), where the
model simply consists of the full LES equations. Such an optimal control framework is
not practicable for real wind-farm control, as the LES model is by orders of magnitude
too expensive for real-time control, further complicated by the need of a state estimation
based on a limited number of wind measurements in the wind farm. However, the current
work can serve as a benchmark framework for controller development. Results also yield
new insights into limits to energy extraction from wind-farm boundary layers, and the
related structure of the turbulent energy fluxes towards individual turbines.
In large-eddy simulations of wind-farm boundary layers, it is computationally not
feasible to fully resolve blades and blade boundary layers on the mesh. Instead, simplified
models are used that provide the turbine forces on the LES flow field. Most common is
the Actuator Disk Model (ADM), in which a uniform force in the turbine disk region is
smoothed onto the LES grid (Mikkelsen 2003; Jimenez et al. 2007; Ivanell et al. 2009;
Meyers & Meneveau 2010; Calaf et al. 2010). We employ such an ADM model, with a disk-
based thrust coefficient that is dynamically controlled per turbine in time, representing
possible turbine pitch and torque control actions (cf. further discussion in manuscript).
Furthermore, for the wind-farm boundary layer, only cases with neutral stratification are
considered, and two important simplifications are made with respect to its representation.
Firstly, following many of the studies in the field (e.g. Lu & Port´e-Agel 2011; Yang et al.
2012; Meyers & Meneveau 2013; Abkar & Port´e-Agel 2013; Andersen et al. 2013), we
consider the asymptotic limit of a very large ‘infinite’ wind farm in a fully-developed
wind-farm boundary layer, allowing the use of periodic boundary conditions, and fast
pseudo-spectral discretization methods. As, e.g., verified in wind-tunnel experiments by
Chamorro & Port´e-Agel (2011), such a regime becomes relevant for wind farms with
horizontal extents that exceed 10–20 times the height of the boundary layer. Secondly, we
consider a neutral pressure-driven boundary layer with symmetry conditions at the top,
instead of a full conventionally neutral Atmospheric Boundary Layer (ABL) that is driven
by a geostrophic balance, includes free-atmosphere stratification, a capping inversion
layer between the ABL and the free atmosphere, etc. Such an approach is relatively
common for simulations of near-surface features in atmospheric boundary layers, and
has been used also in the context of wind-farm simulations by Calaf et al. (2010, 2011);
Yang et al. (2012); Andersen et al. (2013). It presumes that the turbines are situated in
the inner layer of the boundary layer (cf. Calaf et al. 2010 for a more detailed discussion).
This working hypothesis is limited by the fact that our turbines are close to the upper
limit of the inner layer, i.e. the hub-height is 100 meter, while the top tip-height is 150
meter, for a boundary layer height of 1km. Nevertheless, we believe that such an approach
is a good approximation for a first analysis of the optimal control of wind-farm boundary
layers, and results are carefully discussed in view of differences to a real ABL.
The paper is further organized as follows. In Section 2 LES of wind-farm boundary
layers is introduced, and a number of ‘uncontrolled’ reference cases are presented. Sub-
sequently, the optimal-control methodology is discussed in Section 3. In Section 4, we
present results for a first optimal control case. Two additional optimal control cases
4 J. P. Goit and J. Meyers
Figure 1: Computational domain Ω and boundaries Γ
that include penalization of turbulent dissipation are presented in Section 5. Finally, in
Section 6 main conclusions and future research directions are discussed.
2. Large-eddy simulation of a wind-farm boundary layer
2.1. Governing equations and boundary conditions
We consider a thermally neutral pressure-driven boundary layer, with constant pressure
gradient ∇p
∞
/ρ ≡ f
∞
e
1
(with e
1
the unit vector in the stream-wise direction). The
governing equations are the filtered incompressible Navier-Stokes equations for neutral
flows and the continuity equation, i.e.,
∇ ·
u = 0 (2.1)
∂
u
∂t
+
u · ∇
u = −
1
ρ
∇p + f
∞
e
1
+ ∇ · τ
M
+ f (2.2)
where
u = [u
1
, u
2
, u
3
] is the resolved velocity field, p the remaining pressure field (after
subtracting p
∞
), τ
M
is the subgrid-scale model, and where the density ρ is assumed to
remain constant. Furthermore, f represents the forces (per unit mass) introduced by the
turbines on the flow (see discussion in §2.2). Since the Reynolds number in atmospheric
boundary layers is very high, we neglect the resolved effects of viscous stresses in the
LES.
The computational domain is schematically represented in figure 1. In stream-wise and
span-wise directions, periodic boundary conditions are used (i.e. respectively on Γ
1
and
Γ
2
). At the top boundary (Γ
+
3
) symmetry conditions are imposed. At the ground sur-
face (Γ
−
3
), impermeability is imposed in combination with Schumann’s (1975) wall-stress
boundary conditions and Monin–Obukhov similarity theory for neutral rough boundary
layers. It relates the wall stress [τ
w1
, τ
w2
] to the wall-parallel velocity components [u
1
, u
2
]
at the first grid-point using (Moeng 1984; Bou-Zeid et al. 2005)
τ
w1
= −
κ
ln (z
1
/z
0
)
2
u
2
1
+ u
2
2
0.5
u
1
, (2.3)
τ
w2
= −
κ
ln (z
1
/z
0
)
2
u
2
1
+ u
2
2
0.5
u
2
, (2.4)
where z
0
is the surface roughness of the wall, and z
1
the vertical location of the first
grid point. Furthermore, the bar on u
1
and u
2
represents a local average obtained by
filtering the wall-parallel velo city [u
1
, u
2
] in directions parallel to the wall, avoiding an
overestimation of the wall stresses (Bou-Zeid et al. 2005). In the current work, we use
Optimal control of wind-farm boundary layers 5
two successive one-dimensional Gaussian filters with filter width 4∆ (and ∆ the grid
spacing).
In view of the complexity asso ciated with the formulation of the adjoint equations and
adjoint subgrid-scale model required for the optimal control (cf. §3), we opt for a relatively
simple subgrid-scale model, i.e. the Smagorinsky (1963) model with wall damping,
τ
M
= 2ℓ
2
(2S : S)
1/2
S, (2.5)
with S = (∇
u + (∇
u)
T
)/2 the resolved rate-of-strain tensor. The Smagorinsky length ℓ
(= C
s
∆ far from the wall) is damped using Mason & Thomson’s (1992) wall damping
function, i.e. ℓ
−n
= [C
s
∆]
−n
+ [κ(z + z
0
)]
−n
, with κ = 0.4 the Von K´arm´an constant,
and where we take n = 3. Furthermore, ∆ = (∆
1
∆
2
∆
3
)
1/3
is the local grid spacing, and
C
s
the Smagorinsky coefficient. We employ C
s
= 0.14, consistent with the high-Reynolds
number Lilly value for cubical sharp cut-off filters (Meyers & Sagaut 2006). Note that
some other works have used more advanced subgrid-scale models in LES of wind farms,
such as the scale-dependent Lagrangian model of Bou-Zeid et al. (2005) (Calaf et al. 2010,
2011; Abkar & Port´e-Agel 2013). In Calaf et al. (2010) a comparison was made between
this model and the current Smagorinsky implementation, and differences in mean velocity
and Reynolds stress distributions were found to be small.
2.2. Actuator-disk model
Actuator-disk models add the axial forces exerted by the wind turbines on the flow to
the Navier–Stokes equations. Tangential forces are usually neglected (given the tip-speed
ratios at which turbines are operated they are more than an order of magnitude smaller).
In a validation study, comparing ADM with and without tangential forces (Wu & Port´e-
Agel 2011), it was demonstrated that standard ADMs provide an accurate representation
of the overall wake structures behind turbines except for the very near wake (x/D < 3).
Moreover, also Reynolds stresses were found to be accurately predicted, thus yielding
a good representation of the interaction of the wind farms with the boundary layer.
Later this was further corroborated by Meyers & Meneveau (2013) in a detailed analysis
of energy fluxes in wind farms, comparing models with and without tangential forces.
In the current work, we employ a standard actuator-disk model. It corresponds to the
version used by Meyers & Meneveau (2010); Calaf et al. (2010); Meyers & Meneveau
(2013), and is briefly reviewed below.
The axial force of a turbine i (= 1 · · · N
t
) on the flow field can be expressed as
F
i
= −
1
2
C
′
T,i
ρ
V
2
i
A (2.6)
with C
′
T,i
the disk-based thrust coefficient,
V
i
the average axial flow velocity at the
turbine rotor disk (see further below), and A = πD
2
/4 the rotor-disk surface. The disk-
based thrust coefficient C
′
T,i
results from integrated lift and drag coefficients over the
turbine blades, taking design geometry and flow angles into account (cf. Appendix A for
a detailed formulation). For an ideal design, and in absence of any drag forces C
′
T,i
= 2.
Moreover, below rated wind speed, conventional turbines use generator-torque control to
keep the turbine at a constant optimal tip-speed ratio independent of wind speed, while
the blade pitch is kept constant at its optimal design value. In an ADM, this corresponds
to using a constant value for C
′
T,i
(see also Eq. A 5 in Appendix A).
In an ideal turbine design, the force F
i
is uniformly distributed over the disk area.
Therefore, in an actuator disk model, a uniform force (per unit mass) is distributed over
the LES grid cells in the vicinity of the actuator disk using (Meyers & Meneveau 2010;
![Figure 14: (Color online) (a–d) Contours of mean streamwise velocity field averaged over control windows and turbine elements; (a,b) in a horizontal plane through the hub; (c,d) in a vertical plane through the turbine. (e,f): Contours of ũ ′′ 1 ũ ′′ 3 in a vertical plane through the turbine (left)(a,c,e): uncontrolled case; (right)(b,d,f): the optimal control case averaged over time window [5TA, 20TA].](/figures/figure-14-color-online-a-d-contours-of-mean-streamwise-2rmzbkgu.webp)
![Figure 15: (Color online) Contours of Reynolds stresses averaged over control windows and turbine elements. (a,c,e,g,i): uncontrolled case; (b,d,f,h,j): the optimal control case averaged over time window [5TA, 20TA]. Figure (a) and (b) show a horizontal plane at the turbine-tip level; while (c–j) show xz–planes through the rotor center.](/figures/figure-15-color-online-contours-of-reynolds-stresses-e9bjd077.webp)
