Introduction
In recent years, more and more wind turbines have been installed in ever
larger wind farms. On the one hand, this has advantages in logistics, network
connection, maintenance, and the utilization of the limited suitable
locations. On the other hand, this leads to situations where turbines are
considerably more affected by harmful wake conditions. Downstream turbines
are experiencing higher turbulence, which is generally associated with larger
fatigue loads, and lower wind speeds, which results in a lower energy yield.
In order to counteract this, different strategies are being investigated that
try to reduce these negative wake effects. One approach to achieve this is
active wake deflection by intended yaw misalignment, which was already
investigated by in a wind tunnel experiment. While
conventional, so-called greedy turbine control seeks to optimize the
operation of the individual turbines without taking into account the mutual
effects on the other turbines, active wake deflection is the attempt to alter
the trajectory of turbine wakes in order to improve the inflow conditions of
downstream turbines. If two turbines are interacting through a wake, the
deflection is achieved by deliberately introducing a yaw misalignment of the
rotor of the upstream turbine with respect to the wind direction. The rotor
then generates a thrust force component that is perpendicular to the wind
direction, which laterally deflects the wake. The goal is that the power gain
of downstream turbines is higher than the power loss of the misaligned
upstream turbine. More recent wind tunnel experiments
and large eddy simulations (LESs)
demonstrated the potential of such wake steering
strategies for increasing the overall energy yield. Additionally,
investigated the influence of different atmospheric
stabilities on active wake deflection in an LES study and analysed multiple
sources that contribute to the uncertainty in the estimation of the deflected
wake position. However, in both wind tunnel tests and conventional
high-fidelity computational fluid dynamics (CFD) simulations the inflow enters through a defined area, thus
realistic dynamical changes in the wind direction are often not adequately
reproduced. Fluctuating wind conditions and a high sensitivity towards the
wind direction make it difficult to take appropriate account of wakes in wind
farm control in an uncontrolled environment like the free field.
The contribution of this article continues the investigations of
and extends it by using methods of stochastic programming
to take better account of the uncertainties occurring in
the field. This is done to derive a robust control strategy for the yawing of
turbines, which is evaluated below. First results were published in the final
report of BMWi-funded research project CompactWind and
presented at WindTech 2017 . The stochastic programming
approach was also pursued in for optimizing yaw
angles for wake deflection while considering yaw errors only. In contrast
this paper takes into account wind direction dynamics and measurement
inaccuracy.
The objectives of this paper are (1) to analyse the impact of dynamical wind
direction changes on active wake deflection strategies, (2) to introduce a
methodology to optimize the yaw angle adjustment in a wind farm by taking
these fluctuations and measurement uncertainties into account and (3) to
propose open-loop control algorithms for active wake deflection in a wind
farm.
Methods
In this paper, a quantitative analysis of wind direction variability and its
effect on active wake deflection is carried out. Therefore, first the
employed model for wake deflection (Sect. )
and an analytical description of the statistics of wind direction variability
(Sect. ) are introduced. Based on this, an optimization
of the yaw angle adjustments of all turbines in a wind farm with respect to
the relative change in the power output of the farm is established
(Sect. ) and evaluated for a fictitious reference wind
farm (Sect. ).
Wake deflection model
The investigation in this article is based on the FLOw Redirection and
Induction in Steady state (FLORIS) model , which has been
especially developed for active wake deflection at the Delft University of
Technology and the National Renewable Energy Laboratory (NREL;
https://github.com/wisdem/floris, last access: 8 November 2018). For a
more elaborate description of control-oriented models in general, see
. The comparatively low computational complexity of the
steady-state models makes it possible to perform optimization algorithms and
validations on the basis of large datasets.
FLORIS extends the popular Jensen wake model to a more
detailed wake description, containing three zones: a near wake zone, a far
wake zone, and a mixing zone (see Fig. ).
A schematic visualization of the wake model in FLORIS, containing
three discrete wake zones. Furthermore, active wake deflection by purposely
misaligning the turbine rotor with the flow is shown (cf. Fig. 4a in
).
These three wake zones each contain their own set of parameters for wake
recovery and expansion, increasing the model's flexibility and fidelity.
Furthermore, FLORIS uses the simplified analytical model from
which determines the wake deflection as a function of the
turbine's thrust force and yaw angle. For multiple wake overlap situations,
the sum of squares approach for the superposition of wake deficits is
followed, first suggested by .
In short, FLORIS predicts the time-averaged steady-state conditions of the
flow and each turbine's power capture as a function of each turbine's axial
induction (i.e. a parameterization of the generator torque and blade pitch
angles), yaw angle, and atmospheric conditions inside a given farm. The
applicability of the model has been demonstrated in high-fidelity simulations
(e.g. ), wind tunnel tests , and
even to some degree field tests .
Statistical analysis of wind direction measurements
The analyses in this article were carried out on the basis of measurements at
a meteorological mast (referred to as a met. mast hereafter). In ,
measurements from the same device were used.
As described there, the total height of the met. mast is 91.5m. The
wind direction was measured by a wind vane of type 4.3150.00.212
(manufactured by Thies GmbH) at a height of 89.4m. The wind vane
was installed on a boom at the met. mast approx. 1m above the main
structure of the met. mast. As small disturbance could be introduced from wind
directions around 134∘ since another boom was located at a distance
of 1.65m in that direction, to which the highest cup anemometer at
a height of 91.5m was attached. For the evaluation this possible
disturbance was not considered, since this wind direction rarely appeared in
the dataset and the effect should be marginal.
The wind direction angle can be expressed in radians φ∈[0,2π) or in degrees φ∈[0∘,360∘) interchangeably.
Both representations are used in this article. Formulas are generally written
using radians, while illustrations are presented in degrees using the
standard conventions, i.e. 0∘ represents north and the rotation is
clockwise.
The wind direction φ∈[0,2π) as input variable is a decisive parameter for the successful
application of active wake deflection. In the field, however, the wind
direction can change continuously and sometimes abruptly, as can be seen in
the example in Fig. .
Exemplary 5 min time series of wind direction measurements recorded
at an onshore test site in northern Germany sampled at 1Hz
resolution.
The turbulent changes in the wind direction are in contrast to the slowly
reacting yaw mechanism of utility-scale wind turbines. The deviation between the
wind direction and the yaw angle of the turbine is usually averaged over
several minutes and a threshold for the deviation is used to keep the turbine
from constantly yawing . This has the effect that the
turbine is in standstill mode most of the time . Although the
details of the yaw control depend on the manufacturer, and is commonly kept
confidential, in our experience the yaw angle remains constant for about 5 to
10 min in most cases before the yaw control corrects the yaw angle according
to the changed wind direction. For this reason, we have studied the
statistics of 5 min wind direction time series from 1Hz
measurement data denoted as Φt∈{(φ1,…,φ300)∈R300|φτ∈[0,2π),τ=1,…,300}, where the
variable t∈N indexes successive time series.
Figure depicts an exemplary time series Φt from
which a histogram is derived in Fig. . A probability
density function of a normal distribution (in red) with the same mean value
(284.78∘) and standard deviation (4.60∘) as the measurement
data is fitted to the histogram. Since a histogram depends very much on the
binning, we added a quantile–quantile plot in Fig. b,
which is commonly used to compare distributions. Here, the measurement data
are compared with the fitted normal distribution. The measurement data are
sorted by their values and plotted against the respective quantiles of the
normal distribution. The better the distributions match, the more the
values lie on the straight red line.
(a) Histogram of exemplary 5 min wind direction
measurements in blue with the fitted Gaussian normal distribution in red,
(b) quantile–quantile plot of the exemplary 5 min wind direction
measurements in blue with the normal distribution reference represented by
the dashed red line.
For the exemplary time series, both representations demonstrate the
similarity to the normal distribution reasonably well, which agrees with the
findings of that wind direction behaviour is normally
distributed within one averaging period, although Gaumond investigated
10 min time series. However, it should be noted that conventional stochastic
tools for the analysis of directional data are not generally valid, whereas
circular statistics consider that, for example, φ and φ+k⋅2π
for any k∈Z are identical angles on a standard circle. For
example, the von Mises (or Tikhonov) distribution is typically used as an
approximation of a wrapped normal distribution. Further, the directional mean
φDM∈[0,2π) differs from the arithmetic mean. It is
defined as the angle of the sum of all unit vectors of the wind directions
φτ,τ∈N (see Eq. ) , but in
programming commonly the four quadrant inverse tangent (atan2) operator is used
for the computation:
φDM=arg∑τexp-1⋅φτ=atan2∑τsin(φτ),∑τcos(φτ).
Nevertheless, since the wind direction distributions during a short time
period include a relatively small sector compared to the whole circle, the
differences between the normal distribution and a wrapped normal distribution
are negligible. Therefore, and since it is comparatively less complex, we use
the standard tools as mentioned above. But, since each angle can be expressed
in multiple ways, we shift the transition angle at 360∘/0∘
of each 5 min time series to its respective opposite angle of
φDM. This can be achieved by the modulo operator (see
Eq. ), where φraw refers to the original raw
data.
φ=φraw+π-φDMmod2π-π+φDM
In this way, we can apply the conventional calculation for the mean value and
the standard deviation of the wind direction data φ. The validity of
the assumption of an underlying normal distribution for 5 min wind direction
data series is statistically tested in Sect.
Approach for optimization of yaw angles
In order to optimize the yaw settings of the turbines in a wind farm, we
assume that the total power output of a wind farm consisting of n wind
turbines corresponds to the sum of the individual power outputs of the wind
turbines. Conduction losses are therefore ignored. The turbines' power is
estimated by FLORIS for a set of environmental conditions and control
variables, i.e. the axial induction factor aj and the yaw angle γj
of each individual turbine j=1,…,n. With Γ={γj∈[0,2π)|j=1,…,n} we denote the set of yaw angles of all
turbines. We assume that the turbines run at a constant axial induction
factor of aj=13 for all j and the power output Pj of each
turbine is normalized with respect to the power output of a turbine in
undisturbed inflow conditions, since we are focussing on the influence of the
yaw angle on the relative change of turbine power. Therefore, both aj and
the wind speed are omitted in the following equations. Pj depends on its
own control variable γj, the wind direction and the yaw angles
of all other turbines due to the aerodynamic interaction in the wind farm.
Hence, we denote Pj=Pj(γ1,…,γn,φ).
Next we introduce two different optimizations of the yaw angles which
differ in the description of the wind direction variability. Firstly, we are
neglecting wind direction changes within the investigated time period. The
optimization problem formulated in Eq. () aims at
finding the set of optimal yaw angles Γopt(φ)={γ1opt(φ),…,γnopt(φ)},
which maximizes the power output of the wind farm for the prescribed wind
direction φ. We will refer to this as the conventional optimization
in the following.
findΓopt(φ)=argmaxΓ∑j=1nPjγ1,…,γn,φforthewinddirectionφ
Secondly, the optimization of the yaw angles is formulated more robust
towards wind direction dynamics and uncertainties in the measurements.
Instead of considering only one wind direction, the new problem description
aims at finding the set of optimal yaw angles for a distribution of wind
directions weighting the results for every individual wind direction by its
probability of occurrence. This is achieved in
Eq. () by a probability density function
ρ(φ), which represents wind directional variation and
uncertainties stochastically. We will refer to this approach as the robust
optimization in the following.
findΓopt(ρ(φ))=argmaxΓ∫02πρ(φ)∑j=1nPjγ1,…,γn,φdφforaprobabilitydensityfunctionρ(φ).
This formulation is a generalization of the conventional optimization, which
is obtained when we insert the dirac delta function for the probability
density function.
To solve the integral in Eq. () the cost function is discretized. For the
calculation we have chosen a step size of 1∘, which corresponds
reasonably well with the measuring accuracy of wind vanes and still gives a
good representation of the distribution.
A number of algorithms can be used for the computation of these kinds of
optimization problems, including the intuitive game theoretic approach
presented in . This algorithm has the benefit that it works
on complex, nonlinear problems and does not depend on any gradients, but
unfortunately it converges relatively slowly. For this research, therefore, we
used the pattern search algorithm , which has
similar properties but we experienced a faster convergence speed.
Test case
In order to evaluate the control strategies derived from the conventional and
the robust optimization, a case study is performed for a reference test wind
farm. It consists of nine NREL 5-MW turbines with a rotor
diameter of 126 m in a grid layout (see Fig. ). The
turbines are situated relatively close to each other, such that strong wake
effects occur, as the active wake deflection is of special interest for such
situations.
Layout of the reference wind farm. The reference turbine (T32) is
marked in red.
With this layout the distances between adjacent turbines are 3D
horizontally, 4D vertically, and 5D diagonally. These values are comparable
to the dense spacing in the offshore wind farm Lillgrund
.
The focus of the investigation is to determine the sensitivity of the control
strategies with regard to wind direction variability and uncertainty. In
order to perform this investigation with realistic data, wind direction
measurements from a met. mast at 91.5m height at a near-coastal
test site in Brusow, northeastern Germany, were used as input. The
surrounding area was mostly flat, but some complexity was added by a nearby
forest. For more details we refer to . Of the available
measurements, only data with a 5 min average wind speed between 3.5 and
14 ms-1 were used, since power optimization is only of interest
when the turbines operate above cut-in and below rated wind speed. This
corresponds to 87 % of the data collected between 30 June and
22 November 2016, which gives a total of N=35586 of 5 min time series.
Figure illustrates a wind rose of the data showing the
frequencies of occurrence of the directions.
Wind rose of 1 Hz measurement data recorded at a met. mast in
Brusow, north-eastern Germany.
Results
In this section, the method for investigating the statistical properties of
wind direction changes using measurement data is examined
(Sect. ). In
Sect. the two methods for yaw angle
optimization were applied to the reference wind farm and in
Sect. open-loop control algorithms are derived from the
optimizations and are evaluated using the measurement data.
Stochastic properties of wind direction measurements
As mentioned in Sect. , we analysed the stochastic
properties of 5 min time series of 1 Hz wind direction measurements
denoted as Φt. Specifically, we want to verify the hypothesis that
Φt can be approximated statistically by a normal distribution as
indicated by Fig. . Therefore, we performed a
Kolmogorov–Smirnov test on the subject . In
this fitting test, the empirical distribution of Φt is compared to a
normal distribution and a critical value is calculated. This value, together
with the chosen significance level, determines whether to accept or reject
the hypothesis. In our case, 70.58 % of the measurements used for
this investigation passed the test for a significance level of 5 %.
From this we draw the conclusion that in most cases 5 min wind direction
time series can be reasonably represented by normal distributions. For
normally distributed data, the mean value and the standard deviation are
sufficient to describe the distribution completely, so the standard deviation
is of particular importance for our measurements. For 95 % of the
used data the standard deviation was between 0.67 and 12.67∘, with an
average of 5.26∘. For a slightly shorter period, measurements of
atmospheric stability were also possible. A histogram of the standard
deviations of the wind direction divided into the stability classes from this
period is presented in Sect. .
However, the dynamics of the wind direction are not the only uncertainty
factor in rotor alignment. In addition, there are inaccuracies in the
determination of the wind direction and the alignment of the turbine. Such
types of measurement errors are commonly assumed to be independent and
normally distributed . For a random variable that is based
on two or more independent distributions, its distribution can be determined
by the convolution of the individual distributions. In the case of normal
distributions, the convolution results again in a normal distribution, in
which the variances in the underlying distributions are added arithmetically.
For these reasons, a normal distribution is chosen for the probability
density function of the measured wind direction in the robust optimization,
which represents the assumed uncertainty and variability in the wind
direction. This is in accordance with , who used a range
of wind directions together with weightings corresponding to normal
distributions as model input for similar wake models to take into account the
variability of 10 min wind direction time series, and thus could improve the
agreement of model results with measurements.
Since the support of the normal distribution is unrestricted, we limit the
range to φ±2σ, in order to cover the majority (≈95.45%) of the occurring events. The advantage of using a normal
distribution is that the robustness of the optimization is governed by only
one variable, i.e. the chosen standard deviation σ of the
distribution, which we will refer to as the robustness parameter. For
completeness it should be mentioned that the wind direction
measurement by nacelle anemometry is commonly subject to a bias as well. Such an
offset has a clearly degrading effect on active yaw deflection control and
should be reduced by proper calibration of the wind vane. For this purpose
several practical procedures are available as for instance demonstrated by
and . For the analysis, we
assume in our analysis that a wind direction bias is negligible.
Results of the yaw angle optimizations
The solutions of the conventional and the robust optimization
are the optimal yaw angles of all the turbines for all wind directions. In
Fig. the results of the yaw schedule of only one turbine
(T32) is displayed for four cases. In the following, we will refer to this
turbine as the reference turbine. It is located in the centre of the
southernmost row of the wind farm and it is highlighted in red in
Fig. . Different robustness parameters, σ=0, 4, and 8∘,
were chosen and the results are compared to the baseline yaw
schedule, which is the case when there is no intentional yaw misalignment,
represented by the grey diagonal line. The black plot refers to the
conventional optimization corresponding to σ=0∘. The blue and
red plots demonstrate the results of the robust optimization with σ=4
and σ=8∘, respectively.
Optimized yaw angles of the centre turbine in the southernmost row
(T32) for three different robustness parameters σ=0, 4, and 8∘. In
addition, directions are marked and named accordingly, at which neighbouring
turbines are located downstream.
In Fig. it can be seen that the deviations from the baseline
become smaller with increasing robustness parameter. In the black plot
(σ=0∘), the yaw angles have relatively large deviations in the
wind sector from roughly 70 to 290∘. In seven distinct situations the
yaw angle rapidly changes from a positive to a negative misalignment, which
means that the yaw angle rotates contrary to the wind direction. This
increases the overall yawing activity of the turbine and it is generally
undesirable and should therefore be used with caution. These situations
correspond to the angles at which at least one turbine is in the full wake of
the reference turbine. In the remaining wind direction sector (290 to
70∘) there is no yaw misalignment, since in these situations the wake
of the reference turbine does not affect the other turbines in the wind farm.
In the blue plot (σ=4∘), the yaw misalignment is reduced
compared to the first case. This applies in particular for wind directions
where the downstream wind turbine is further away (e.g. at around
159∘ T11 and 201∘ T13). The number of fluctuations, where the
yaw misalignment rapidly changes from positive to negative, is reduced to
five. This correspond to the angles at which one of the directly neighbouring
turbines is in the full wake of the reference turbine.
In the red plot (σ=8∘), the deviations from the baseline
decrease further. Only at angles around 90 and 270∘, for which the
turbines affected by the wake of the reference turbine are the closest, the
yaw angle visibly rotates contrary to the wind direction. The other
fluctuations are smoothed out to plateau-like segments. At these plateaus
the reference turbine maintains a nearly constant yaw angle for a wind sector
of about ±7 to ±9∘ around the direction of maximum
interaction. In these special situations, the orientation of the reference
turbine points almost exactly to the adjacent downstream turbine. This
ensures that the wake is deflected to the correct direction (away from the
downstream turbine) without adjusting the yaw angle, even if the wind
direction changes within the specified sector. We hereinafter refer to this special case
as passive wake deflection. It will be further discussed in
Sect. .
We elaborate an exemplary case to better understand the impact of wind
directional variation and uncertainty on active wake deflection for the
different robustness parameters. For this purpose, we select an arbitrary
wind direction that is assumed to predominate at the moment and to which the
turbines adjust according to the respective optimization. We call this wind
direction the estimated wind direction φest. Then we
analyse how performance of the different optimized yaw settings changes when
the actual wind direction deviates from the estimated wind direction. As an
illustrative example we choose φest=172∘, which is a
situation where strong wake effects occur. The optimized yaw angles of the
turbines are determined by the robust optimization for the robustness
parameters σ=0, 4, and 8∘. The resulting yaw angles
of all turbines in the reference wind farm for this particular case are
displayed in Fig. .
Illustration of the yaw angles of the reference wind farm for an
estimated wind direction φest=172∘ according to the
different optimizations.
For a better comparison of the optimized yaw angles to the baseline, the yaw
angles according to the baseline are depicted in each of the four
illustrations in Fig. by a grey line.
Again, it can be seen that the deviation becomes smaller with increasing
robustness parameters. However, the results of the robust optimization with
σ=4 and 8∘ appear unexpected at first in regard to
two aspects.
Firstly, the yaw angles of the northernmost turbines (T11, T12, T13) slightly
deviate from the inflow direction, although there are no downstream turbines
to consider. The observed small positive offset is opposite to the negative
yaw misalignment of the other turbines. The reason for this is that the
robust optimization considers all inflow directions around the estimated wind
direction in its objective function. The turbines in the northernmost row
align themselves towards inflow directions from where less wake effects
occur, so they can produce more power in these situations. Consequently, the
power output is reduced for inflow direction in the opposite direction.
However, the relative power loss is smaller, since in this case the turbines
already produce less power due to the stronger wake effects.
Secondly, the optimized yaw angle of the T31 differs slightly from the
optimized yaw angle of the turbines T32 and T33. This is due to the location
of this turbine at the edge of the wind farm. If the wind would turn
anticlockwise, the wakes of the turbines T32 and T33 affect the rest of
the wind farm, e.g. T12. This is not the case with T31, so this turbine does
not have to take this into account and applies a larger yaw misalignment.
In order to further analyse the results of the optimizations, we look
in detail at the reference turbine T32. The set points for the yaw angle of
the reference turbine (T32) are displayed in Fig. , which
is an enlarged part of Fig. for angles around
φest.
Optimized yaw angles of the reference wind turbine for
φest.
The yaw angle of the reference turbine according to the conventional
optimization (σ=0∘) is γ=189∘, which equals an
intentional misalignment of 17∘. The robust optimization with σ=4∘ results in a yaw angle γ=186∘ and the robust
optimization with σ=8∘ gives a yaw angle γ=177∘.
For these settings, we can now calculate the power output of the reference
wind farm for different wind directions with the help of FLORIS. The
normalized power difference
Pdiff(φ)=Popt(φ)-Pbaseline(φ)maxφ(Pbaseline(φ))
for the robustness parameters σ=0, 4, and 8∘are displayed in
Fig. .
Normalized power gain of the conventional optimization (black) and
of the robust optimization (blue and red) compared to the baseline (grey) for
φest=172∘.
The graph illustrates the power gain of the robust optimizations and how it
is affected if the wind direction φ deviates from
φest. All three yaw settings achieve power increase close
to φest, which is the design point for the optimization in
this case. As expected, at the design point the conventional optimization
(σ=0∘) has the highest gain of the optimizations, but it also
has the largest drop further away from φest. While the
robust optimization with σ=4∘ is only slightly below the
conventional optimization at the design point, it performs better in the
outer regions (e.g. around 160 or 185∘). For the robust
optimization with σ=8∘ the maximum gain is lower, but the
losses in the outer regions (e.g. around 160 or 185∘) are strongly
reduced.
For wind directions above roughly 191∘ the power difference
Pdiff becomes positive again for all robustness parameters. This
is due to the fact that the turbines that follow the baseline (γj=φest for j=1,…,n) get large yaw misalignments and
thus the power output Pbaseline is significantly reduced. In
summary, the graphs show that for all three cases, a deviation of the wind
direction from the estimated wind direction can lead to significant power
losses, rather than power increases.
Evaluation of control algorithms
In this section, the following four time-dependent yaw control algorithms are
introduced and evaluated with the help of FLORIS on the basis of wind
direction measurements.
Greedy yaw control: this reference control is derived from the baseline, it refers to the situation
that every individual turbine tries to locally maximize its power output by yawing directly into the wind direction without
any intentional yaw misalignment. The term greedy control was introduced by
in the scope of
induction-based wind farm control.
Conventional wake deflection: this active wake deflection control scheme applies the yaw angles
calculated by the conventional optimization (σ=0∘). For a given φest
the yaw control algorithm uses the precalculated yaw angle of the conventional optimization in the form of a look-up table.
Robust wake deflection (σ=4∘): the open-loop control of the robust wake deflection
works in the same way as the conventional wake deflection, but this time the yaw angle settings of the
turbines from the robust optimization with σ=4∘ are used.
Robust wake deflection (σ=8∘): analogous to the case before, but σ=8∘.
In the next step we are extending our evaluation of the four abovementioned
control strategies based on the actual time series from the wind direction
measurements. Two test cases, A and B, are each analysed for three different
robustness parameters (σ=0, 4, 8∘). The evaluation process and
the test cases are described in the following and illustrated in
Fig. .
Evaluation process of the control schemes with wind direction
measurements. Red boxes mark the steps in which the measured data are used as
input.
First, the 1 Hz wind direction data φ was split up in 5 min
time series Φt∈R300, t=1,…,N, with N=35586,
in the same manner as it was done in Sect. . For test
case A the estimated wind direction is defined as the mean wind direction of
the time series φest=Φt‾, indicated by the
⋅‾-operator, and passed as input to the open-loop
control schemes. For test case B, an additional Gaussian error θ with
a standard deviation of 4∘ is added to the mean wind direction
φest=Φt‾+θ to simulate additional
inaccuracies like, for example, measurement uncertainties, yaw deviations
through the thresholds of the yaw control, and alignment errors. The output
of the optimized wake deflection are the optimized yaw angles,
γjopt(φest)=fj(φest), of
all turbines j=1,…,n. The function fj represents the yaw schedule of
the jth turbine according to Sect. . To
evaluate the success of the control strategy we compare it to the baseline,
which is the greedy yaw control. Therefore, the estimated wind direction is
also passed to the greedy yaw control scheme,
γjgreedy(φest)=φest, for
all j. Next, the output of both control schemes γj and the next
time series of wind direction Φt+1 are passed to FLORIS and the
average power outputs, Popt and Pgreedy, for the
5 min of wind direction data are computed for the optimized and the greedy
yaw control. This step simulates that the yaw mechanism of the turbines does
not constantly correct the yaw angle and can only react retroactively to
changes in the wind direction. Also, according to , the
wind direction distribution as model input gives a better agreement with
measured data. Therefore we use the empiric distribution for every time
series.
Test case A: evaluation of the relative power gain over the
estimated wind direction φest from the measured time series
of 5 min averaged wind direction during approx. 5 months. The
yellow plot marks the median of the power gain for the respective wind
direction and the dashed yellow plot above and below the median are the
upper and lower quartiles, respectively. (a) Conventional wake
deflection (black), (b) and (c) robust wake deflection for
σ=4 (blue) and σ=8∘ (red),
respectively.
Finally, the power output for the optimized wake deflection Popt
is compared to the power output of the greedy yaw control Pgreedy
and the results PoptPgreedy are displayed in
Fig. and Fig. for test case A and B,
respectively. Additionally, the average power gain Pgain=PoptPgreedy‾-1 is displayed in the
figures.
Figures and illustrate the relative
power change (y axis) over the estimated wind direction (x axis) in a
scatter plot, where one dot represents one 5 min time series Φt each.
As mentioned before, this evaluation has been carried out for the three
different active wake deflection controls with robustness parameters of
σ=0, 4, and 8∘. Therefore, both figures consist of three graphs
each.
Test case B: evaluation of the relative power gain over the
estimated wind direction φest from the measured time series
of 5 min averaged wind direction during approx. 5 months. The
yellow plot marks the median of the power gain for the respective wind
direction and the dashed yellow plot above and below the median are the
upper and lower quartiles, respectively. (a) Conventional wake
deflection (black), (b) and (c) robust wake deflection for
σ=4 (blue) and σ=8∘ (red),
respectively.
Starting with test case A, the upper graph of
Fig. a displays
the result of the conventional wake deflection. It can be seen that the
relative power change is highly scattered around the neutral value of 1. This
happens mainly for wind directions with strong wake effects. In these cases,
both relatively large power gains and losses occur. A surplus in performance
is achieved if the wake deflection works as intended, indicated by a value
above 1.0. A value below 1.0 means a power loss, which arises when the
fluctuations in the wind direction or inaccuracy in its determination are too
large. Overall, an average relative performance gain of 0.6% was
achieved in this test case.
In the middle graph the result of the robust wake deflection for σ=4∘ is presented in blue, while the result of the upper graph is
displayed in black in the background. In the comparison one can see that the
spread decreases slightly. Fewer extreme cases occur, both in terms of
performance increase and losses. Moreover, the centre of the distribution is
shifted towards a power increase resulting in a higher average relative
performance gain of 1.44%. Similar to before, the lower graph
illustrates the difference between the robust wake deflection with σ=8∘ (in red) to the conventional wake deflection (in black). The
scattering of the values decreases further here, while the average relative
performance gain is 1.39%.
Figure depicts the results of test case B in the same
manner as before. An additional Gaussian error was added to the estimated
wind direction in this case simulating measurement noise. As a consequence,
lower values are achieved on average for each wake deflection approach.
Although this is hard to see in the upper graph, the average relative
performance gain is reduced to -0.49%. This example proves that
the active wake deflection can fail its objective on average if uncertainties
are not taken properly into account in the control strategy.
The introduction of additional uncertainties also affects the robust wake
deflection, but the effects are not as strong as with the conventional wake
deflection. The robust wake deflection with σ=4∘, presented in
the middle graph still reaches a positive average relative performance gain
of 0.61%. As expected the robust wake deflection with the highest
robustness parameter (σ=8∘) proves to be less affected by the
given additional uncertainties. In the lower graph results have the smallest
change in performance and achieve the best average value of
1.05% in this scenario.
The achieved results of the robust control algorithms for the different
robustness parameters are summarized for test cases A and B in
Table .
Summarized power gains for test cases A and B and the different
robustness parameters.
Power gain
σ=0∘
σ=4∘
σ=8∘
Test case A
0.6 %
1.44 %
1.39 %
Test case B
-0.49 %
0.61 %
1.05 %
Discussion
The evaluation of the yaw angle optimization and of the associated yaw
control algorithms are based on real dynamic wind direction measurements, but
for the calculation of the wake losses and the power output, a simplified
steady-state wake model is used, which approximates the average wake flow. In
addition, we have limited our investigation to the partial load range, which
we consider to be the most important. In this case, we assumed a constant
thrust coefficient of the turbines for the analysis. Furthermore, we have
assumed that all uncertainties that occur can be estimated by a normal
distribution. This assumption proved to be sufficient in the evaluation, but
individual sources of uncertainty can still be further investigated. The
deviations of the mean values from successive wind direction time series
denoted by ΔΦt‾, for example, seem to be well described
by a t distribution (see Fig. ). This finding could be
used for the selection of the weightings in the optimization to improve the
optimization. We have decided against this at this point, since an important
aspect of this yaw control is its relative simplicity. By integrating a
further distribution (by convolution with the normal distribution) the
robustness could no longer be described by the robustness parameter alone.
Histogram of the changes of the mean values of successive wind
direction time series ΔΦt‾ with a fitted
t distribution.
Hence the work here is intended to serve as a proof of concept and as the basis
for further investigations. Following this research, qualitative
investigations based on LESs and free field experiments are
proposed. When LESs are used one should take care that they
properly reproduce real wind direction dynamics.
(a) Histogram of 5 min standard deviations of the wind
direction separated in the categories of atmospheric stabilities stable,
neutral and unstable. (b) Normalized histogram of the
data.
The relative power gain used here as a key performance indication must not be
confused with a pure increase in the annual energy production (AEP). For a
reliable estimation of the AEP, a time series of both wind speed and direction
of an average entire year together with the wind turbine power curve and
availability as well the wind farm layout have to be available. The direct use
of the commonly applied Weibull distribution of the mean wind speed would be
insufficient. Since in this paper we wanted to focus on comparing the
efficiency of the different control strategies in the partial load range, we
used the relative power increase instead of the AEP.
Exemplary illustration of the yaw angle set points of the reference
turbine T32 according to the passive wake deflection.
In this study, the robustness parameter was deliberately set to a fixed value
for the entire evaluation period in order to demonstrate its influence and
effects. A meaningful refinement of the algorithm would be to utilize a
variable robustness parameter and adapt it to the ambient conditions, e.g.
the mean wind speed, turbulence intensity, and atmospheric stability.
Observations and LESs indicate that
with a stable stratification and associated low turbulence intensity,
considerably lower wind direction changes occur. This is supported by
stability measurements during the measurement campaign for this evaluation.
In the period from 26 July to 22 November 2016, the Monin–Obuhkov length
(MOL) could be derived from measurements of a meteorological measuring
station at the location. The classification from MOL into stability classes
was done according to and the histogram of the 5 min
standard deviations of the wind directions divided into the stabilities
unstable, neutral, and stable is shown in
Fig. a. Figure b shows the respective
empirical probability for each bin.
The histogram shows two pronounced maxima. One by approx. 1∘ and the
second by approx. 5.25∘. It appears that the distribution consists of
two composite distributions. The first, with the focus around 1∘, is
dominated by measurements in stable atmospheric estimation and the second,
with the focus around 5.25∘, mainly consists of neutral ones. The
higher the standard deviation gets, the more likely it is to have unstable
stratification, as can be seen in Fig. a.
The presented results are potentially of significant importance for
implementing active wake deflection in the field. The simplicity of the
presented open-loop robust control algorithm makes it easy to integrate it
into a real yaw control system, which offers the possibility to obtain
further insights with the assistance of field campaigns. For this purpose,
the wind farm layout and the turbine characteristics must be known for the
calculations with the wake model; in addition, the global alignment of the
turbines should be as correct as possible and the wind measurements must be
relatively reliable. If these requirements are met, the optimized yaw
schedules can be calculated for the individual turbines and the robust wake
deflection can be used. In principle, the robust wake deflection is even a
decentralized control system, since each turbine follows its own optimized
yaw set points independently of the others. However, in practice a wind farm
regularly undergoes topology changes. This means that turbines change their
status and are switched off if necessary. In such a case, the optimized wake
deflection of at least the adjacent turbines should be deactivated for the
corresponding wind direction sector and the greedy control should be used. A
straightforward adjustment would be, for example, the switch to the greedy
control for these turbines in the respective wind sector.
Given the industry's interest in easy and robust solutions, a particular
implementation could be the so-called passive wake deflection. This means
that the yaw angle of an upstream turbine is set to a constant value for
certain wind direction sectors. “Passive” in this context refers to the
strongly reduced yaw activity in comparison to the large yaw amplitude in the
case of the conventional yaw angle optimization discussed in
Sect. . This idea is derived from the example
in Fig. and is further illustrated in
Fig. .
In this case, according to the robust optimization with σ=8∘,
the yaw angle of the reference turbine for a wind direction sector from
φest=170 to φest=188∘ remains
between γ=175 and γ=178∘. The adjustment of the
control would be to fix the yaw angle for such sectors to a constant value,
e.g. γ=176.5∘. Within such a sector, the orientation of
the turbine would be directed almost exactly to the next neighbouring downstream
turbine. The wake deflection is not particularly strong in such a case, but
the advantage is that the wake for all wind directions within this sector is
automatically deflected away from the downstream turbine, making the
application very reliable. Another benefit of such an implementation of the
robust control would be the possibility to significantly reduce the yawing activities of
the turbine by keeping the intended yaw misalignments
relatively small; this may be possible if integrated correctly into the existing yaw control.
This is in contrast to the increased yaw activity, which is associated with the
conventional wake deflection control (σ=0∘, see
Fig. ).
The consideration of the aerodynamic interactions in wind farm control has
some critical requirements that must be met as best as possible. This
includes the absolute orientation of the wind turbine and a bias in the
measurements. While the absolute orientation of the wind turbine is not
important for turbine control, it plays a decisive role in wind farm control,
as it is required to derive the aerodynamic interactions of the turbines. A
bias in wind direction measurement has negative implications for both wind
turbine and wind farm control. For this reason, the risk of a significant
bias needs to be minimized. Therefore, great care must be taken during
installation and alignment of the wind vane. If possible, additional
measuring instruments for determining the wind direction should be
considered, such as nacelle-mounted lidar or the consideration of blade loads
for the determination of the inflow as described in .