Addressing deep array effects and impacts to wake steering with the cumulativecurl wake model
 National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, 80401, USA
 National Wind Technology Center, National Renewable Energy Laboratory, Golden, CO, 80401, USA
Abstract. Wind farm design and analysis heavily rely on computationally efficient engineering models that are evaluated many times to find an optimal solution. A recent article compared the stateoftheart Gausscurl hybrid (GCH) model to historical data of three offshore wind farms. Two points of model discrepancy were identified therein. The present article addresses those two concerns and presents the cumulativecurl (CC) model. Comparison of the CC model to highfidelity simulation data and historical data of three offshore wind farms confirms the improved accuracy of the CC model over the GCH model in situations with large wake losses and wake recovery over large interturbine distances. Additionally, the CC model performs comparably to the GCH model for single and fewerturbine wake interactions, which were already accurately modeled. Lastly, the CC model has been implemented in a vectorized form, greatly reducing the computation time for many wind conditions. The CC model now enables reliable simulation studies for both small and large offshore wind farms at a low computational cost, thereby making it an ideal candidate for wakesteering optimization and layout optimization.
Christopher J. Bay et al.
Status: closed

CC1: 'Comment on wes202217', Benoit Foloppe, 18 Mar 2022
Hello,
In equation (3), could you define y_{i }and z_{i }please ?
Best regards,
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer. AC1: 'Reply on CC1', Christopher Bay, 01 Sep 2022

CC2: 'Comment on wes202217', Blondel Frédéric, 28 Mar 2022
Dear authors,
I quickly went through the preprint (sorry if I missed some elements), and have some remarks:
 it is my understanding that equation (7) is only valid for a Gaussianshaped velocity deficit, but not for a superGaussian one. I tried to do the maths on my side and unfortunately ended up with an undetermined integral (I would be happy to share the first results if you want to, and eventually try to find a way around).
 there is something I do not understand: are you using a blending between the superGaussian and the Gaussian model, or the superGaussian model alone? In the first case, could you please provide the constants used for the Gaussian model? (as mentioned I went through the paper very quickly, maybe I just missed this part!)
 could you provide the model constants you have been using in the simulations? Are they taken from Cathelain et al. (https://halifp.archivesouvertes.fr/hal02995695)?
 in the paper, you show very nice comparisons against LES and SCADA data for different wind farms. However, in a recent publication from Lanzilao and Meyers (https://onlinelibrary.wiley.com/doi/full/10.1002/we.2669), the agreement between the superGaussian model and SCADA data is very poor. Do you have any idea what is going on? Is it due to the new superposition model? In such a case, maybe the comparison with the locallinearsum model would help analyze the impact of Majid's model.
 Could you indicate which wakeaddedturbulence model you have been using in these simulations?
Best,
Frédéric.
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.
AC2: 'Reply on CC2', Christopher Bay, 01 Sep 2022
Thank you for the comments.
1) The blending of the cumualtive wake model and the Blondel & Cathelain models in this work is done in a heuristic sense that provided improved wake prediction performance without a therotical derivation. I would love to discuss more about your efforts and work towards a theoretical derivation for this combination.
2) We are using a blending. For the values of the constants, we used the values given from your respective papers, shown below:
a_s: float = field(default=0.179367259)b_s: float = field(default=0.0118889215)c_s1: float = field(default=0.0563691592)c_s2: float = field(default=0.13290157)a_f: float = field(default=3.11)b_f: float = field(default=0.68)c_f: float = field(default=2.41)3) That is correct4) I do believe it is due to the implicit inclusion of the wake superposition in Majid's cumualtive wake model. We are planning a more indepth model comparision in future works.5) We are using the Crespo Hernandez wakeaddedturbulence model.Cheers,Chris

AC2: 'Reply on CC2', Christopher Bay, 01 Sep 2022

RC1: 'Comment on wes202217', Luca Lanzilao, 05 Apr 2022
In this article, the authors develop a new analytical windfarm flow model which accounts for larger wake losses deep in a wind farm and more persistent wake losses in the farwake region. The model, which is named the cumulativecurl model, is then extensively validated against LES results (from SOWFA) and SCADA data for three different offshore wind farms. Two main conclusions can be drawn from the validation campaign. First, the model compares reasonably well with numerical data and measurements in all cases considered. Second, the new model outperforms the GCH one in cases with large wake losses and wake recovery over large turbine distances while it performs similarly in the other cases. I believe that this paper is of interest to the wind energy community as it shows the potential of a new analytical flow model which partially solves a longstanding problem, that is the mismatch in predictions in case of deeparray effects. Moreover, I enjoyed reading this work, which is well presented and well structured. Here below, you can find some scientific questions and technical comments.
Scientific comments/questions:
 Abstract: it could be more descriptive. For instance, the authors write “Two points of model discrepancy were identified therein. The present article addresses those two concerns and presents the cumulativecurl (CC) model.” Which are these two concerns? They will become clear once reading the text, but I would find it useful to mention them here. Also, would it be possible to translate the “improved accuracy” or “greatly reducing the computational time” into percentage and speedup values?
 Section 2: I understand that it is more practical to cite others' works instead of rewriting the full model. Eventually, this also makes the article easier to read. However, I find it difficult to follow at times. For instance, the authors mention that the original cumulative wake model proposed in Bastankhah et al. (2021) does not include a nearwake model. However, in the current work, this deficiency is overcome by representing the nearwake region with a superGaussian wake model. How is this done? Is there an analytical derivation? I think that the reader would benefit from a more indepth description of the model, which could also be provided in the appendix.
 Section 3.1 (line 167): Which is the horizontal resolution used in the precursor simulation? Also, for how long the precursor simulations have been advanced in time?
 Section 3.1 (line 197): From where does this formula come? Consider adding a reference. Also, I would consider a different notation for the wind speed (ws can be seen as w*s in a mathematical expression).
 Section 3.1 (line 230): For how long the windfarm simulations are run in SOFWA? In the article, it is reported only the spinup time (1200 s), but not the time over which statistics are collected.
 Section 3.2: Over which region the velocity is spatially averaged to produce the power trends observed in Fig. 1? Is the heightdependent velocity profile given by the precursor simulations taken into account in the analytical models (GCH and CC)? Or do the analytical model assumes a uniform inflow velocity profile? If so, how is the velocity magnitude estimated? Does it refer to the velocity at hub height? This information can improve the interpretation of the validation results.
 Section 3.2 (line 243): The authors mention that “The TI assumed in the GCH and CC models was selected to yield perfect agreement with SOWFA at a 7D distance downstream.” I believe that if the TI value was tuned so that a zero error would occur at 5D or 10D, the validation would look different. Therefore, why not use the ambient TI reported in table 1 (i.e. the ambient TI values given by the precursor simulations)? Please, comment on this.
 Section 3.2.2: How is the ambient TI evaluated in this section? Are the authors using the ambient TI shown in table 1? I’m concerned about the ambient TI because usually, the analytical wake model predictions are strongly dependent on this value. Moreover, it is important to mention the ambient TI in case of the reader would like to reproduce some of the results. Note that the same question holds also for sections 3.3 and 3.4. Finally, it is not clear to me how the added TI is computed. Would it be possible to include this in the text?
 Figure 2: I find this plot difficult to read. Have you tried using different symbols instead of different lines for the various precursor cases? That is just a suggestion since it may make it worst.
 Figure 5: Why the GCH and CC firstrow turbine power is lower than the one predicted by SOFWA in all cases? This mismatch (although limited to a few percentage points) could lead to a bias in the measurements further downstream. Moreover, I would find it very interesting to include the prediction of, for instance, the Jensen model in the current figure. This could highlight how much better the models have become at matching LES results. However, I also understand that this could be out of the scope of the current manuscript.
 Figure 7: Which are the yaw angles applied at every turbine row? The authors mention that “This pattern assumes a large yaw misalignment angle for the first row of turbines, which then decreases linearly to zero for the last row of wind turbines.”. However, they do not provide the yaw misalignment for the first row of turbines. This information is necessary in case of the reader would like to reproduce the results.
 Section 4: Very nice and strong validation of the model. I have only a minor question here. In figure 15 (top panel), both models predict a lower energy ratio for a wind direction of 140 degrees than 320 degrees. Why is this happening? In fact, in the first case, turbine 22 operates in the wake of one turbine while in the second case it operates in the wake of six upwind turbines.
 I noticed that in many cases the caption of the figures also contains interpretations of the results. I would use the caption only for describing the figure, therefore moving the interpretation of the results in the main text.
Technical comments:
 Line 247: Typo “largely largely”
 line 320: rephrase the sentence

RC2: 'Comment on wes202217', Carl Shapiro, 06 May 2022
This paper gives adds the superGaussian model of Blondel and Cathelain and the cumulative wake superposition of Bastankhah et al. to the Gausscurl hybrid model to address issues with deep array effects. The comparisons to highfidelity models and field data are comprehensive. The paper is a worthwhile addition to the considerable research on wind farm wake modeling that is necessary for wind farm design and control.
Introduction: I enjoyed this clear discussion of the complications of wake modeling (including wake super position and near and far wake models) in various wind farm configurations. Two issues that could use some discussion are (1) momentum conserving models and linearized momentum conserving models (often called mass conserving models) and (2) the choice of wake expansion rate through turbulence characteristics.
Section 2.2: It is hard to decipher where each of these equations come from and how they have been modified in this implementation. Is there a consistent theoretical basis for adding the Blondel & Cathelain model and cumulative model of Bastankhah et al. to the GCH model? Or are the additions heuristic?
Section 2.2: This model has a large number of free parameters, which makes it more difficult to use. Could you discuss in more depth how these parameters are selected to make the model more widely useable.
All graphs: Please use vector formats for these images and use consistent font sizes. The resolution is fairly low and the font is sometimes hard to read. Use more descriptive titles without using underscores.
Figure 1: The improvement here is not as apparent to me as claimed in the text. I would have assumed that the superGaussian near wake model would improve the agreement in the near wake. In fact, the opposite seems to be the case. Furthermore, the choice to tune the results at x/D=7 affects the model accuracy. If the tuning had been done at x/D=3 the results might be quite different. A better approach would be to minimize the error over all measurements.
Section 3.2.2: Since these results are for a single turbine, they are only including the effect of including the Blondel & Cathelain model in the GCH model and the Bastankhah model does that have an impact, correct? Or am I misunderstanding that? I suggest adding some discussion on what aspects of the model are being tested here.
Section 4: The paper has a lot of great data for the comparison. While the graphs are very instructive to understand the differences between the models, it’s hard to compare the average error. Could you provide average error results for each of these graphs in a table?
Figure 10: What do the color plots on the right represent? There is no label and they are difficult to read.
Sections 3&4: I suggest changing the “Reflection” subsections to “Discussion."
 AC3: 'Response to Referee Comments', Christopher Bay, 01 Sep 2022
 AC4: 'Second Response to Referee Comments', Christopher Bay, 24 Nov 2022
Status: closed

CC1: 'Comment on wes202217', Benoit Foloppe, 18 Mar 2022
Hello,
In equation (3), could you define y_{i }and z_{i }please ?
Best regards,
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer. AC1: 'Reply on CC1', Christopher Bay, 01 Sep 2022

CC2: 'Comment on wes202217', Blondel Frédéric, 28 Mar 2022
Dear authors,
I quickly went through the preprint (sorry if I missed some elements), and have some remarks:
 it is my understanding that equation (7) is only valid for a Gaussianshaped velocity deficit, but not for a superGaussian one. I tried to do the maths on my side and unfortunately ended up with an undetermined integral (I would be happy to share the first results if you want to, and eventually try to find a way around).
 there is something I do not understand: are you using a blending between the superGaussian and the Gaussian model, or the superGaussian model alone? In the first case, could you please provide the constants used for the Gaussian model? (as mentioned I went through the paper very quickly, maybe I just missed this part!)
 could you provide the model constants you have been using in the simulations? Are they taken from Cathelain et al. (https://halifp.archivesouvertes.fr/hal02995695)?
 in the paper, you show very nice comparisons against LES and SCADA data for different wind farms. However, in a recent publication from Lanzilao and Meyers (https://onlinelibrary.wiley.com/doi/full/10.1002/we.2669), the agreement between the superGaussian model and SCADA data is very poor. Do you have any idea what is going on? Is it due to the new superposition model? In such a case, maybe the comparison with the locallinearsum model would help analyze the impact of Majid's model.
 Could you indicate which wakeaddedturbulence model you have been using in these simulations?
Best,
Frédéric.
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.
AC2: 'Reply on CC2', Christopher Bay, 01 Sep 2022
Thank you for the comments.
1) The blending of the cumualtive wake model and the Blondel & Cathelain models in this work is done in a heuristic sense that provided improved wake prediction performance without a therotical derivation. I would love to discuss more about your efforts and work towards a theoretical derivation for this combination.
2) We are using a blending. For the values of the constants, we used the values given from your respective papers, shown below:
a_s: float = field(default=0.179367259)b_s: float = field(default=0.0118889215)c_s1: float = field(default=0.0563691592)c_s2: float = field(default=0.13290157)a_f: float = field(default=3.11)b_f: float = field(default=0.68)c_f: float = field(default=2.41)3) That is correct4) I do believe it is due to the implicit inclusion of the wake superposition in Majid's cumualtive wake model. We are planning a more indepth model comparision in future works.5) We are using the Crespo Hernandez wakeaddedturbulence model.Cheers,Chris

AC2: 'Reply on CC2', Christopher Bay, 01 Sep 2022

RC1: 'Comment on wes202217', Luca Lanzilao, 05 Apr 2022
In this article, the authors develop a new analytical windfarm flow model which accounts for larger wake losses deep in a wind farm and more persistent wake losses in the farwake region. The model, which is named the cumulativecurl model, is then extensively validated against LES results (from SOWFA) and SCADA data for three different offshore wind farms. Two main conclusions can be drawn from the validation campaign. First, the model compares reasonably well with numerical data and measurements in all cases considered. Second, the new model outperforms the GCH one in cases with large wake losses and wake recovery over large turbine distances while it performs similarly in the other cases. I believe that this paper is of interest to the wind energy community as it shows the potential of a new analytical flow model which partially solves a longstanding problem, that is the mismatch in predictions in case of deeparray effects. Moreover, I enjoyed reading this work, which is well presented and well structured. Here below, you can find some scientific questions and technical comments.
Scientific comments/questions:
 Abstract: it could be more descriptive. For instance, the authors write “Two points of model discrepancy were identified therein. The present article addresses those two concerns and presents the cumulativecurl (CC) model.” Which are these two concerns? They will become clear once reading the text, but I would find it useful to mention them here. Also, would it be possible to translate the “improved accuracy” or “greatly reducing the computational time” into percentage and speedup values?
 Section 2: I understand that it is more practical to cite others' works instead of rewriting the full model. Eventually, this also makes the article easier to read. However, I find it difficult to follow at times. For instance, the authors mention that the original cumulative wake model proposed in Bastankhah et al. (2021) does not include a nearwake model. However, in the current work, this deficiency is overcome by representing the nearwake region with a superGaussian wake model. How is this done? Is there an analytical derivation? I think that the reader would benefit from a more indepth description of the model, which could also be provided in the appendix.
 Section 3.1 (line 167): Which is the horizontal resolution used in the precursor simulation? Also, for how long the precursor simulations have been advanced in time?
 Section 3.1 (line 197): From where does this formula come? Consider adding a reference. Also, I would consider a different notation for the wind speed (ws can be seen as w*s in a mathematical expression).
 Section 3.1 (line 230): For how long the windfarm simulations are run in SOFWA? In the article, it is reported only the spinup time (1200 s), but not the time over which statistics are collected.
 Section 3.2: Over which region the velocity is spatially averaged to produce the power trends observed in Fig. 1? Is the heightdependent velocity profile given by the precursor simulations taken into account in the analytical models (GCH and CC)? Or do the analytical model assumes a uniform inflow velocity profile? If so, how is the velocity magnitude estimated? Does it refer to the velocity at hub height? This information can improve the interpretation of the validation results.
 Section 3.2 (line 243): The authors mention that “The TI assumed in the GCH and CC models was selected to yield perfect agreement with SOWFA at a 7D distance downstream.” I believe that if the TI value was tuned so that a zero error would occur at 5D or 10D, the validation would look different. Therefore, why not use the ambient TI reported in table 1 (i.e. the ambient TI values given by the precursor simulations)? Please, comment on this.
 Section 3.2.2: How is the ambient TI evaluated in this section? Are the authors using the ambient TI shown in table 1? I’m concerned about the ambient TI because usually, the analytical wake model predictions are strongly dependent on this value. Moreover, it is important to mention the ambient TI in case of the reader would like to reproduce some of the results. Note that the same question holds also for sections 3.3 and 3.4. Finally, it is not clear to me how the added TI is computed. Would it be possible to include this in the text?
 Figure 2: I find this plot difficult to read. Have you tried using different symbols instead of different lines for the various precursor cases? That is just a suggestion since it may make it worst.
 Figure 5: Why the GCH and CC firstrow turbine power is lower than the one predicted by SOFWA in all cases? This mismatch (although limited to a few percentage points) could lead to a bias in the measurements further downstream. Moreover, I would find it very interesting to include the prediction of, for instance, the Jensen model in the current figure. This could highlight how much better the models have become at matching LES results. However, I also understand that this could be out of the scope of the current manuscript.
 Figure 7: Which are the yaw angles applied at every turbine row? The authors mention that “This pattern assumes a large yaw misalignment angle for the first row of turbines, which then decreases linearly to zero for the last row of wind turbines.”. However, they do not provide the yaw misalignment for the first row of turbines. This information is necessary in case of the reader would like to reproduce the results.
 Section 4: Very nice and strong validation of the model. I have only a minor question here. In figure 15 (top panel), both models predict a lower energy ratio for a wind direction of 140 degrees than 320 degrees. Why is this happening? In fact, in the first case, turbine 22 operates in the wake of one turbine while in the second case it operates in the wake of six upwind turbines.
 I noticed that in many cases the caption of the figures also contains interpretations of the results. I would use the caption only for describing the figure, therefore moving the interpretation of the results in the main text.
Technical comments:
 Line 247: Typo “largely largely”
 line 320: rephrase the sentence

RC2: 'Comment on wes202217', Carl Shapiro, 06 May 2022
This paper gives adds the superGaussian model of Blondel and Cathelain and the cumulative wake superposition of Bastankhah et al. to the Gausscurl hybrid model to address issues with deep array effects. The comparisons to highfidelity models and field data are comprehensive. The paper is a worthwhile addition to the considerable research on wind farm wake modeling that is necessary for wind farm design and control.
Introduction: I enjoyed this clear discussion of the complications of wake modeling (including wake super position and near and far wake models) in various wind farm configurations. Two issues that could use some discussion are (1) momentum conserving models and linearized momentum conserving models (often called mass conserving models) and (2) the choice of wake expansion rate through turbulence characteristics.
Section 2.2: It is hard to decipher where each of these equations come from and how they have been modified in this implementation. Is there a consistent theoretical basis for adding the Blondel & Cathelain model and cumulative model of Bastankhah et al. to the GCH model? Or are the additions heuristic?
Section 2.2: This model has a large number of free parameters, which makes it more difficult to use. Could you discuss in more depth how these parameters are selected to make the model more widely useable.
All graphs: Please use vector formats for these images and use consistent font sizes. The resolution is fairly low and the font is sometimes hard to read. Use more descriptive titles without using underscores.
Figure 1: The improvement here is not as apparent to me as claimed in the text. I would have assumed that the superGaussian near wake model would improve the agreement in the near wake. In fact, the opposite seems to be the case. Furthermore, the choice to tune the results at x/D=7 affects the model accuracy. If the tuning had been done at x/D=3 the results might be quite different. A better approach would be to minimize the error over all measurements.
Section 3.2.2: Since these results are for a single turbine, they are only including the effect of including the Blondel & Cathelain model in the GCH model and the Bastankhah model does that have an impact, correct? Or am I misunderstanding that? I suggest adding some discussion on what aspects of the model are being tested here.
Section 4: The paper has a lot of great data for the comparison. While the graphs are very instructive to understand the differences between the models, it’s hard to compare the average error. Could you provide average error results for each of these graphs in a table?
Figure 10: What do the color plots on the right represent? There is no label and they are difficult to read.
Sections 3&4: I suggest changing the “Reflection” subsections to “Discussion."
 AC3: 'Response to Referee Comments', Christopher Bay, 01 Sep 2022
 AC4: 'Second Response to Referee Comments', Christopher Bay, 24 Nov 2022
Christopher J. Bay et al.
Christopher J. Bay et al.
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