Introduction
Wake interactions between turbines within a wind farm cause reduced power
extraction and increased turbine loading in downstream rows. The current
control paradigm in such farms optimizes performance at the wind turbine
level and does not account for these interactions, resulting in sub-optimal
wind-farm efficiency. In contrast, control strategies at the farm level allow
the wake interaction to be influenced and promise to improve overall wind-farm
performance by improving wind conditions for downstream turbines. This can be
achieved by redirecting propagating wakes (yaw control; see, e.g.,
) or by
affecting the induced wake velocity deficits (axial induction control; see,
e.g., ).
A more exhaustive survey of wind-farm control in a broader context can be
found in and .
In contrast to the studies cited above that all focus on the static set point
optimization of wind farms, introduced a dynamic
induction control approach based on large-eddy simulations (LES) and adjoint
gradient optimization. In this study, individual turbines were used as
dynamic flow actuators that influence the wind-farm boundary layer flow in
such a way as to optimize aggregate wind-farm power extraction. The
methodology was applied to the asymptotic case of a fully developed
“infinite” aligned wind farm, and power gains of about 16 % were
quantified. Later, this approach was also used in induction control studies
of wind farms with entrance effects in and more
recently in , in which similar gains of the order of
15–20 % were achieved. It is important to note that the computational
cost of this LES-based dynamic induction control methodology is orders of
magnitude too high for direct implementation as a practical control strategy.
However, the methodology allows us to assess the theoretical potential for
wind-farm control, and increased understanding of the flow physics can lead
to simplified control strategies that can be applied in practice.
Recently, the methodology of was generalized to
include dynamic yaw control in . In this study,
induction control and yaw control were compared for a relatively small
aligned wind farm, and yaw control was found to yield higher power gains for
this setup. Furthermore, the high potential of combined induction and yaw
control was quantified, and analysis of the yaw control signals allowed for
the
identification of practical simplified dynamic yaw control strategies. The search for
similar practical control strategies for induction control has remained
unsuccessful to date.
The current paper presents efforts on understanding optimal control dynamics
observed in the optimal induction control simulations by
(further denoted as MM17). The outline of the
paper is as follows: first, Sect. discusses
the numerical setup and optimal control simulations of MM17 that will be
further analyzed in the current paper.
Section presents an analysis of the
control and thrust force dynamics and performs some numerical experiments to
elucidate the characteristics of the optimal controls. It will be shown that the
coherent behavior of turbines situated in the first row of the wind farm
stands out from their downstream counterparts. Thereafter,
Sect. identifies the shedding of vortex rings
from the first row based on a flow visualization. Further, a simple
sinusoidal thrust control approach is presented that successfully mimics this
process with a robust increase in power extraction in the second
row. Next, Sect. shortly discusses the
behavior of the intermediate rows, i.e., turbines that have both upstream and
downstream neighbors, for which similar simple control strategies remain
elusive thus far. In conclusion, Sect. summarizes the
main findings of this paper.
Description of optimal control simulations in MM17
The current section describes the optimal control simulations performed by
MM17, the results of which are further analyzed in the current paper. First,
the methodology is introduced. Then, the numerical setup is detailed.
Afterwards, the optimization results on power extraction and time-averaged
flow field features are discussed.
Control methodology
A schematic overview of the wind-farm control methodology is shown in
Fig. . Figure a
illustrates the control block diagram: an iterative optimization loop updates
the wind-farm control vector φ(t) until a set of optimized
controls φ⚫(t) is found. This optimization is based
upon an unsteady turbulence-resolving LES wind-farm flow model, and
sensitivities of the cost functional J (i.e., the total wind-farm power
extraction) are calculated using an adjoint formulation of this flow model.
In this way, a priori simplifications to the turbulent boundary layer and
wake representation are avoided as much as possible, and the control signals
are designed in a such a way that turbines actively tap into the dynamics of
the turbulent flow. The optimization is performed using a receding-horizon
control framework, as illustrated in Fig. b. In
this framework, wind-farm controls φ(t) are optimized for a
finite time horizon T, involving a sequential set of LES and adjoint LES
simulations. Upon convergence of the optimization, optimized control signals
are applied in a flow advancement simulation for a time TA<T,
after which a new optimization window is initiated.
Schematic overview of wind-farm optimal control methodology from
MM17. (a) Control block diagram with adjoint gradient-based
optimization and LES flow models illustrating data flow of (optimal) controls
φ(⚫), system state q, cost functional
J, and its gradient ∇J. (b) Receding-horizon framework
subdividing time into discrete flow advancement windows of
length TA with prediction horizon T. Each arrow represents a
forward or adjoint LES. Every window consists of an optimization stage (blue
and red lines) followed by a flow advancement stage with optimal controls
φ⚫ (green lines).
Within each optimization window, the total wind-farm power is optimized by
solving the following optimization
problem constrained by partial differential equations.
minimizeφ,qJ(φ,q)=-∫0T∑i=1NtPidts.t.∂ũ∂t+ũ⋅∇ũ=-∇p̃+p̃∞/ρ-∇⋅τsgs+∑i=1NtfiinΩ×(0,T),∇⋅ũ=0inΩ×(0,T),τdC^T,i′dt=CT,i′-C^T,i′i=1…Ntin(0,T),0≤CT,i′≤CT,max′i=1…Ntin(0,T),
The cost functional that is optimized in Eq. () is the total
wind-farm energy extraction over time horizon T. The control variables are
the time-dependent thrust coefficient set points CT,i′ of every turbine
i (=1…Nt), i.e., φ=[CT,1′(t),…,CT,Nt′(t)], and the state variables are denoted as
q=[ũ(x,t);p̃(x,t);C^T,1′(t),…,C^T,Nt′(t)], with ũ the filtered
velocity, p̃ the filtered pressure, and C^T,i′ the
(time-filtered) thrust coefficient for turbine i (see below).
The filtered Navier–Stokes momentum and continuity state equations in
Eqs. ()–() are
solved using an in-house LES solver (see, e.g., for a detailed discussion of the solver).
The time-filtering state equation in Eq. () applies
a one-sided exponential time filter to the thrust coefficient set points
CT,i′ with a characteristic wind turbine response timescale τ
. Finally, the box constraints in
Eq. () limit the variations in the turbine thrust
coefficients to technically feasible values.
The forces exerted by turbine i on the boundary layer flow are parameterized
using a standard nonrotating actuator disk model as fi(x,t)=-(1/2)C^T,i′(t)Vi(t)2Ri(x)e⟂,i, where
Ri is a smoothed representation of the geometric footprint of
the turbine on the LES grid and e⟂,i is the
rotor-perpendicular vector. Further, the disk-averaged velocity is defined as
Vi=(1/Ai)∫ΩRi(x)ũ⋅e⟂,idx, with Ai the rotor disk area. Mechanical power
captured by the wind turbine is calculated as Pi=(1/2)CP,i′(t)Vi(t)3Ai, with CP,i′=0.9CT,i′, resulting from a fit of LES
results to momentum theory, eliminating the overprediction of wind turbine
power on typical wind-farm LES grid resolutions .
Case setup
The wind farm considered in MM17 has an aligned pattern of 12 rows by
6 columns. The wind turbines have a hub height zh=100 m with a rotor
diameter D=100 m and are spaced apart by 6D in both the axial and
transversal directions. The flow is simulated on a domain of 10×3.6×1 km3 discretized on a simulation grid of 384×192×144 grid points. A snapshot of the streamwise velocity field is shown in
Fig. . The wind farm was controlled over a total of
NA=15 time windows with a prediction horizon T=240 s
(i.e.,
the time it takes for the flow to pass four rows of turbines) and a flow
advancement time of TA=T/2=120 s, resulting in a total
control time Ttot=NATA=30 min.
Instantaneous streamwise velocity ũx for the 12×6 aligned wind farm considered in MM17. Black lines indicate wind turbine
locations. The black dashed line near the end of the domain indicates a
buffer region used for the imposition of inflow boundary conditions. Figure
originally published in under a CC-BY 4.0
license.
A conventionally (greedily) controlled wind farm with steady CT′=2 was
defined as a reference case. Note that this would correspond to a farm with
ideal turbines for which generator torque is being controlled dynamically to
track the maximum power point at the Betz limit perfectly. In a real turbine
controller this may, e.g., be achieved with the extremum-seeking control
proposed by . Several different optimal control cases
were defined based on the wind turbine response time τ=0, 5, or 30 s
(instantaneous, fast, or slow response; see Eq. )
and the maximal thrust coefficient CT,max′=2 or 3, with
thrust forces that can respectively only be reduced (underinductive) or
increased (overinductive) compared to the Betz optimum at CT′=2 (see
Eq. ). Cases are denoted as C<X>t<Y>, where X
and Y represent CT,max′ and τ, respectively, e.g., C3t30
for the case with CT,max′=3 and τ=30 s. The choice of
(and sensitivity to) setup parameters is further elaborated in MM17.
Simulation results
Figure illustrates the energy extraction of the
optimally controlled wind-farm cases normalized by the greedy reference
control case. Figure a shows that the adjoint LES-based
control approaches achieve energy gains ranging from a minor 2 % in the
most restrictive C2t30 case to over 20 % in the C3t0 case. From
Fig. b it can be seen that, for all cases except C3t30,
power is curtailed in the first row to a limited degree, whereas the
downstream rows compensate for this loss by extracting significantly more
energy. Furthermore, not taking into account the first row, the last row
achieves the highest energy extraction in every case, as it can act greedily
without compromising power extraction in downstream neighbors.
Energy extraction EO of optimally controlled wind-farm
cases from MM17 normalized by a greedy reference case ER.
(a) Total energy extraction. Error bars indicate confidence
intervals of ±2 standard deviations. (b) Energy extraction by
row normalized by first-row reference power. C3t0, red line; C3t5, yellow
line; C3t30, blue line; C2t0, red dashed line; C2t5, yellow dashed line;
C2t30, blue dashed line. Figure originally published in
under a CC-BY 4.0 license.
In the remainder of this section, time-averaged wind-farm flow properties
will be investigated. Here and throughout the remainder of this paper, we
focus on case C3t5, as it produces similar energy gains as the highest-yield
case C3t0 (see Fig. a), but achieves this with smoother
thrust coefficient signals. In the following discussion, the time-averaging
operation is denoted by an overline, and flow field variables are decomposed
into mean and fluctuating components as ũ=ũ‾+ũ′≡Ũ+ũ′.
Figure illustrates the time-averaged flow field
quantities of the reference case (left panels, a1–g1) and the differences
between the optimized C3t5 case and the reference case (right panels,
a2–g2). Simulation results are averaged over the different columns and are
shown as either top views at hub height (Fig. b,
f, g) or side views through a turbine column
(Fig. a, c, d, e).
Figure a and b illustrate side views and
top views of the axial velocity throughout the wind farm. It can be seen that
downstream turbines in the controlled case experience consistently higher
incoming velocities, which explains the increased energy extraction discussed
above. Furthermore, a larger drop in streamwise velocity over the turbine
disk can be observed, most notably in the first-row turbines, indicating
deeper wakes with enhanced recovery before hitting the next row of turbines.
Furthermore, it can be observed that the axial velocity in the flow above and
beside the wind turbine column is reduced, indicating that the mean flow
kinetic energy is depleted in these regions to the benefit of the flow
passing through the wind turbines.
Figure c shows side views of turbulence kinetic
energy k. The figure shows an increase in turbulence throughout the entire
wind farm, spreading to the internal boundary layer above the turbines. Note
specifically the sharp increase in turbulence in the core wake region behind
the first-row turbine, for which an enhanced recovery was found as discussed
above. The turbulence intensity TI≡(2k/3)1/2/U∞ at hub
height (not shown in the figure) is 10 % at the inlet for both the
reference case and the controlled case. The combination of reduced near-wake
mean velocities and increased velocity fluctuations in the controlled case
increase local TI in the turbine wakes (ranging from ≈2 % points in the wakes of the middle rows to ≈12 % points in
the first and last rows). This increase in turbulence intensity dissipates to
below a 1 % point difference at 10D downstream of the last row.
Figure d and e show side views through the rotor
centerline of top-down turbulence and mean flow transport of axial momentum,
i.e., -ũx′ũz′‾ and -ŨxŨz,
respectively. It can be seen that, although mean flow vertical transport is
virtually unaffected, the turbulence top-down transport of axial momentum is
increased significantly in the upper part of the wakes, indicating increased
turbulent mixing with the internal boundary layer above the wind-farm canopy.
The effect is somewhat more pronounced in the wake behind the first row of
turbines, for which a slight increase in the upwards transport of momentum
can also be observed in the lower part of the wake.
Figure f and g show plan views at hub height of
the transversal turbulence and mean flow transport of axial momentum, i.e.,
ũx′ũy′‾ and ŨxŨy,
respectively. The sign convention is such that positive values correspond to
transport in the positive y direction in the figure. A slight increase in
turbulent transversal transport towards the wake centerline can be observed
behind every row. The mean flow transversal momentum transport into the wake
region is increased significantly behind the first two turbine rows.
Downstream of these rows, the difference between these cases and the
reference case is far less coherent. The latter can be explained by the fact
that Ũx in the inter-column channels starts to deviate
significantly from the reference case as shown in
Fig. b.
The analysis of flow features given above indicates that the optimal controls
in case C3t5 influence the wind-farm flow field in such a way as to provide
better flow conditions for downstream turbines. Increased axial velocities
are observed for all downstream turbines, and enhanced momentum transport
towards the turbine region is achieved. Furthermore, many of the observed
flow features are most salient for the first-row turbines. In the following
section, the optimized thrust coefficients themselves will be investigated.
It will be shown that, also from a controls perspective, first-row turbines
stand out from their downstream counterparts.
Time-averaged flow field quantities of simulation results from MM17.
Left: time averages of reference simulations with CT′=2 for all
turbines. Right: difference Δ between reference and optimal control
(C3t5) simulation, defined as Δ=XC3t5-XREF for any
variable X. (a) Axial velocity Ux (plan view at hub height).
(b) Axial velocity Ux (side view through turbine column).
(c) Turbulence kinetic energy k=(1/2)(ũx′ũx′‾+ũy′ũy′‾+ũz′ũz′‾) (side view). (d) Turbulence top-down transport -ũx′ũz′‾. (e) Mean flow top-down
transport -ŨxŨz (side view). (f) Turbulence
horizontal transport ũx′ũy′‾ (plan view).
(g) Mean flow horizontal transport ŨxŨy
(plan view). Black lines indicate wind turbine locations. Simulation results
are averaged over the six different wind turbine columns.
Thrust coefficient analysis and numerical experiments
The current section focuses on the optimal thrust coefficients generated by
the optimal control simulations in MM17 and performs numerical experiments
to uncover some of the characteristics of these control signals. Note that
the conclusions drawn within this section should be interpreted as
observations of the current C3t5 optimal control cases given specific
wind-farm layout and flow conditions, and hence cannot just be generalized
for any wind-farm control in general.
First, the thrust coefficient signals themselves are analyzed in
Sect. . Second, the optimized thrust coefficients
are applied only to subsets of turbine rows in
Sect. . In this way, the interdependency of
optimized thrust coefficients in different rows can be evaluated. Third,
additional optimal control simulations, in which only one single active row
is optimized, are discussed in Sect. . These
optimizations provide an indication of how increased power potential is
distributed among the rows and allows us to compare the resulting single-row
optimized thrust coefficients with the fully cooperative coefficients from
case MM17. Fourth, Sect. evaluates the dependency
of optimized thrust coefficients on the actual turbulent flow realization.
Finally, Sect. discusses the main conclusions
from the abovementioned sections and summarizes the lessons learned.
Analysis of thrust coefficient signals
Figure illustrates the time evolution of some of the
thrust coefficients C^T′ in the C3t5 case. The figure shows that,
for all rows but the last one (i.e., R12), C^T′ varies significantly
in time and that the amplitudes and frequencies of these variations are
somewhat higher in the upstream rows of the farm. In contrast, row 12
features only minor unsteadiness at lower frequencies and has an increased
mean value of C^T′. This relatively steady behavior of the last row
can be explained by the fact that there are no further downstream turbines
that can benefit from row 12 actively influencing local flow conditions;
hence, the row optimizes its own power only. The increase in mean
C^T′ in row 12 can be explained based on Fig. ,
which shows the power extraction as a function of steady C^T′ for
unwaked turbines, subject to identical turbulent inlet as in case C3t5.
Although momentum theory predicts maximal power extraction for steady uniform
inflow at C^T′=2, the actual optimal steady value for the ADM at
the current spatial resolution lies somewhat higher at C^T′≈2.4, for which power extraction is about 1.4 % higher than at
C^T′=2. This behavior is related to the overprediction in ADM
power due to the diffuse turbine representation on typical simulation grids:
the mass flow through the rotor disk at C^T′>2 is slightly too
high compared to momentum theory, resulting in a shift of optimal
C^T′ towards somewhat higher values. Although the linear fit CP′=aC^T′ introduced in , Appendix A,
eliminates the error in maximal power extraction, it does not correct the
value of the optimal C^T′ (note that this could be achieved through
a more complex relation between CP′ and C^T′). Returning to the
more complex thrust coefficients in the other rows, it is worth noting that
based on the current dataset no statistically significant correlations
between the thrust coefficients of different turbines could be found.
Furthermore, attempts towards linking thrust coefficient dynamics to upstream
flow measurements (e.g., velocities, shear or kinetic energy) through linear
regression models and random forest regressors have been unsuccessful to
date.
Figure a and b show row-averaged power spectral
densities of the thrust forces and thrust coefficients, respectively. The
figure shows that the variances of both the thrust coefficients and their
resulting forces are highest in row 1. Further downstream, rows 2 to 11 have
very similar spectral behavior, and row 12 shows significantly lower
variability. The high-frequency slopes of around -5 observed both for fT
and C^T′ indicate that force variability on short timescales is
caused mainly by thrust coefficient variations, whereas the slower thrust
force dynamics tend more to a -5/3 slope, suggesting that these are
governed by the unsteadiness in the turbulence instead. Note that, even
though the spectra for all rows except row 12 collapse at frequencies below
0.05 Hz, the first-row spectrum shows a small peak at f≈0.02…0.03 Hz (fD/U∞≈0.2…0.3) as indicated by
the purple arrow. It will be shown later in this paper that variations in the
thrust coefficient around this frequency are directly related to increased
power extraction.
Time evolution of the thrust coefficient C^T′ for a
selection of optimally controlled turbines in case C3t5. (a) Total
time horizon. (b) Zoomed view including set point CT′ in gray.
Normalized power extraction as a function of steady thrust
coefficient C^T′ for wind turbines subject to the same
free-stream turbulent inflow as in case C3t5. Every dot corresponds to one
LES.
Power spectral density (PSD) estimates of the row-averaged thrust
force fT (a) and thrust coefficient C^T′ (b)
as a function of frequency f (bottom axis) and nondimensional Strouhal
number St (top axis).
Application of optimal thrust coefficients to subsets of turbine rows
In order to further study how the optimal controls increase overall wind-farm
power, Fig. shows power extraction resulting from applying
a subset of the optimal controls to specific turbine rows only.
Figure a depicts simulation results for which the optimized
controls are applied only to one specific row, with the thrust
coefficient in all other rows kept at the reference value of C^T′=2. From the figure it can be seen that only for the controls of the first
row (R1) does this result in a significant power increase in rows 2 and 3. This
indicates that the optimal controls, as generated by the optimization at the
wind-farm level, react to the precise flow conditions caused by upstream
control actions and can hence only be applied independently for the first
row, which has no upstream dependence on other controls.
Figure b shows results from simulations in which the
controls are applied for all rows up to a certain row; i.e., R1–R3
indicates the application of optimized controls generated by case C3t5 to
rows 1, 2, and 3. An interesting observation from this figure is that for
any row i except the last one, the power potential as observed in case C3t5
is almost fully recovered by only applying the optimal controls up to row
i-1. This suggests that self-optimization is very limited: the optimal
controls for a given turbine are designed to create favorable flow conditions
in the downstream rows instead of increasing local power. Furthermore,
although the discussion in the previous paragraph has shown that downstream
controls are optimized with the upstream actions in mind, the converse is not
true: upstream control actions do not require a specific downstream response
in order to increase power in that downstream row.
Normalized row-averaged power extraction for the reference case,
optimal control case C3t5, and subset control cases up to optimization
window 3. (a) Subset control cases with optimal controls applied
only in one specific row. (b) Subset control cases with
optimal controls applied for all rows up to specific row.
Optimization of single active turbine rows
The previous section has shown that, based on the full-farm optimization
case, the first-row controls can be applied independently from other turbine
controls, whereas this does not work for the downstream rows. To further
quantify the potential for increasing wind-farm power in each row of
turbines, the current section considers a set of additional optimal control
cases in which only a single active row is optimized, with all other
rows remaining passive. Furthermore, by comparing the optimized controls of
these cases with the full-farm optimization case C3t5, the degree of
cooperation between turbines can be assessed. Note that the current
single-row optimal control is not equivalent to greedy control: the optimizer
still aims to increase aggregate farm power by taking into account wake
interactions with downstream turbines. Furthermore, in contrast to the
single-row control simulations from the previous section (i.e., in
Fig. a), the current optimizations will yield controls that
are explicitly designed to increase power given that all other rows
are passive. To limit computational costs, the additional optimizations are
only performed for a single time window.
Figure shows the relative increase in wind-farm
power extraction for each of the 12 individually optimized control cases.
The optimization is run until the continuous adjoint gradient accuracy
prevents further progress in the optimization. Upon interpreting the actual
values from the figure, it is important to note that the reported power gain
covers the full optimization horizon T and is hence affected by
finite-horizon effects. Furthermore, the first window of an optimal control
simulation as considered here contains an initial dead zone corresponding
to the wake advection lag before upstream turbines start influencing their
downstream neighbors. This tends to reduce gains compared to later time
windows. Nevertheless, the relative order of the different cases still
provides information that can be generalized to full optimal control studies
with multiple windows and longer time horizons.
The figure shows that the first row (R1) holds by far the most promise for
optimizing wind-farm power. This is not surprising as R1 produces the most
power of all rows and typically leaves the deepest wakes, causing second-row
turbines to perform poorly in aligned wind-farm layouts (see, e.g.,
; and
). At the other end of the spectrum, the last row
(R12) is the least useful. The intermediate rows (R2–R11) lie closer
together, with the general trend being that the potential is somewhat
decreased with downstream distance into the wind farm, although this decrease
is not monotonous.
Figure illustrates the row-wise relative power
increase matrix for each of the single-row optimization cases. The figure
indicates that, for each of the optimization cases, the largest power
increase is observed in the first row downstream of the active turbine (i.e.,
Ri+1) and that the influence on row Ri+3 is limited. This is
explained by the fact that the finite optimization horizon used in MM17
(i.e.,
T=240 s) allows for more interactions with directly neighboring turbines
than with those located further downstream. Furthermore, except for the
optimal control case of the last row (R12), self-optimization is virtually
nonexistent: power gains are achieved by modifying the flow to yield more
favorable conditions for downstream rows.
Modification of thrust coefficient signals
The observations from previous sections illustrate that, at least to some
degree, the optimized thrust coefficients are tuned to local flow conditions.
In the current section, the possibility of whether the coefficients contain
traits that are independent of flow conditions is investigated. To this end,
the optimized thrust coefficients are modified in such a way that
correlations between them and specific flow events are eliminated. This is
done in two independent test cases.
In the first case, the controls, which were specifically generated for
selected turbines, are reassigned to other turbines by randomly swapping the
control sets of different turbine columns. In doing so, each turbine will
receive controls that were specifically designed for another turbine in the
same row. To avoid erroneous conclusions based on coincidence, the column
swap is performed in two random independent ways. The variability of flow
conditions for different columns can be qualitatively observed in Fig. 2. To
further strengthen the hypothesis of the current experiment, we verified that
the correlation between flow conditions in different columns is small, i.e.,
with an average Pearson correlation coefficient of 0.12 between columns for
the incoming velocity fluctuations 6D upstream of the first row.
The row-averaged power for these cases is shown in Fig. a.
Increase in wind-farm power extraction for the first (and only)
time window of the optimal control cases in which only a single row is
optimized.
In the second case, controls remain assigned to their original turbines, but
are shuffled in time by randomly swapping optimal controls generated for
different control windows. In this way, the spectral thrust characteristics
for timescales smaller than the control horizon TA=120 s
remain unchanged, whereas the time synchronization of control actions to
specific flow events is eliminated. Similar to the first case, this is done
in two random ways, and the limited correlation between velocity fluctuations
in different time windows was quantified at 0.07.
Figure b illustrates the row-averaged power for these
cases.
The figure shows similar behavior for each of the modified control cases:
only in the second row (R2) can a consistent (though small) increase in power
extraction be observed. This suggests the presence of flow-invariant
features in the control signals of the first row. Note, however, that the full
power gain in the second row is only partially attained, indicating that
the first turbine row also reacts to the specific flow conditions.
Relative power increase matrix for downstream rows in the single-row
optimization case for row Ri, indicated in the horizontal axis. N/A
indicates nonexisting downstream rows. Finite-horizon effects are eliminated
by only reporting a power increase up to t=TA for the active
row Ri.
Normalized row-averaged power extraction for the reference case,
optimal control case C3t5, and modified control cases. (a) Modified
control cases with controls swapped between wind turbine columns.
(b) Modified control cases with controls swapped randomly by time
window.
Discussion
The observations and experiments from previous sections have revealed
information that increases the understanding of the optimized thrust
coefficients and can be used as a starting point towards designing practical
wind-farm controllers that do not require computationally expensive LES-based
optimal control simulations.
A first conclusion is that wind turbines can be classified into three
distinct categories based on their position within the farm: first-row
turbines, last-row turbines, and intermediate turbines. The most salient
behavior can be found in the first-row turbines (R1). It was shown that these
turbines exhibit the largest variability in thrust forces and hold the
greatest potential for power optimization. Furthermore, they are not
influenced by upstream turbine control action and are the only turbines that
retain part of the power gains after eliminating possible correlation between
controls and specific flow events. The characteristics of the last-row
turbines (R12) also stand out from the rest due to the fact that by
definition the last row has no downstream turbines and hence holds no
further potential for coordinated control. The remaining intermediate rows
(R2–R11) have similar spectral thrust characteristics and potential for
power increase, as they are situated between but clearly separated from the first- and
last-row turbines. Further, it is worth noting that the behavior and analysis
of control actions in these turbines is most complex: not only do they
influence downstream turbines, but they in turn are also dependent on the controls of
upstream turbines.
A second conclusion is that, whether or not the wind farm is controlled with
the possibility of active response and cooperation between turbines, the
resulting power and thrust characteristics are very similar. It was shown
that self-optimization is very limited and that for any row i the full
potential in power increase is virtually attained by applying controls only
for the upstream turbines up to row i-1. These observations strongly
suggest that the optimized thrust coefficients are designed in a parabolic
manner, i.e., with a unidirectional propagation of control information from
the first row to the last and very little upstream influence of downstream
turbine actions. With this in mind, the following section of this paper will
focus on the first and most promising link in the control chain: the turbines
situated in the first row of the wind farm.
First-row turbine behavior
The current section further focuses on the analysis of the first-row
turbines. First, a qualitative analysis of the instantaneous flow field is
performed in Sect. , resulting in the observation
of vortex rings being shed from first-row turbines. Thereafter, this
mechanism is mimicked by imposing sinusoidally varying thrust coefficients in
these turbines in Sect. , with the aim of increasing power
through similar mechanisms as in the computationally expensive optimal
control cases.
Flow field visualization
Figure shows snapshots of the vorticity and
velocity fields at t=300 s for the reference case (left) and the optimal
control case C3t5 (right). Figure a and b show
isosurfaces of vorticity magnitude colored by streamwise velocity
ũx. Figure a shows that in the
reference case the first-row turbines shed relatively stable vortex sheets
that demarcate the wake from the free-stream flow. The sheets destabilize and
break up as they are advected downstream, resulting in complex
three-dimensional vortical structures. Furthermore, as also shown in
Fig. c, wake mixing is limited, and downstream
turbines experience reduced velocities. In contrast, the optimized case shows
coherent vortex rings being shed from the first-row turbine. As indicated by
the black arrows in Fig. b, the locations of the
rings in the controlled case coincide with naturally occurring bulges in the
vortex sheet of the reference case: the controlled turbines further
destabilize the sheet through well-timed temporal variations in its thrust
coefficient. Figure c shows that this results in
smaller velocity deficits in the wake region. Note that downstream of the
second turbine the vorticity field becomes much more complex and differences
in the flow fields are less coherent.
The observed shedding of ring vortices seems to occur at specific
flow-synchronized times to exploit the natural instabilities in the original
vortex sheets. Therefore, the remainder of this paper will attempt to
accomplish the same effect through simple sinusoidal thrust variations.
Instantaneous snapshots at t=300 s of a portion of the wind-farm flow
field for the reference (left) and optimized (right) case. (a, b) Isosurface of vorticity magnitude colored by streamwise velocity
ũx. Deep-wake regions (with ũx<4.5 m s-1) are
rendered in black. Black arrows indicate the naturally occurring unstable
bulges in the reference case and the accompanying vortex rings in the optimized
case. (c) Contours of streamwise velocity ũx. Coloring is
in units of m s-1. Wind turbines are represented as gray disks.
Sinusoidal thrust variations
The aim of the current section is to mimic the quasi-periodic shedding of
vortex rings by upstream turbines as observed above through the use of simple
periodic variations in the thrust coefficient. Instead of optimizing a
high-dimensional control signal that can evolve freely in time as in MM17, we
impose a sinusoidal perturbation on the Betz-optimal coefficient
C^T′=2, parameterized by its amplitude A, and its frequency in
the form of a nondimensional Strouhal number St=fD/U∞,
with f the dimensional frequency, D the turbine diameter, and U∞
the unperturbed time-averaged upstream velocity:
C^T′(t)=2+Asin(2πSttU∞D).
Parameter sweep
Instead of optimizing A and St using a similar gradient-based
optimization setup as in MM17, we perform a parameter sweep to find optimal
parameter combinations. The reason for this is that we would need a rather
long optimization horizon T to find a robust parameter combination that is
independent of specific flow realizations. Unfortunately, the chaotic nature
of turbulent flow fields makes long-time optimization using adjoint LES
practically infeasible to date (see, e.g., ). However,
the fact that we have only two parameters renders a parameter sweep
computationally feasible. The sweep is performed for a reduced-size wind-farm
LES, as illustrated in Fig. . The farm consists of 4×4 turbines in an aligned layout with S=6D in both the streamwise and
span-wise directions, geometrically equivalent to the optimally controlled
wind farm in MM17. The simulation is performed on a domain of 4×2.4×1 km3 with a simulation grid of 192×256×144
grid points. A wall roughness length z0=10-1 m is used. A set of
wind-farm flow simulations is advanced in time by 30 min, during which
the front row is controlled using a sinusoidally varying thrust coefficient
C^T′, as defined in Eq. (). Within this set,
the amplitude A is varied between 0.5 and 2, with increments of 0.5.
Furthermore, the Strouhal number St is varied between 0.05 and
0.6, with increments of 0.05. In total, this leads to 48 LES cases within the
set.
Reduced 4×4 wind-farm simulation setup for sinusoidal
variation parameter study showing instantaneous contours of streamwise
velocity ũx. Coloring is in units of m s-1. The dashed line
indicates the start of the fringe region for the imposition of unwaked inflow
conditions.
Figure illustrates the power extraction for all
cases considered. Figure a illustrates the relative
power gains over the reference case. From the figure it can be seen that
there is a well-defined range of values for A and St for which
wind-farm power can be increased substantially through upstream sinusoidal
thrust variations, with a maximal power increase of ≈5 % at
(St∗,A∗)=(0.25,1.5). For instance, a Strouhal
number St =0.25 corresponds here to a sine wave period of
≈50 s for a turbine with diameter D=100 m and a free-stream
velocity U∞=8.5 m s-1. For instance, considering the NREL
5MW blade profiles, the maximum thrust coefficient of 3.5 can be attained by
slightly changing the rotor design, e.g., using a 50 % increase in blade
chord length and an operational tip speed ratio 25 % higher than the
original design value (see Appendix A in ).
Furthermore, given such redesign, dynamic reductions from this value could be
realized through blade pitch control, for which actuation rates of the order
of 10∘ s-1 are possible (see, e.g.,
). Figure b
illustrates normalized power extraction by row for the reference case, the
best sinusoidal case, and the first four rows of the optimal control case
C3t5 from MM17. The figure shows that the power gain in the sinusoidal cases
originates mostly from the second row and that power in the first row is
decreased by approximately 5 %. In contrast, optimal control case C3t5,
in which all rows are active, also increases power in rows 3 and 4 and
reduces first-row power by only 1 %.
Power extraction of baseline sinusoidal thrust case (S=6D, z0=10-1 m). (a) Relative gain in mean wind-farm power extraction
over reference case as a function of sine amplitude A and frequency
St. (b) Row-averaged mean power extraction for the best
sinusoidal case, the optimal control case C3t5 from MM17, and the reference
case normalized by first-row reference power.
Figure illustrates instantaneous vorticity
and axial velocities for a set of wind turbines of the aforementioned
reference case (a), the best sinusoidal case with (St,A)=(St∗,A∗)=(0.25,1.5) (b), and a sinusoidal case that
does not lead to increased power extraction with (St,A)=(0.6,2) (c). The figure illustrates the fact that sinusoidal variations in the first-row
thrust coefficient indeed cause the periodic shedding of vortex rings.
Figure b shows that at the optimal
frequency this leads to increased wake mixing, providing the second-row
turbine with a higher incoming velocity. In contrast,
Fig. c shows that even though higher-frequency thrust oscillations also result in the
periodic shedding of vortex
rings, this does not automatically lead to more favorable flow conditions for
downstream turbines. Therefore, it can be concluded that the correct timing and
spacing of vortex rings is essential for increased wake mixing.
Snapshots of wind-farm flow fields at t=1800 s. Left:
isocontours of vorticity colored by streamwise velocity. Right:
contours of streamwise velocity. (a) Reference. (b) Best
sinusoidal case with (St,A)=(St∗,A∗)=(0.25,1.5). (c) Sinusoidal case with (St,A)=(0.6,2).
In order to assess whether the same strategy can be used in the downstream
turbines as well, Fig. illustrates the results from an
identical parameter sweep as discussed above, except that here the second
turbine row is controlled using a sinusoidal thrust coefficient.
Figure a indicates that the sinusoidal actuation of the
second row invariably leads to losses in wind-farm power.
Figure b shows that, for the optimal combination of
parameters of (St,A)=(0.25,1.5) as reported for first-row
actuation above, the minor power increase in the third row does not
compensate for additional losses in the second and fourth row. This shows
that the proposed simple control strategy does not work when applied to waked
turbines and that more elaborate control strategies are required to harness
the gains achieved by the optimal control simulation in the downstream
regions of the farm.
Power extraction of second-row sinusoidal thrust case.
(a) Relative gain in mean wind-farm power extraction over reference
case as a function of sine amplitude A and frequency St.
(b) Row-averaged mean power extraction for the sinusoidal case with
the optimal parameters as for row 1 sinusoidal thrust and the reference
case normalized by first-row reference power.
It is interesting to note that, even though the current parameter sweep is
performed using different initial and inlet conditions than those applied in
MM17, the optimal frequency of sinusoidal variations in C^T′ at
St∗=fD/U∞=0.25 corresponds to the location of
the peak in the first-row thrust coefficient spectrum of C3t5 in
Fig. . In the following paragraphs, the
robustness of the best parameter pair for first-row thrust variations, i.e.,
(St,A)=(0.25,1.5), is investigated with the aim of assessing
the general applicability of this control strategy. To this end, similar
parameter sweeps are performed for cases with varying turbine spacings and
turbulence intensities.
Robustness with respect to turbine spacing and turbulence intensity
Figure shows the power extraction resulting from two
parameter sweeps with streamwise turbine spacings of 5D and 7D. The results are promising: for the given cases,
St∗ and A∗ do not depend on streamwise turbine
spacing. Furthermore, even in the 7D spacing case
(Fig. c–d), which naturally features lower overall
power losses in downstream rows, power extraction in the second row can be
significantly increased through sinusoidal variations in the first-row thrust
coefficient.
Power extraction of sinusoidal thrust cases with varying streamwise
spacing. (a, b) Decreased spacing at S=5D. (c, d)
Increased spacing at S=7D. (a, c) Relative gain in mean
wind-farm power extraction over reference case as a function of sine
amplitude A and frequency St. (b, d) Row-averaged mean
power extraction for the best sinusoidal case and the reference case
normalized by first-row reference power.
Figure depicts the power extraction results from a
parameter sweep with the same wind-farm layout as in the baseline case, but
with a 10-fold increase in roughness length, i.e., z0=1 m. This results
in a turbulence intensity of approximately 16 % compared to 10 % in
the baseline case. Again, the best parameter combination of
(St∗,A∗)=(0.25,1.5) remains unchanged. Further,
even for this higher turbulence case, in which downstream losses are lower
due to naturally better wake mixing, power is increased in the second row,
leading to a relative gain in wind-farm power of around 2 %.
Power extraction of sinusoidal thrust case with increased wall
roughness z0=1 m. (a) Relative gain in mean wind-farm power
extraction over reference case as a function of sine amplitude A and
frequency St. (b) Row-averaged mean power extraction for
the best sinusoidal case and the reference case normalized by first-row
reference power.
As evidenced above, periodic sinusoidal variations in first-row thrust
coefficients substantially increase power extraction in the second row,
resulting in a net gain in total power for the considered 4×4 wind
farm. Moreover, different simulation sets indicate that, at least for the
range considered here, the best values for the Strouhal number and the amplitude of
these variations, i.e., (St∗,A∗)=(0.25,1.5), are
robust with respect to turbine spacing and turbulence intensity.
Full-scale wind-farm LES
In the remainder of this section, the sinusoidal variation strategy will be
tested in a full-scale wind-farm LES corresponding to the full 12×6
aligned wind farm of MM17. Simulations are performed for a reduced range of
amplitudes and Strouhal numbers corresponding to the most favorable region
identified in the parameter sweeps above. In order to increase statistical
convergence, the time horizon for each simulation is extended to a physical
time of 10 h.
Figure shows the power extraction of the full-scale
LES. Figure a shows the relative power gains over
the reference case for the full wind farm. It can be seen that the total
power gain or loss is below 0.5 % for each of the sinusoidal control
cases. Figure b shows the row-wise power extraction
for the reference case, the sinusoidal thrust case with (St,A)=(0.25,1.5), and the optimal control case C3t5. It is shown that, although
the second row of the sinusoidal thrust case achieves similar power gains as
those observed above, from the fifth row onwards power is slightly reduced in
the sinusoidal case.
Power extraction of full-scale sinusoidal thrust case (S=6D,
z0=10-1 m). (a) Relative gain in mean wind-farm power
extraction over reference case as a function of sine amplitude A and
frequency St. (b) Row-averaged mean power extraction for
the sinusoidal case with parameters from previous sections, the optimal
control case C3t5 from MM17, and the reference case normalized by first-row
reference power.
The top panel of Fig. shows cross sections of
time-averaged axial velocities Ũx at the rotor disk locations for
the reference case. Further, the middle and bottom panels illustrate
deviations from the reference velocity for the (St∗,A∗) sinusoidal case and the optimal control case C3t5, respectively.
The figure shows that both controlled cases show similar characteristics at
the second turbine row, with an increased axial velocity at the rotor disk
accompanied by decreased velocities above and below. Downstream, it can be
seen that the passive turbines of the sinusoidal case fail to retain
increased velocities at the rotor disks, instead resulting in slightly lower
disk velocities starting from the fifth row. In contrast, case C3t5, in which
all turbines are actively controlled, succeeds in attaining similar cross
section characteristics with higher rotor velocities in the downstream as
well. Note also that for the fifth row the disk velocity is slightly lower
for the sinusoidal control case than for the reference case, consistent with
the decreased power extraction observed in Fig. .
This can be explained by the fact that first-row control actions cause
enhanced entrainment of momentum from the internal boundary layer above the
turbine canopy that would otherwise be entrained by natural turbulent mixing
in passive downstream rows. In consequence, lesser entrainment occurs for
downstream rows, resulting in a slight decrease in disk velocities from the
fifth row onwards.
Cross section of time-averaged axial velocity Ũx at rotor
locations in rows 1, 2, 3, 4, 5, and 12. (a) Reference case.
(b) Difference between best sinusoidal perturbation case (with A=1.5, St=0.25) and reference case. (c) Difference
between optimal control case C3t5 and reference case. Coloring is in units
of m s-1.
As shown throughout the current section, a qualitative analysis of
instantaneous flow features in the optimal induction control wind farm from
MM17 has led to the identification of a sinusoidal thrust control strategy
for first-row turbines, resulting in increased power extraction in the second
row. However, important comments should be made. First, sustained sinusoidal
thrust variations with a large amplitude could contribute significantly to
turbine fatigue loading of the first-row turbines. Furthermore, partial wake
alleviation and unsteady passing of the abovementioned vortex rings could also
increase fatigue loading in downstream rows. Hence, structural aspects should
be taken into account upon evaluating the practical viability of the
approach. Second, even though experiments have shown that, for practically
relevant tip speed ratios, wind turbines shed vortices in a similar way as
disk-like bluff bodies , the current behavior
could still be an artifact of the relatively simple ADM used throughout this
study. Further verification using higher-fidelity wind turbine models, such
as actuator line models, and wind tunnel testing is hence necessary.
Intermediate-row turbine behavior
The current section discusses the intermediate rows. It was shown that,
without active participation in these rows, upstream gains are lost in
downstream rows, and only full optimal control succeeds in achieving
significant gains in downstream rows as well (see
Figs. , ). It was
already mentioned that the analysis and behavior of turbines within
intermediate rows is more complex than in the first row: they aim to
influence the flow to the benefit of downstream rows but are also dependent
on the actions of upstream rows. The remainder of the current section aims to
illustrate the additional difficulty of power increase in downstream rows
and speculates on possible future paths for the identification of simplified
control strategies as found for the first row.
First, as shown in the top panels of Fig. , even
in the uncontrolled case the kinetic energy of the flow in the vicinity of
the turbine rotor is depleted more and more in the downstream rows. This
complicates control strategies for these rows as the opportunity for
increased mixing with high-energy flow is decreased. Furthermore,
intermediate turbines are subjected to increased turbulence levels and more
complex vorticity dynamics, as illustrated in
Fig. . This could explain why sinusoidal thrust
control did not lead to increased power when applied to the second row:
whereas the first row produces increased mixing by destabilizing relatively
stable vortex sheets into vortex rings, the second row is already
continuously immersed in complex vorticity patterns for which this simple
approach does not work. However, note that R7 in
Fig. , for instance, also seems to show quasi-periodic sinusoidal
variations in C^T′ at a time period of approximately 50 s. This
is an indication that, also for intermediate rows, vortex ring shedding could
amount to part of the power increase observed in the optimal control
simulations, albeit at specific moments in time, synchronized with the local
flow conditions.
Second, it is important to note that the vortex ring shedding mechanism
constitutes only part of the power increase caused by the first row.
Figure illustrates that the first-row optimized thrust
coefficient also results in a significant power increase in the third row,
which is not observed using the sinusoidal thrust strategy. Furthermore, the
analysis of the modified control cases in Fig. proves
that the first-row controls are also partially synchronized with the flow.
This shows that other mechanisms, dependent on specific flow events for
increasing wind-farm power, are at play as well. Even though the application
of regression algorithms in an attempt to link turbine actions to
low-dimensional flow measurements (e.g., local velocity, shear and kinetic
energy) has been unsuccessful thus far, similar analysis based upon more
complex flow features (e.g., vorticity structures, high-speed turbulent
streaks, or downdrafts) might be more promising. This requires further
optimal control simulations over an extended time, as the total control time
horizon of 30 min in the current dataset is insufficient for robust
statistics in this kind of analysis. This is an important remaining challenge
to be addressed in future research.
Conclusions
The current paper provided an analysis of the thrust coefficient control
characteristics for the C3t5 optimal control case featured in
.
Analysis of the thrust coefficients and numerical experiments have shown a
clear distinction between first-row turbines, last-row turbines, and
intermediate turbines. Furthermore, observations strongly suggest that the
optimization works in a unidirectional way: upstream turbines influence the
flow field, resulting in favorable conditions for their downstream neighbors,
yet information on the possibility of active response and cooperation in the
latter has no influence on upstream control actions.
Qualitative analysis of instantaneous flow fields led to the observation of
the quasi-periodic shedding of vortex rings from first-row turbines in the
optimal control case. This flow feature was successfully mimicked using simple
sinusoidal thrust actuation of the first row. The best parameter set for
these sinusoidal variations proved robust to both wind turbine spacing and
turbulence intensity, with an amplitude A∗=1.5 and a
nondimensional frequency at St∗=0.25. Interestingly, this
frequency corresponds to the peak at St=0.2…0.3 observed
in the first-row thrust coefficient spectra of the optimal control case.
Although the first-row sinusoidal control led to a robust increase in total
power for a reduced-size 4×4 wind farm, a full-scale test indicated
that downstream turbine activity is required to obtain increased power at
larger farm scales. It was also shown that the simple sinusoidal strategy
does not lead to increased power extraction when applied to downstream
intermediate turbines. Identifying the mechanisms for power increase in these
turbines hence remains an important open research question. Finally, it is
important to remark that all current simulations were performed using a
standard nonrotating actuator disk model without the inclusion of mechanical
turbine loading. Therefore, wind tunnel testing and/or simulations with more
advanced turbine models (such as the actuator line model) including
the assessment of turbine loading are essential to evaluate the real-life
applicability of the sinusoidal thrust strategy.