Dynamic robust active wake control

. Active Wake Control (AWC) is a strategy for operating wind farms in a way to maximize the overall power production and/or reduce structural loading on the wind turbines. Many recent studies indicate that this technology, and more specifically the so-called wake redirection approach to AWC, have a significant potential for increasing the annual energy production (AEP) by up to a few percentage points (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) percent. The current state-of-the-art approach is to optimize AWC for a range of static wind conditions, which is expected to perform sub-optimally in real-life due to the continuous variations of the wind 5 resource and the very slow yaw dynamics of the turbines. Recent work has addressed this variability in a robust design setting with the focus on maximizing the energy capture (robust AWC). This paper continues on this line of research, and develops a dynamic robust AWC strategy that aims to optimize the balance between maximum power production (requiring increased level of yawing) and minimum loads on the yaw drive (requiring limited yaw motion). It is shown with a (cid:58)(cid:58) To (cid:58)(cid:58)(cid:58)(cid:58) this (cid:58)(cid:58)(cid:58)(cid:58) end, (cid:58)(cid:58)(cid:58) an (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) uncertainty (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) quantification

e. Would be nice to clearly define the novel contributions in this article vs. what was done in previous work.FarmFlow already existed, but has been made dynamic: that is new, no? Uncertainty quantification is novel, at least for that exhaustive of a parameter set.Robust AWC and hysteresis already existed in literature, right?Response: Good point.As already mentioned in our response to 1 a), we added a heading "Contribution of this work" in the introduction.We have now also added a summary of the main contributions of the paper for more clarity: "In summary, the main contributions in this work are as follows: • Development of dynamic wake model, based on the originally static FarmFlow tool, suitable for design and evaluation of AWC solutions.• Exhaustive uncertainty quantification analysis, pinpointing the most important uncertainty contributors that need to be considered in a robust AWC design setting.
• Optimization of the parameters of a dynamic AWC algorithm using a wide range of dynamic simulations, with the purpose of maximizing the power gain and minimizing the yaw duty.• Design and evaluation of a dynamic robust AWC algorithm for a realistic case study with a full scale wind farm."

Minor comments
1.The abstract contains the general outline of the paper but misses the actual contributions and results.It currently does not suffice as a standalone summary of the paper.Please include the core findings, qualitatively but also quantitatively.For example, depict the parameters that were found to be the most important from the sensivity analysis, depict the potential AEP gain in percent, and so on.

Response:
We have extended the abstract as suggested, by including the following text: "To this end, an uncertainty quantification analysis has first been performed for a range of variables (wind speed, wind direction, yaw error, turbulence intensity, wind shear, air density, power curve, thrust curve, power loss coefficient due to yawed error), which indicated the wind direction, yaw error, turbulence intensity and the wind velocity as the highest uncertainty contributors.Robust AWC has next been synthesized by including stochastic uncertainties in these parameters.A stationary analysis through stochastic averaging indicated that the robust AWC design only slightly outperforms the nominal one in terms of power gain.For the dynamic design and analysis, the originally stationary FarmFlow wake model has been extended to enable dynamic simulations, including wake dynamics and a dynamic yaw control model.By selecting a certain dynamic adaptation algorithm structure (a low-pass filter, hysteresis, and sample and hold mechanism), a wide range of dynamic simulations has been performed to optimize its parameters for achieving the best balance between power gain and yaw duty.Dynamic simulations for a realistic case study with a full-scale wind farm indicated that the developed dynamic robust AWC results in a large reduction of the yaw duty (30-50% lower) while at the same time improving the overall power gain (2.05% vs. 0.56%), as compared to the conventional nominal AWC." 2. Generally, and especially when citing literature, you should clarify the test environment used in that publication.The differences between a FLORIS-based simulation study, a SOWFAbased simulation study, a field experiment or a wind tunnel experiment are very significant.
Response: Of course, and we believe we have done that in many places in the original manuscript, such as on Line 55 "…in recent field studies with wake redirection (Fleming et.al., 2020(Fleming et.al., , 2019)", Lines 67-69 "Above-mentioned studies on robust AWC were all performed using a simplified control-oriented wake model, namely the FLOw Redirection and Induction in Steady State (FLORIS) model -an understandable choice given the computational requirements for robust optimization.",Lines 70-71 "…utilizing a different steady-state wake model, called the lifting line model.",Line 83 "In Smiley et.al (2020), for instance, dynamic simulations have been performed using the stationary FLORIS wake model", etc.It is unclear to us what the point of this comment is.
3. For literature review: similar work is from M. Sinner et al., 2021, but this only appeared in April 2021.I can understand that the authors had already finished this publication mostly by then.You could consider including it in a revision."Power increases using wind direction spatial filtering for wind farm control: Evaluation using FLORIS, modified for dynamic settings", Sinner et al., 2021, JRSE Response: Thanks for pointing us out to this relevant recent publication, we have of course included a citation in the revised manuscript through the following text in the introduction: "Combining these techniques with wake steering was considered recently in the work of Sinner et al. (2021) using a modified FLORIS model." 4. Line 96: "This analysis ... the wind velocity."This is a conclusion and should not be part of the introduction.Rather, the introduction should be limited to what topics will be addressed in the article.The same goes for the sentence starting at line 99: "A stationary analysis ... of power gain."Nice, but should go to conclusion.Response: Agreed.The sentence on Line 96 of the original manuscript has been removed, and the next sentence modified to: "Based on the results from this analysis, robust yaw misalignment set-points have been optimized with respect to the most significant uncertainty sources, modelled as independent stochastic processes with selected PDFs." The sentence commencing at Line 99 has been modified to: "A stationary analysis based on stochastic averaging has been carried out to evaluate the performance of the robust AWC design as compared to the nominal one in terms of power gain." 5. Line 182: "wake generated by a wind turbine is propagated downstream based on the local wind direction variations in its way", I do not understand this.

Response:
The sentence has been rephrased as follows: "To this end, the wake generated by a wind turbine is propagated downstream in such a way that it follows on its way the local wind direction variations in the wind field.This way, both time delay and meandering effects are modelled." 6. Line 183: "because the traveling time ... current time window."I do not understand this.
Response: This sentence has been rephrased as follows: "Since the travel time of a wake between two turbines takes longer than the simulation sample time, …" 7. Line 202: "written in an output file", this seems inefficient.Can this not directly be exchanged through memory or over a network protocol?Response: This certainly can, and might be implemented in future updates of the software tool.
8. Line 260: On what signal does the LP filter work?Response: The following clarification was added to the LP filter description: "As visualized in Fig. 1, the LP filter acts on the wind speed and wind direction signals." 9. Line 314: You mention that the PDF for turbulence intensity is based on historical data.Does your definition of TI (i.e., being the standard deviation in streamwise direction, match up with the definition in the data?I can imagine that the historical data considers the TI to include both streamwise and cross-stream turbulence. Response: Our definition of TI in the wake model and in the wind field generator is, in fact, pretty standard, and as such are strictly speaking not exactly matching the definition TI measurements based on 10 minute statistical met mast data (mean value and standard deviation of the measured wind velocity).However, notice that these data is used to construct a rough, though realistic, statistical model of the turbulence intensity variations, which serves primarily as an example to based the robust analysis and design on.We do not believe that tinkering around the edges here will improve the value of the paper.
10. Line 349: What optimization algorithm is used?How confident are you that the solution has converged?What are the bounds, e.g., have you limited the minimum and maximum yaw angles? Response: The following text has been added in Section 3.2 to explain the optimization algorithm used in this study: "To solve the underlying optimization problems, a tailor made algorithm has been used that requires a minimum number of function evaluations (farm simulations) to converge.The algorithm is similar to the conventional bisection method, but generalized to multivariate objective functions.By confining the optimization variable to lie within an initial ndimensional box, the gradient of the objective function is evaluated at the centre point at each iteration and the box is reduced in size by keeping only that part that is oriented opposite to the gradient.While this algorithm has no theoretical guarantees to converge to an optimum solution for general nonlinear functions, has been successfully used for many years by the authors and works pretty well for the application at hand, its low calculation effort being its main advantage over alternative algorithms.This allows it to be used in combination with relatively complex wake models such as FarmFlow.To reduce computation time even more, the number of optimization variables is limited to the yaw setpoints of the two most upstream turbines in each row of turbines oriented downstream.The yaw set-points for the remaining turbines in the row are linearly decreased between the second turbine and the last one, which has zero yaw misalignment set-point.No limitation has been applied to the yaw set-points in this section."To clarify that this algorithm is used in all optimizations throughout the paper, the following like is added at the end of Section 3.3: "All optimization problems are solved by using the algorithm, described in Sect.3.2.The yaw misalignment angles have been limited to ±30 degrees." 11. Table 3: the wind speed range seems so high, while in reality you could feed in the wind speed measurements into the LUT, perhaps with an uncertainty bound but definitely smaller than an uncertainty of 8 m/s.How do you defend this decision?Also, how do these findings line up with your earlier work stating that wind speed can be ignored in yaw optimizations?Response: In fact, the idea is to not use the wind speeds as an argument for the LUT (other than for switching AWC on and off), but rather to have the LUT robust with respect to the whole range of wind speed variations.This is mentioned in the first paragraph of Section 3, as well as in the second bullet point in Section 3.1, where the modeling of the wind speed uncertainty is described.Nevertheless, to better clarify this, the following line is added to the mentioned bullet point: "Instead, the LUT will be designed to be robust to wind speed variations." 12. Line 381: I would have expected the yaw-induced power loss coefficient to have a larger effect on the optimal yaw angles, since it directly impacts the energy lost by yawing an upstream wind turbine.Can you reason why this is not the case in this study?Response: You are absolutely right.Large variations in the yaw-induced power loss coefficient result in significant variations in the optimal yaw set-points.Take, for instance the result for Case 2 in Figure 5, depicted by the middle (green) bar plot.The optimal yaw misalignment of the first turbine is around 22° for yaw-induced power loss coefficient of 2.3, and 35° for a coefficient of 1.3.Computing the corresponding power losses, cos()  , one gets 0.84 for =2.3 and =22°, and 0.77 for a =1.3 and =35°.The larger power loss at the first turbine in the later case is then compensated by the corresponding higher yaw misalignment, leading to increased power production downstream.So the results make perfect sense.Including the assumed distribution in Figure 4, however, narrows down the most probable range of variations of the yaw-induced power loss coefficient to below 1.7, leading to a quite small size of the green box in Figure 5.To summarize, the yaw-induced power loss coefficient does affect the optimal yaw set-points, but for the assumed uncertainty model (PDF) this impact remains limited and the parameter does not need to be considered in the robust optimization.
13. Figure 5: yaw angles of -40 deg and + 45 deg seem excessive.Can you explain your choice for allowing yaw angles to go all the way to these values?Since we would never optimize the yaw angles until those limits in practice, this may skew the sensitivity analysis somewhat, no?Perhaps certain parameters are important at high misalignment angles, but really are not that important in the range we expect to yaw the turbines to.
Response: This is, in fact, quite a good point, and not touching upon it is indeed an omission.The optimizations in Section 3.2 have been consciously performed without applying limitations on the yaw misalignment, as the exact limitations that will apply in future applications are not exactly known and the results from the uncertainty quantification analysis should be generic.However, as limitations will apply in practice, we agree it is important to discuss their impact.We have added the following paragraph to Section 3.2 to discuss this important aspect: "Notice that, due to the fact that no limitations have been imposed, the yaw misalignment set-points are getting quite high in some cases, raising to values of 40 degrees and even higher.Such a high yaw misalignments are currently considered unrealistic in a real-life application.Usually, they are limited to around 30 degrees in many research studies, or to even lower values in the first field tests with wake redirection (Flemming et al., 2017(Flemming et al., , 2019;;Doekemeijer et al., 2021).It becomes clear from Fig. 5 that by imposing a limitation of ±30 degrees one would significantly limit the variation in the yaw misalignment set-points.Nevertheless, the main conclusions drawn above in terms of the most significant uncertainty contributors will still hold, with probably only the wind shear disappearing from this list."Response: This has to do with the nonlinearily of the objective function, the chosen optimization algorithm, and the termination criterion, indeed.It is, of course, possible to improve the optimizer to get smoother curves, but that would require increasing the calculation time significantly.We believe, however, that this is not really necessary since resulting power gain is not very sensitive on such relatively small variations of the yaw misalignment angles.This fact can be appreciated from Figure 12, noticing that the power gain barely changes between nominal and robust AWC.Smoothing of the yaw set-points can be easily done after the optimization, which does save a lot of calculation effort, especially when designing AWC for the complete spectrum of wind conditions in a full scale wind farm.Robust design does, however, deliver smoother yaw angles in general.
18. Figure 9: "robust AWC" and "nominal AWC (with uncertainty)" are not the same thing, yet it is hard to distinguish them in their definitions.Can you clarify?Response: We have added the following text at the end of Section 5.1 to clarify the different curves in the figure (now Figure 12 in the revised manuscript due to the change explained in the Response to Major comment 1h): "For the sake of clarity, the difference between the red curve "nominal AWC (no uncert.)"and the black curve "nominal AWC (with uncert.)" in Fig. 12 is that the former one depicts the power gain evaluated just for the nominal values of the uncertainty parameters (^), while the later one represents the gain evaluated by including the whole uncertainty set  through the joint PDF _ ().In both cases, the yaw misalignment setpoints are the same, namely _, optimized for the nominal values of the parameters ^, i.e. neglecting the uncertainty, as defined in Eq. (4).The blue curve "robust AWC" corresponds to the case when both the optimization of the yaw set-points and the evaluation of the power gain are performed by accounting for the uncertainty through _ ()." 19. Figure 11: 11% Energy gain is very substantial and not particularly realistic for AEP.Maybe repeat that this is for particular 3-turbine case.Also, these figures are hard to see.I would suggest turning them into top-view (2D) contour plots instead.Same goes for Figure 12.
Response: The 11% gain is not for AEP but for the performed simulation conditions only, mean wind velocity of 8 m/s and wind direction varying around 270 degrees.11% power gain is not at all unusual in such a scenario, as you would agree.We added a clarification about the simulation conditions in Section 4 as follows "The average wind velocity is 8 ms -1 , and the wind direction varies around 270° (see light grey line in Fig. 10)."20.Line 555: Just to clarify, so dynamic FarmFlow runs 1:1 (6 hours of simulation means 6 hours of computing in real time on a single core)?If so, it may be worth evaluating the potential for a full year of operation (~9,000 CPU hours).
Response: Correct.A full year can easily be calculated on a computer cluster.
• This sentence: "This Kaimal spectrum is used for frequencies above 10−3 150 Hz, i.e. time scales of 30 minutes and slower".If the range is above a frequency, do you mean lower and not slower?Response: Yes, slower is now changed to lower.
• Page 6: "The parameter c(αrs) is the decay factor".Decay of what?Response: We have added the following clarification regarding this parameter: "The parameter c(αrs) is the decay factor, a parameter characterizing the decay rate of the coherence function." • Page 10: Recommend to explain figure 2 in more detail in the caption Response: We modified the caption of the figure as follows to make it more explanatory: "Illustration of the impact of wake on wind measurements and yaw motion of two commercial wind turbines, when the second turbine (T2) is in the wake of the first one (T1).The thin lines represent LP filtered wind direction measurements at the two turbines, while the think ones -the nacelle position measurements." • Page 13: Don't need to revise the paper, but wanted to note I think some recent papers might point to a distribution for yaw loss exponent centered somewhat higher, or even dependent on wind speed: "Related to that is the work of Annoni et al. (2019) focused on constructing consensus wind direction estimates." • Page 15. "Variations in the thrust curve and the yaw-induced power loss exponent have generally limited impact on the optimal yaw set-points, which suggests that they could be left out from the robust optimization."This is surprising, at least for the power curve exponent, it would seem that at some loss level it would start to have a strong impact,?Response: You are absolutely right.Large variations in the yaw-induced power loss coefficient result in significant variations in the optimal yaw set-points.Take, for instance the result for Case 2 in Figure 5, depicted by the middle (green) bar plot.The optimal yaw misalignment of the first turbine is around 22° for yaw-induced power loss coefficient of 2.3, and 35° for a coefficient of 1.3.Computing the corresponding power losses, cos()  , one gets 0.84 for =2.3 and =22°, and 0.77 for a =1.3 and =35°.The larger power loss at the first turbine in the later case is then compensated by the corresponding higher yaw misalignment, leading to increased power production downstream.So the results make perfect sense.Including the assumed distribution in Figure 4, however, narrows down the most probable range of variations of the yaw-induced power loss coefficient to below 1.7, leading to a quite small size of the green box in Figure 5.To summarize, the yaw-induced power loss coefficient does affect the optimal yaw set-points, but for the assumed uncertainty model (PDF) this impact remains limited and the parameter does not need to be considered in the robust optimization.
• Page 19: "The yaw set-points for the remaining turbines in the row are linearly decreased between the second turbine and the last one, which has zero yaw misalignment set-point."This is a great idea!Is this novel to this paper or has it the concept been used elsewhere?
Response: Well, it's not really new, this is an approach we use for years to reduce the computational time for the optimization.It is first mentioned in our work: Kanev, S.; Savenije, F. & Engels, W. Active wake control: an approach to optimize the lifetime operation of wind farms, Wind Energy, 2018, 21, 488-501 • Page 21: Metrics are really useful, the power gain per unit yaw travel increase is very interesting, is this also a novelty of this paper or something used in other papers or other contexts Response: Well, we haven't seen these metrics in other publications, but they are probably also not too difficult to invent when one tries to capture the wish to optimize the balance between yaw duty and power gain into a cost function.

Introduction
During the last decade, the field of active wake control (AWC) has been widely studied by many researchers worldwide.AWC is an approach to operate the wind turbines in a wind farm in a collaborative manner with the aim of reducing the negative effects of the wakes behind wind turbines on the overall power production and the individual wind turbines' structural loading (Andersson et al., 2021;Kheirabadi and Nagamune, 2019;Boersma et al., 2017).More specifically, the wake redirection 1 approach to AWC, employing intentional yaw misalignment to steer wakes away from downstream turbines, is currently considered as the most potential technology with respect to power production increase, with possible power gains of up to a few percent on annual basis (Kanev et al., 2018;Gebraad et al., 2017;Fleming et al., 2016). ::::: AWC :::::::::::::: implementation ::::::::: challenges At present, large scale implementation of wake redirection AWC is hampered by two main challenges.The first one concerns the increased risk perception due to the required operation under significant yaw misalignment, driven by the lack of comprehensive understanding of the impact on the structural loading on the turbines, often in combination with restrictive obligations in service contracts.The second challenge is related to the uncertainty in the predictions for the expected annual energy production (AEP) increase, caused by the simplistic static approach that is currently used to optimize AWC on the one side, and the underlying uncertainties in the modelling used for that purpose on the other side.Within the past few years, some initial results addressing these challenges have started to appear in the literature.With respect to the first one, for instance, most existing studies are limited to just one or two turbines (Fleming et al., 2013(Fleming et al., , 2015;;Croce et al., 2020;Zalkind and Pao, 2016).
A more detailed study involving a utility scale wind farm was presented in Kanev et al. (2020), where the impact of wake steering AWC on the structural loads of the turbines during their complete lifetime has been investigated using the so-called loads lookup table (LUT) approach (Reyes et al., 2020).The results demonstrate that, even though by itself yaw misalignment does increase the structural loads of some turbines in specific wind conditions, the wake-induced loading is decreased even more, so that the accumulated loads over the whole lifetime of each wind turbine generally remain lower than without AWC.
This conclusion is implicitly confirmed by the fact that the industry starts to develop this technology into commercial products (Siemens Gamesa Renewable Energy, 2019).
To sort out the second challenge, however, further research is required.The uncertainty in the AEP predictions is driven by a wide range of factors: the variability in the wind resource, wake meandering, uncertainties in the environmental conditions (atmospheric stability, turbulence, shear, veer, etc.), the slow dynamics of the yaw system, wake model uncertainties, measurement uncertainties, etc. Disregarding these, as done in the current state-of-the-art approach to AWC design that relies on optimization under stationary conditions, gives rise to suboptimal performance in real-life, i.e. lower power gain or, possibly even loss of power.To ensure maximum power gain in the field, it is essential to move from the current static approach to dynamic robust AWC design that properly accounts for the mentioned variabilities and uncertainties.One way to achieve that is by means of a two-stage approach, where the term "robust" relates to the selection of static yaw misalignment set-points that maximize power production in the presence of uncertainties (either deterministic, or stochastic), while the "dynamic" part deals with the adaptation of the set-points based on the highly varying wind resource measurements aiming to optimize the balance between maximal power gain and minimal loading on the yaw drive.The importance of achieving both robustness and optimized yaw dynamics has become apparent in recent field studies with wake redirection (Fleming et al., 2020(Fleming et al., , 2019)). :::: State :: of :::: the ::: art The topic of robust AWC design has already attracted the attention of some researchers.Quick et al. (2017) considered yaw angle uncertainty into the optimization for the yaw misalignment set-points.The uncertainty was modelled in a stochastic setting assuming Gaussian probability density function (PDF).This work was followed by Rott et al. (2018), where robustness with respect to wind direction variability and wind direction measurement uncertainty was considered, both modelled through a single PDF.Later on, in 2020, two relevant publications appeared.In the first one, Simley et al. (2020) presented results on robust AWC optimization including uncertainty in both the wind direction and the yaw position.They evaluated the performance of the robust AWC in case studies with different wind turbine spacings and turbulence intensities.The second work was published by Quick et al. (2020), and considers robustness with respect to a range of uncertain parameters, namely wind speed, wind directions, turbulence intensity, wind shear and yaw angle.The authors used a polynomial chaos expansion approach to deal with the underlying high computational complexity of the resulting optimization, and demonstrated that the wind direction is the most significant contributor to uncertainty in the power predictions.Above-mentioned studies on robust AWC were all performed using a simplified control-oriented wake model, namely the FLOw Redirection and Induction in Steady State (FLORIS) model -an understandable choice given the computational requirements for robust optimization.Finally, Howland (2021) studied AWC robustness with respect to model parameter uncertainty and wind direction uncertainty, utilizing a different steady-state wake model, called the lifting line model.The author uses an Ensemble Kalman filter to estimate the relevant wake model parameters and the corresponding probability distributions.In the simulation setup considered there it appeared that including model uncertainty gives rise to a more significant improvement than wind direction uncertainty.It should also be pointed out that, as an alternative to the robust design of yaw offset set-points that are valid for a range of conditions, one could consider adaptive solutions in which the yaw set-points are updated in real-life based on (estimates of) the actual operating conditions.Such an approach is pursued in Howland et al. (2020), where a gradient ascent algorithm updates the yaw offsets at each iteration based on analytically derived gradients for the lifting line wake model, the parameters of which are estimated online.
With respect to dynamic AWC, there is only very limited research published so far.There are, however, numerous publications in which dynamic (or quasi-static) simulations are performed with AWC, that include the yaw system dynamics and variations is the wind conditions.However, the parameters of the AWC adaptation algorithm in these studies, typically a low pass filter acting on the wind direction and wind speed, are not optimized with respect to the power gain and/or the yaw duty.
In Simley et al. (2020), for instance, dynamic simulations have been performed using the stationary FLORIS wake model by propagating the low-frequency components of the wind through the model (neglecting spatiotemporal inflow variations), and adding high-frequency components on top of those at turbine level to feed the yaw system model.In terms of dynamics, a more realistic wake model is used by Bossanyi (2018) that includes turbine and wake dynamic effects and utilizes a stochastic wind field correlated across the wind farm.The dynamic model has been used to simulate AWC and evaluate the impact on farm power and turbine loads.However, even though the authors state to have done a few iterations in choosing parameters of the dynamic AWC algorithm that appear to work reasonably, the presented study does not extend to the point of optimizing the AWC parameters with respect to energy production and yaw duty.In a different work, the same author demonstrates that a centralized yaw control strategy, in which information from surrounding wind turbines is used in the yaw control algorithm, can lead to a drastic reduction in the yaw duty and increase the power capture at the same time (Bossanyi, 2019).Even though AWC is not considered in that study, it is mentioned by the author that the proposed yaw control strategy would be beneficial in AWC as well :::::: Related :: to ::: that :: is ::: the :::: work ::: of ::::::::::::::::::::::: Annoni et al. ( 2019) focused :: on ::::::::::: constructing :::::::: consensus ::::: wind ::::::: direction ::::::::: estimates.
FLORIS ::::: model.Finally, in Kanev (2020) some initial results with optimizing the parameters of a dynamic AWC have been presented.Although the findings there clearly support the necessity of properly optimizing the dynamics of the AWC algorithm, the conclusions there remain of limited value due to the simplified modelling approach employed.More specifically, the stationary FLORIS model was used there, extended with a simple time delay model representing wake dynamics.As such, this model is quite unrealistic as it completely neglects spacial inflow variations.Also missing in the modelling approach there is the impact of the increased turbulence in front of waked turbines on their wind measurements, and the resulting increased turbine yawing at downstream turbines. ::::::::::: Contribution :: of :::: this ::::: work The present work extends on above mentioned research in focusing on dynamic robust AWC.The first part of the study concerns robust design and analysis using stationary simulations with FarmFlow, a 3D parabolized Reynolds-averaged Navier Stokes code with prescribed pressure gradients and k − ϵ turbulence model (Bot and Kanev, 2020).The starting point is the selection of varying or uncertain quantities that can affect the performance of the AWC.To this end, an uncertainty quantification analysis has been performed for a range of variables (wind speed, wind direction, yaw error, turbulence intensity, wind shear, air density, power curve, thrust curve, power loss coefficient due to yawed error).This analysis indicated that the highest uncertainty contributors are the wind direction, yaw error, turbulence intensity and the wind velocity.Subsequently, ::::: Based :: on ::: the :::::: results ::::: from ::: this ::::::: analysis, : robust yaw misalignment set-points have been optimized with respect to uncertainties in these parameters ::: the :::: most ::::::::: significant :::::::::: uncertainty :::::: sources, modelled as independent stochastic processes with selected PDFs.
With the robust AWC in place, the second part of this study continues with the dynamic design and analysis.To this end, the originally stationary FarmFlow wake model has been extended to enable dynamic simulations, including wake dynamics and a dynamic yaw control model.The dynamic simulation model is fed by a realistic wind field including temporal and spacial inflow variations, that include both micro-scale (fast turbulence variations ranging up to several hundreds of meters) and meso-scale (slow variations extending to ten kilometres and more), with corresponding coherence functions and cross power spectra that relate the stochastic properties between different points in space, and including terms to model the flow advection.The yaw dynamics are modelled at a faster sample rate than the wake model, and an additional higher frequency stochastic signal is added to the yaw error to model the increased noise in the yaw error measurements that enter into the yaw position controller.The size of this added noise is made dependent on the turbulence intensity in the flow in front of the turbine.
This gives rise to an increased yaw activity of downstream wind turbines, as seen in real-life measurements.Next, an AWC dynamic adaptation algorithm is considered, consisting of a low-pass (LP) filter, a hysteresis, and sample and hold mechanism, similar to (Kanev, 2020).Numerous dynamic simulations have been performed with different dynamic adaptation parameters, both with nominal and robust AWC yaw set-points.Based on the results from these simulations, the optimal parametrization of the dynamic AWC are determined.The resulting dynamic robust AWC is shown to deliver a large reduction in the yaw duty in combination with increase in the power gain as compared to the nominal AWC solution.
Sect. 3 outlines the the uncertainty quantification analysis, the selection of the dominant uncertainties, the design of robust AWC with respect to these and, finally, gives the results from a stationary robust analysis.Next, the optimization of the AWC dynamic adaptation algorithm is discussed in Sect. 4. Sect. 5 goes on with demonstrating the benefits from the developed dynamic robust AWC methodology on a case study with a model of an existing offshore wind farm.The manuscript is concluded with some final remarks in Sect.6. AWC ::: are ::::::::::: implemented :: in : a :::::::: dynamic ::: link :::::: library :::::: (DLL), ::::: which :: is ::::: called ::: by ::: the ::::::: dynamic ::::::::: FarmFlow :::: code :: at :::: each ::::::::: simulation :::: step. ::: The : different line colors are meant to indicate different sample times.The base sampling rate (black lines) is set by the wind field time series (typically 0.1Hz :: 0.1 ::: Hz : or slower).The yaw model operates at faster sampling rates (green lines) to enable realistic modelling of the yaw motion (e.g.1Hz : 1 ::: Hz), which is important to assess the impact of AWC on the yaw duty.The actual simulation model has a much more extensive interface between the wake model and the AWC algorithm enabling a wide range of possible future applications, but is visualized here in a simplified way, sufficient for the present discussion.Finally, part of the AWC algorithm may operate at slower sampling rates (red lines).The main components are explained :::::::: separately in more detail in the remainder of this section.
For the complex cross power spectrum between two points in space, r and s, the following expression is used for both the micro and the meso scale spectra (denoted below shortly as S) and is adopted from Sørensen et al. ( 2002) where d rs is the distance between the two points, U 0 is the average wind velocity, τ rs (U 0 ) = (cos(α rs )d rs )/U 0 is the time delay, i.e. the time it takes to move downstream from point r to point s at the average wind velocity U 0 , is the coherence function between the points r and s, and α rs denotes the angle between the line through the points and r and s and the wind velocity vector.The advection of the airflow downstream is modelled by the exponent term in Eq. ( 1).
The generation of time series from auto and cross power spectra follows the standard approach of generating frequency domain signals with amplitudes complying with the specified spectra and random phases, and applying inverse fast Fourier transform to construct the time series (Veers, 1988).

Dynamic wake model
In this work a dynamic wake model is developed based on TNO's wake model FarmFlow (Bot and Kanev, 2020) wake : between two turbines takes much longer than the time window of a simulation period ::::::::: simulation :::::: sample :::: time, the arriving wakes from upstream turbines need to be time synchronized with the departure of wakes in the current time window.Because the streamlines of the two-dimensional wind fields are curved and are varying in time, the trajectory of each wake needs to be corrected at the location of arrival, based on the trajectory of the undisturbed flow.After the correction, the wakes are stored in memory including the time and location of arrival.In summary, the quasi-dynamic wind farm simulation is realized as follows: 1.A simulation for the current time window of the wind field starts with the most upstream turbine and ends with the most downstream turbine, as seen from the average wind direction.
2. Before the wake simulation for a turbine starts, arriving upstream wakes at the current time instant are first determined, if any, using the wake information stored at previous time instances (Step 5).
3. From the undisturbed wind field and arriving wakes, the rotor averaged wind speed and wind direction is calculated.
4. Given the determined rotor averaged wind speed and the nacelle direction (coming from the yaw system model, see Fig. 1), the yaw misalignment angle is computed, the power and thrust values of the turbine are determined and a static wake calculation using the (stationary) wake model in FarmFlow is started for the given turbine only.
5. The wake is calculated until it hits a downstream turbine, if any, in which process the precise location of the wake is corrected for the time varying wind direction from the undisturbed wind field.The wake information is then stored in memory, including the time of arrival at the downstream turbine for use in the wake simulation of that turbine later on (performed in step 2).
6.The simulation continues with the next upstream turbine until all turbines are finished within the current time instant.
7. The input and output data for all turbines are written in an output file, which forms the interface to a dynamic link library (DLL) that implements the yaw system dynamics and the dynamic robust AWC algorithm.
8. The DLL is called, which updates the yaw position for each turbine and communicates this information back to Farm-Flow, and the next simulation step is started (step 1).
This process is repeated until the simulations for all time periods are finished.

Yaw model
Similarly to Bossanyi (2018), the yaw system model is simplified to a constant rate motion at 0.35 and a simple yaw controller activating the yawing motion when the LP filtered difference between the yaw misalignment and its set-point exceeds 8 • .A second order Buttherworth LP filter with cut-off frequency of 1/60Hz :::: 1/60 ::: Hz : has been used.As depicted in Fig. 1, the outputs of the yaw model are the nacelle position and the yaw duty (yaw travel and number of yaw on-off events).
The inputs are the yaw misalignment set-point and the measured yaw misalignment.The later is constructed as the difference between the measured local wind direction (coming from the wake model) and the measured nacelle direction.The measured values are formed by perturbing the actual quantities with uncertainties, i.e. terms representing variability of the measurand, measurement noise and uncertainty.These uncertain terms can be either stochastic or deterministic (e.g.measurement bias).
As pointed out already, the yaw system model operates at faster sampling rates than the wake model.This allows to model the yaw dynamics properly to enable realistic assessment of the impact of AWC on the yaw duty.To account for the higher variability of wind direction measurements that are taken at higher sampling frequencies, additional noise is superimposed on the wind directions coming from the wake model.Besides the higher sampling frequency, there is another source of increased noise on the wind directions, namely the turbulence intensity of the air flow impinging the turbine.Operating in the wake of other turbines, a wind turbine will measure an increased level of wind direction variations.These, together with the additional noise due to higher sampling frequency, are represented by the block named "TI-driven variations" in Fig. 1.The additional noise signal, added by this block, is generated based on the standard deviation of the wind velocity in front of the wind turbine.
More specifically, denoting φ(t) = tan v(t) u(t) as the angle between the longitudinal (u(t)) and lateral (v(t)) components of the wind vector, representing variations of the wind direction around its average value ϕ 0 , it can be shown using the Taylor series approximation of degree one around the point This implies that the standard deviation of the wind direction ϕ(t) can be expressed as σ W D = σ v /u 0 , where σ v is the standard deviation of the lateral wind component.In free stream, IEC 61400-1-3:2005 (2005) recommends for σ v the expression σ v = 0.8σ u , which allows one to approximate the standard deviation of the wind direction to the turbulence intensity, namely Assuming that this expression is representative for both the ambient flow as well as the wake, one can write σ W D,amb = 0.8T I amb for the ambient wind direction, and σ W D,wake = 0.8T I wake for that in the wake.In the simulation model, the ambient turbulence intensity, T I amb , comes from the wind field generator, while the turbulence intensity of the wake in front of a wind turbine, T I wake , is calculated by the wake model FarmFlow.The purpose of the block "TI-driven variations" in Fig. 1 is to add additional noise to the higher frequency wind directions used in the yaw system model to obtain σ W D,wake = 0.8T I wake .
This additional noise is generated using the Kaimal spectrum with standard deviation σ W D,add according to the following  This results in increased variations of the measured wind direction at turbines operating in wake condition, which in turn gives rise to increased yaw motion.This fact is observed in real-life as well, as supported by the data presented in Table 1.These data are obtained using high frequency measurements of wind speed, wind direction and nacelle position on two commercial wind turbines in the 2.5MW range, located on flat terrain at a distance of around four rotor diameters.The turbines operate both in free stream for Southern winds, while the second turbine (T2) operates in the wake of the first one (T1) for Western winds.The last row in the table shows that, in wake, T2 experiences lower wind velocity and higher turbulence intensity than the free stream turbine T1, as expected.It also shows that T2 measures increased variation in the measured wind direction and, unsurprisingly, higher yaw duty.This can be observed in the time series of LP filtered wind direction and raw nacelle position measurements for the two turbines in Fig. 2, which is for the Western wind situation.For Southern winds the responses are comparable (not shown in the Figure ).These results support the concept of modelling increased wind direction variability and yaw activity for waked turbines.Notice that the developed model cannot be easily validated by only using such SCADA data, since the reported turbulence intensities in the table are computed from measurements disturbed by the rotor.Nevertheless, the model produces comparable results in terms of yaw motion and is considered sufficient for the purpose of this work.

Uncertainty modelling
The uncertainties considered in this work are listed in Table 2, together with their type, assumed range and the PDF ::::: PDFs used for their modelling.For instance, the ::: The ::::: latter ::: are :::::::: described :: in ::: the ::: list :::::: below: -:: the : wind direction is required in the AWC algorithm as input, and since it is derived from the measured nacelle direction and yaw error, it will be subjected to measurement uncertainty.Next to that, the actual wind direction will vary with respect to the one entering the LUT, denoted in the table as ϕ LU T , due to the applied signal processing in the AWC algorithm (see Fig. 1).The table indicates that the measurement uncertainty and variability are modelled together with a normal distribution with PDF -The PDF parametrizations for yaw error and wind shear exponent have been adopted from Quick et al. (2020).The Laplace distribution, assumed for the yaw error uncertainty is defined as -The uncertainty on the air density has been based on the results in Ulazia et al. (2019), where a study is carried out on the variability of the air density offshore.The statistical indicators, reported there, are a first quartile value of 1.21 and third quartile value of 1.25.The resulting interquartile range of IQR = 0.02 is then used here to determine the standard deviation for which the normal distribution exhibits the same IQR, i.e. σ = IQR/1.35≈ 0.015.
-The parameter denoted as "yawed power loss exp." in Table 2 refers to the exponent a in the power reduction factor cos(β) a by which the power production of a non-yawed turbine is scaled to model the production of a turbine operating at yaw misalignment angle of β.There is a large variety of values for a in the literature, ranging between 1.3 and 2.3.In Fleming et al. (2017) a value ot 1.44 has been fitted to field measurements.Similar value has been derived for another commercial wind turbine, used in the study Doekemeijer et al. (2021).For this reason, a skewed PDF is selected here, defined as Yaw induced power loss model coefficient [-] with highest probabilities clustered at the lower side of the range, see Fig. 4. The selected PDF is such that for p biG (x + 1, µ, λ) it becomes equivalent to the inverse Gaussian distribution, and is therefore referred to as "biased inverse Gaussian" in Table 2.The mode of the PDF, at 1.47, aligns quite well with mentioned field results.
-Finally, the uncertainties on the thrust and power curves are modelled relative changes (percentages) of their nominal curves, i.e. ±5% uncertainty range on the power curve, and ±10% on the thrust curve.The modification of the power curve is realized through scaling of the wind speed to ensure that rated power remains unchanged.Normal distributions are then assumed, centred at nominal values, and with standard deviations such that ±3σ coincide the selected uncertainty ranges.
Now that the possible sources of uncertainty are described and modelled, the next section continues with analysing the impact of these on the performance of the AWC algorithm.

Uncertainty quantification
The uncertainty quantification analysis has been carried out with the purpose to identify the uncertainties with the most significant impact on the performance of the AWC algorithm, measured in terms of optimal yaw misalignment set-points and power gain.The most dominant uncertainties are then to be considered in the robust AWC design framework, discussed in the next section.The reason to look for reduction in the number of uncertainties in the robust optimization is to lower the computational complexity to a manageable level.
The yaw misalignment angles are optimized for one uncertainty parameter at a time, keeping all other parameters at their nominal values.More precisely, let p i be a given parameter from the list in Table 2, U i be the corresponding uncertainty range, and D i (p i ) -its PDF.For a given sample p (r) i ∈ U i of the parameter p i , and keeping the remaining parameters at their nominal values (i.e.p j = p (nom) j ∈ U j , j ̸ = i), conventional (non-robust) AWC design solves the optimization problem of finding the vector of best yaw misalignment set-points γ = [γ 1 , . . ., γ N ], N being the number of turbines, with respect to the total power production of the wind farm, i.e.
Notice that each individual turbine's power production, P t , may depend on the yaw misalignments of other turbines through the wake effects.The optimal power gain for sample p (r) i is then defined as Table 3 gives for each parameter p i its uncertainty set U i , the selected step size in the uncertainty sampling (resulting in samples p (r) i , r = 1, 2, . . .for which the yaw misalignments are optimized), as well as the nominal values of the parameters (p (nom) j ).The resulting optimal yaw misalignment angles γ det (p   The following observations can be made from the figure -Variations in the power coefficient and the air density have no impact on the optimal yaw misalignment set-points, which is expected as these parameters have little to no influence on the wake deficits behind the turbines -Wind direction variability has by far the largest impact on the optimal yaw misalignment set-points which, of course, is due to their very pronounced influence on the wake locations with respect to downstream turbines.Clearly, this parameter is the most important one to consider in a robust AWC setting. -Other quantities, variations of which lead to significant changes in the optimal yaw set-points, are the wind speed, yaw error, turbulence intensity, and wind shear.Notice that for some turbulence intensity and wind speed cases the minimum yaw misalignment values are equal to zero, which occur for uncertainty samples around the edges of their ranges of variation.For measurable quantities (such as the wind speed), such values can better be excluded from the robust optimization when possible and be used instead to deactivate AWC.
The power gain sensitivity analysis is carried out as follows.For each parameter, its most probable sample (modal value), p , is selected and the yaw misalignment angles are optimized for that value, while keeping all other parameters at their nominal value (p j = p (nom) j , j ̸ = i), i.e.
For each parameter sample (p r i , r = 1, 2, . . .), the power ratio is then computed as By optimizing for one single uncertainty value (p (mod) i ), evaluating the power gain with the resulting (fixed) yaw set-points )) for the whole uncertainty range (δP f ix (p (r) i )), and comparing these to the optimized power gains for each single uncertainty sample separately (δP opt (p (r) i )), one gets insight into the maximum amount of power gain improvement achievable by means of robust AWC.If there is little to no difference between δP f ix and δP opt for a given parameter, then it can be left out from the robust design even if it has significant impact on the optimal yaw set-points.The difference (δP opt − δP f ix ) is depicted statistically with the box plots in Fig. 6.A value of 0.05 in the figure, indicates that for some uncertainty realization the optimal power gain δP opt is 5% higher the power ratio δP f ix for fixed yaw set-points.Therefore, values close to zero in the figure indicate that the uncertainty on the corresponding parameter is not important to include in the robust AWC design because that will not lead to a significant improvement in the power gain compared to the case when the AWC design is performed with the parameter kept at its modal value.This is the case for the first four parameters depicted in Fig. 6 (air density, yaw-induced power loss exponent, power curve, and wind shear), and to a lesser extend for the sixth one (thrust curve).
It is interesting to observe that wind shear can now be excluded from the robust optimization, while it passed the first relevance test based on the optimal yaw misalignment set-points.
In summary, the conclusion is that the uncertainty on the following parameters is to be considered in the robust AWC design framework: wind direction, wind speed, yaw error and turbulence intensity.

Robust design and stationary analysis
As discussed in the previous section, the robust AWC optimization will account for uncertainties in the wind direction, wind speed, yaw error and turbulence intensity, with the uncertainty modelled stochastically by means of PDFs (see Sect. 3.1).Then, for a given stationary wind direction ϕ LU T , the robust AWC design problem, will ideally require the solution to the following stochastic programming problem wherein the four element vector p represents the considered uncertain parameters, the set U defines their range of variation (in accordance with Table 2), and D -their joint PDF which, due to assumed dependency between the parameters, equals here the  (2020).This is done by selecting a number of discrete samples for each uncertain parameter and calculating the probability for a given sample through integration of the continuous PDF over the interval that corresponds to this sample (see Fig. 7).In this discretization process, an attempt has been made to limit the total number of samples as much as possible while still trying to reasonably approximate the PDFs.Since uncertainty on the wind direction was shown to have the largest impact on AWC, it was discretized using more points (five) than for the wind speed and turbulence intensity PDFs (for which, respectively, three and two points have been used).Due to the symmetry in the PDF for the yaw error, the number of points used there is also five, giving a total number of 150 (= 5 × 5 × 3 × 2) joint uncertainty samples.Due to the computational complexity of the FarmFlow wake model, with the simulations run on 100 cores in this study, it was beneficial to reduce the number of cases even further.This was done by removing all parameter combinations for which the joint cumulative probability distribution function is lower than 0.05, resulting in a final number of 121 samples.Denoting U d as the set containing these 121 uncertainty samples for the vector parameter p, and D d (p) as the discretized joint PDF, the initial robust AWC optimization problem in Eq. ( 2) is approximated as follows which constitutes the optimization problem considered in this work for synthesizing robust AWC.For solving the problem numerically, a modified pattern search optimization algorithm is used in which only decent directions are evaluated in order to save computational effort.For the same purpose, the number of optimization variables is limited to the yaw set-points of the two most upstream turbines in each downstream oriented row of turbines.The yaw set-points for the remaining turbines in the row are linearly decreased between the second turbine and the last one, which has zero yaw misalignment set-point.
To exemplify the robust AWC design, consider the single-row layout consisting of five 3 MW wind turbines, with 90 m rotor diameter, separated at a distance of 7D (equivalent to the setup in Cases 2 and 3 in :: All ::::::::::: optimization :::::::: problems ::: are :::::: solved :: by ::::: using ::: the ::::::::: algorithm, :::::::: described :: in : Sect.3.2).The robust optimization problem in Eq. is solved for wind directions ranging from 248°to 292°at a step of 1°, an interval centred around the row orientation of 270°.In addition to that, a nominal AWC optimization is performed for the nominal values of the uncertainties p (nom) ∈ U, nom) , ϕ LU T , while evaluating the robust power gain, δP (γ nom , ϕ LU T ), including the uncertainties.The robust power gain for given yaw misalignment set-points γ and wind direction ϕ LU T is computed using the discretized joint PDF The results are summarized in Fig. 11 and 12, depicting the first turbine's optimized yaw misalignment set-point and power gain for the complete turbine array, respectively, for the considered range of wind directions.The results show that the robust yaw misalignments exhibit lower maximum values, extend over a larger interval of wind directions and are smoother.This is expected to have a positive effect on the yaw duty, which will be evaluated with dynamic simulations in Sect. 4. In terms of robust power gain under robust AWC, δP rob (γ rob , ϕ LU T ) (blue curve in Fig. 12), as compared to a nominal AWC, δP rob (γ nom , ϕ LU T ) (black curve in 12), however, the improvement by robust AWC is quite limited .The red curve in Fig. 12 gives the nominal power gain (excluding uncertainties) under nominal AWC.Yaw misalignment angles for the first turbine in the row ::: The ::: yaw :::::::::::: misalignment ::::: angles ::::: have :::: been :::::: limited :: to :::::: ±30 • .
Power gain by nominal and robust AWC for a row of five turbines

Dynamic adaptation algorithm optimization
Now that the robust AWC optimization has been discussed in the previous section, the focus here is on the optimal selection of the parameters of the dynamics adaptation algorithm, described in Sect.2.4.The single-row layout considered in Sect.3.2 and 3.3 is considered sufficient for this purpose, while a more realistic assessment will be performed using a full-scale wind farm model in Sect. 5.For the simulations here, wind field time series have been generated using the approach in Sect.2.1.The wind field has a duration of 6 h and sample time of 10 s, and average turbulence intensity of 7%.The simulations have been carried out with different combinations of dynamic adaptation parameters (see Sect. 2.4) from the sets: -LP filter time constant: 20, 30, 45, 60, 120, 300, 600 s.Notice that, since the sample time is 10 s, time constant of 20 s implies no filtering at all as the filter cut-off frequency then coincides with the Nyquist frequency.
hysteresis size: 0,1,2,3,4 • The following key performance indicators (KIPs) have been evaluated for each simulation: energy gain: the relative increase of the wind farm energy production achieved by AWC average yaw travel: the amount of angular displacement travelled by all wind turbines' nacelles on the average worst-case yaw travel: the highest amount of angular displacement travelled by any nacelle average number of yaw events: the amount of start/stop yaw actions performed by all nacelles on the average Figure 9 provides comparison between the energy gains achieved by robust and nominal AWC for different adaptation parameters.The left-hand side plot corresponds to the situation with no hysteresis (DynP ars(a, 0, c)), while the right-hand side one is for the maximal considered hysteresis size (DynP ars(a, 4, c)).It can be observed that, without hysteresis (left plot), the energy gain by robust AWC is higher than that with nominal AWC, and the improvement remains consistent over the whole range of adaptation parameters considered.Interestingly, the energy gain slightly increases for higher LP filter time constants, which is probably due to the stochasticity in the simulated wind field: too fast an adaptation results in the yaw misalignments trying to follow local variations of the wind direction, giving suboptimal AWC performance in terms of energy gain.This effect is strongly reduced when the largest hysteresis is in place, as seen in the right-hand side plot in Fig. 9.The local wind direction variations falling within the ±4 • hysteresis zone will not appear in the yaw set-points any more, so any further LP filtering does not improve the energy gain.In fact, in that case a slight decrease in the energy gain is observed with increasing the LP filter time constant, which is attributed to the increased delay and decreased alertness of the yaw set-points to global wind direction changes.
Another interesting observation from Fig. 9 is that the case with large hysteresis gives rise to higher overall energy gain, while at the same time difference between nominal and robust AWC gets smaller.
Next, the KPIs related to yaw duty are discussed using the plots in Fig. 10, where they are expressed relatively with respect to the reference case without AWC.With respect to the four considered measures for yaw duty (yaw travel and number of start/stop events, each expressed either as an average or as a worst-case over the different turbines), the general conclusion can be drawn that increasing the hysteresis size significantly reduces the yaw duty, where huge reductions are observed for the cases with lower LP filter time constants and AWC sample times (i.e. for faster adaptation strategies).With the largest hysteresis sizes considered, the yaw duty is lowest and practically independent on the remaining adaptation parameters.This is a welcome result since, as explained above, these hysteresis sizes also turned out to improve the energy gain.Because of that, the power gain per unit yaw travel (or yaw event), depicted in the two plots on the bottom of Fig. 10, is highest for the largest considered hysteresis sizes.
With these adaptation parameters, the following results are achieved by the optimized dynamic robust AWC algorithm in terms of energy gain and yaw duty for the considered case study, all expressed relatively with respect to reference case (AWCfree): energy gain: relative increase of 12% in the wind farm energy production average yaw travel: relative increase of just 1.13 (i.e.13% increase), which is seems quite acceptable given the fact the AWC requires the nacelle to travel substantially between positive and negative offsets as the wind direction changes.
worst-case yaw travel: relative change of 0.95 (i.e. 5% reduction) in worst-case yaw travel.Since in the reference case it is the last turbine in the row that gets worst-case yaw travel in the simulation, this result shows that it remains higher than the yaw travel of the first four turbines even under dynamic robust AWC.(compare the solid black line with the dashed black line).For the remaining turbines (not plotted here) this effect is even more pronounced as they are yawed more often in the reference case.
worst-case number of yaw events: relative change of 0.67 (i.e.33% reduction) in worst-case number of yaw events.Altogether, it can be concluded that the results are rather positive with dynamic robust AWC achieving high energy gain and overall reduction in the number of yaw events, at quite limited negative impact on the average yaw travel.
The wind farm is located at a distance of around 14 km off the Dutch coast, and consists of 36 Vestas wind turbines of 3 MW each.The turbines are modelled in FarmFlow through their power and thrust curves (CERC, 2016).
Three dynamic simulations have been carried out, one without AWC (serving as reference case), one with nominal AWC and one with robust AWC.Each of these simulations took around 6 hours to complete on a single core.As an illustration, the local wind direction and the nacelle orientation for two selected wind turbines, those encircled in the layout plot in Fig. 13, are given in Fig. 14 for the three mentioned cases.Turbine T7 (left-hand side plot) is operating in free stream conditions for the simulated wind direction, while turbine T27 is in double wake situation and, hence, experiencing larger variations in the wind direction in accordance with the modelling described in Sect.2.3.Because of that, the nacelle of T27 makes more often excursions than that of T7 in the reference case without AWC (black lines in the figure), resulting in higher yaw duty.This becomes even more pronounced when looking at the yaw motion under nominal (blue curves) and robust AWC (red curves).
Finally, the results from the simulations have been evaluated in terms of the KPIs, defined in Sect. 4. The results are summarized in Table 4 for the three simulated cases.Besides the absolute values of the KPIs for the three scenarios, the table provides between brackets the relative increases with respect to the reference case without AWC.The following observations can be made from the table : -In terms of energy production, dynamic robust AWC improves significantly over the nominal AWC (2.05% vs 0.56% energy gain).Notice that the overall gains are lower than one might expect, which is to a large extend due to the somewhat irregular layout, especially in the lower right and upper left parts of the farm.
-The yaw travel, both average and worst-case, is significantly lower under robust AWC, which is partially due to the lower yaw misalignment set-points under this strategy as compared to nominal AWC.Compared to the reference case without AWC, the yaw travel under dynamic robust AWC is higher (factor 2.5-2.7),which is primarily due to the transitions between positive and negative misalignment.Notice that the reported values are only representative for the simulated wind conditions (with, on the average, wind direction aligned with turbine rows), and will be significantly lower on annual basis.
-The number of start/stop yaw events with the robust controller are also lower than with the nominal one.The increase with respect to the reference case is relatively low in this case (1.1-1.5), which on annual basis is expected to be even lower.

Conclusions
This paper considers the design of dynamic robust AWC that aims at optimizing the balance between the yaw duty and the power gain in realistic conditions, i.e. in the presence of wind resource variability and measurement and model uncertainty.The starting point was an uncertainty quantification analysis performed for the following variables: wind speed, wind direction, yaw error, turbulence intensity, wind shear, air density, power curve, thrust curve, power loss coefficient due to yawed error.This analysis indicated that the wind direction, yaw error, turbulence intensity and the wind velocity are the most important quantities to include in a robust AWC optimization.To this end, the variabilities and uncertainties are modelled as stochastic processes with corresponding PDFs, and the yaw misalignment set-point optimization is addressed as a stochastic programming problem through discretization of the probability distributions to arrive at a finite number of scenarios.A stationary analysis based on stochastic averaging using the PDFs of the uncertain parameters indicated that the robust AWC design slightly outperforms the nominal one in terms of power gain.
Subsequently, the design of the dynamic adaptation algorithm is considered, for the purpose of which the originally stationary FarmFlow wake model has been extended to enable dynamic simulations, including wake meandering and dynamic yaw control.The dynamic simulation model is fed by a realistic wind field including temporal and spacial inflow variations, with variations ranging from the micro-scale to meso-scale.Additional wind measurement noise is added for turbines operating in wake conditions, dependent on the local turbulence intensity, to model the increased yaw activity of downstream turbines, as observed in real-life measurements.Next, an AWC dynamic adaptation algorithm is considered, consisting of a low-pass filter, a hysteresis, and sample and hold mechanism.The parameters of these three building blocks have been optimized through a range of dynamic simulations with a five turbine array.It is shown that large-size hysteresis (±4 • ) in combination with a low-pass filter with 60 s time constant and 10 s AWC sample time achieve the best trade-off between power and yaw duty.
With these parameters, a reduction in the average number of yaw start/stop events of almost 50% and a power gain of 12% with respect to the reference scenario without AWC was achieved for the considered simulation.The yaw travel increased on the average by 13%, but its worst-case value over the turbines decreased by 5%.
Finally, the dynamic robust AWC approach is evaluated on a full-scale offshore wind farm model, for which both nominal and robust AWC controllers have been designed.The same (optimized) adaptation parameters have been used with both nominal

Figure 1 .
Figure 1.Block scheme of the simulation model
, a ParabolizedReynolds-averaged Navier-Stokes code with prescribed pressure gradients to calculate the flow in wind farms.Based on the rotor averaged wind speeds, the power and induced velocities are determined from measured power and thrust curves.The pressure gradients in the near wake region are prescribed as a function of the thrust force coefficient.To this end, a database is used containing precomputed pressure gradients obtained from a panel method with an actuator disk model in which the wake is represented by discrete constant strength vortex rings.The basic background flow is modelled by an atmospheric wind shear model based on Monin-Obukhov similarity theory.The original FarmFlow model solves stationary flow throughout the wind farm.For the development and analysis of dynamic AWC algorithms, FarmFlow has been extended to model a quasi-dynamic flow in two-dimensional wind fields at hub height (see Sect. 2.1) with spatiotemporal variations.To this end, the wake generated by a wind turbine is propagated downstream based on :: in :::: such : a :::: way :::: that : it ::::::: follows ::: on :: its :::: way the local wind direction variations on its way, thereby modelling : in ::: the ::::: wind :::: field.:::: This :::: way, : both time delay and meandering effects ::: are :::::::: modelled.The simulation time is equal to the time step in the wind field time series.Because the traveling time of the wakes ::::: Since ::: the ::::: travel :::: time :: of :: a ::::

Figure 3 .
Figure 3. Histogram of turbulence intensity computed using 20 months of wind measurements on an offshore met mast, and a Weibull PDF fit.Only data corresponding to wind speeds in the interval [4,12] ms −1 have been used in constructing the histogram

Figure 4 .
Figure 4. PDF used for representing model uncertainty in the yaw-induced power loss coefficient

Figure 5 .
Figure 5. Box plots for the optimal yaw misalignment angles of the most upstream turbine, evaluated for the considered parameters and setup cases

Figure 6 .
Figure 6.Box plots for the difference between the power gains obtained for the considered uncertainty samples with fixed (optimized for the model value of the uncertain parameter) and varying (optimized for each uncertainty sample individually) yaw misalignment set-points

Figure 7 .
Figure 7. PDFs for the uncertain parameters considered for robust optimization

-
figure, the blue and red plots give the results with maximum AWC sampling time (DynP ars(20, 0, 600)) and maximum LP filter time constant (DynP ars(600, 0, 10)).It can be concluded from the plots that the hysteresis size has the most pronounced impact on the yaw duty (red and dashed black lines to the left).Long AWC sampling time (blue line in the right plot) does reduce the yaw excursions, but also results in the yaw misalignment angles being held constant at possibly suboptimal values for long periods of time, which may detrimental for the power gain.Long LP filter time constant appears to result in some seemingly unnecessary yaw excursions during changes in the set-points.

Figure 8 .
Figure 8. Yaw motion with AWC under different dynamic adaptation parameters

Figure 9 .
Figure 9. Energy gain by dynamic robust AWC and dynamic nominal AWC without hysteresis (left) and with 4 • hysteresis (right)

Figure 10 .
Figure10.KPIs for the simulations with dynamic robust AWC: average and worst-case yaw travel, average and worst-case number of yaw start/stop events, and power gain per unity yaw travel and per unity yaw start/stop events

Figure 13 .
Figure13.Layout of the OWEZ wind farm with the rotors oriented towards the average wind direction in the simulation, and with two selected for scoping wind turbines encircled.

Figure 14 .
Figure 14.Wind direction and nacelle orientation for two selected wind turbines, T 7 and T 27, and three cases: without AWC, with dynamic nominal AWC and dynamic robust AWC.The sizes of the rotors are exaggerated by 50% for better readability of the figure.
This sentence is not precent in the revised manuscript.Instead, as explained in the Response to question 10 above, the description of the optimization algorithm has been explained in detail in Section 3.2.
17. Figure8: neither line is particularly smooth.Does this suggest that the optimization has not converged?

Table 1 .
Impact of wake on wind measurements and yaw motion of two commercial wind turbines

Table 3 .
Sampling of the uncertainties for the purpose of uncertainty quantification