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. ).

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 .

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. .

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.

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. 

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.

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.

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.

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.

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.

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. .

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.

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)max⁡φ(Pbaseline(φ)) for the robustness parameters σ=0, 4, and 8∘are displayed in Fig. .

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.

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.

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. .

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.

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.

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 .