Wake interactions between wind turbines in wind
farms lead to reduced energy extraction in downstream rows. In recent work,
optimization and large-eddy simulation were combined with the optimal dynamic
induction control of wind farms to study the mitigation of these effects,
showing potential power gains of up to 20 % (Munters and Meyers, 2017, Phil. Trans.
R. Soc. A, 375, 20160100,

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.,

In contrast to the studies cited above that all focus on the static set point
optimization of wind farms,

Recently, the methodology of

The current paper presents efforts on understanding optimal control dynamics
observed in the optimal induction control simulations by

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.

A schematic overview of the wind-farm control methodology is shown in
Fig.

Schematic overview of wind-farm optimal control methodology from
MM17.

Within each optimization window, the total wind-farm power is optimized by
solving the following optimization
problem constrained by partial differential equations.

The filtered Navier–Stokes momentum and continuity state equations in
Eqs. (

The forces exerted by turbine

The wind farm considered in MM17 has an aligned pattern of 12 rows by
6 columns. The wind turbines have a hub height

Instantaneous streamwise velocity

A conventionally (greedily) controlled wind farm with steady

Figure

Energy extraction

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.

Figure

Figure

Figure

Figure

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

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.

Figure

Figure

Time evolution of the thrust coefficient

Normalized power extraction as a function of steady thrust
coefficient

Power spectral density (PSD) estimates of the row-averaged thrust
force

In order to further study how the optimal controls increase overall wind-farm
power, Fig.

Figure

Normalized row-averaged power extraction for the reference case,
optimal control case C3t5, and subset control cases up to optimization
window 3.

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.

Figure

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.,

Figure

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 6

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

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

Normalized row-averaged power extraction for the reference case,
optimal control case C3t5, and modified control cases.

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

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.

Figure

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

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

Instead of optimizing

Reduced

Figure

Power extraction of baseline sinusoidal thrust case (

Figure

Snapshots of wind-farm flow fields at

In order to assess whether the same strategy can be used in the downstream
turbines as well, Fig.

Power extraction of second-row sinusoidal thrust case.

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

Figure

Power extraction of sinusoidal thrust cases with varying streamwise
spacing.

Figure

Power extraction of sinusoidal thrust case with increased wall
roughness

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

In the remainder of this section, the sinusoidal variation strategy will be
tested in a full-scale wind-farm LES corresponding to the full

Figure

Power extraction of full-scale sinusoidal thrust case (

The top panel of Fig.

Cross section of time-averaged axial velocity

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 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.

First, as shown in the top panels of Fig.

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

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

Datasets supporting this article can be found in a figshare
repository. These datasets include time-averaged flow fields, optimized
controls and optimized wind-farm power extraction for the case reported in
this paper (Munters and Meyers, 2016, Figshare data collection repository,

The authors declare that they have no conflict of interest.

The authors received funding from the European Union's Horizon 2020 research and innovation program under grant agreement no. 727680 (TotalControl). The authors also acknowledge funding by the European Research Council under grant agreement no. 306471 (ActiveWindFarms). The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation – Flanders (FWO) and the Flemish Government, department EWI. Edited by: Sandrine Aubrun Reviewed by: two anonymous referees