the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 24 Apr 2026
Comment on "A theoretical upper limit for offshore wind energy extraction" by Simão Ferreira et al. (2026)
Abstract. A theoretical limit for the energy extraction of offshore wind farms has been suggested by Simão Ferreira et al. (2026) based on a simple analytical model that was originally designed to provide an estimate of the wake loss of an infinite wind farm. Simão Ferreira et al. (2026) validated the model with 72 offshore wind farms using an ad hoc and undocumented method to correct the model for application to finite wind farms. In this work, we discuss a number of concerns regarding the reproducibility of the finite wind farm correction and its sensitivity to the model results and validation, as well as the application of the model to assess national policies. We conclude that the limit proposed in Simão Ferreira et al. (2026) is not a theoretical limit but a model limit that is strongly dependent on the finite wind farm correction.
Status: final response (author comments only)
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RC1: 'Comment on wes-2026-59', Anonymous Referee #1, 02 May 2026
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CC2: 'Reply on RC1', Jens Nørkær Sørensen, 18 May 2026
Dear reviewer
In the comment to our article, it is argued that the derived theoretical limit is not a theoretical limit, but rather a model limit strongly dependent on the finite wind-farm correction methodology. Furthermore, it is claimed that the model is validated using an ad hoc and undocumented approach for application to finite wind farms. Unfortunately, several of the results presented in the comment are based on misreadings, conceptual misunderstandings regarding the use of our model, and misinterpretations of the original paper. We have now uploaded a pdf-file in which we systematically show that the conclusions in the comment is based on false assumptions and errors. Although we are aware that reading both our orginal paper, the comment and now also the somewhat lengthy rebuttal on the comment constitute a big effort, we humbly ask you to take this into consideration in your review.
Best regard
The authorsDisclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.Citation: https://doi.org/10.5194/wes-2026-59-CC2
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CC2: 'Reply on RC1', Jens Nørkær Sørensen, 18 May 2026
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RC2: 'Comment on wes-2026-59', Anonymous Referee #2, 13 May 2026
The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes-2026-59/wes-2026-59-RC2-supplement.pdf
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CC3: 'Reply on RC2', Jens Nørkær Sørensen, 18 May 2026
Dear reviewer
In the comment to our article, it is argued that the derived theoretical limit is not a theoretical limit, but rather a model limit strongly dependent on the finite wind-farm correction methodology. Furthermore, it is claimed that the model is validated using an ad hoc and undocumented approach for application to finite wind farms. Unfortunately, several of the results presented in the comment are based on misreadings, conceptual misunderstandings regarding the use of our model, and misinterpretations of the original paper. We have now uploaded a pdf-file in which we systematically show that the conclusions in the comment is based on false assumptions and errors. Although we are aware that reading both our orginal paper, the comment and now also the somewhat lengthy rebuttal on the comment constitute a big effort, we humbly ask you to take this into consideration in your review.
Best regard
The authorsDisclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer.Citation: https://doi.org/10.5194/wes-2026-59-CC3
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CC3: 'Reply on RC2', Jens Nørkær Sørensen, 18 May 2026
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CC1: 'Comment on wes-2026-59', Jens Nørkær Sørensen, 16 May 2026
In their comment submitted to WES, van der Laan and Watson direct a sharp criticism against our article ‘A theoretical upper limit for offshore wind energy extraction’. In their comment they claim that the derived theoretical limit is not a theoretical limit, but an analytical model limit. Furthermore, they question the reproducibility of the results, and claim that the model is validated using an ad hoc and undocumented method to correct the method for application to finite wind farms.
Unfortunately, the results presented in the comment are based on misreadings, conceptual errors when using our model and misunderstandings of the paper.
We are nevertheless grateful for the effort by vdLW to reproduce, analyze, and challenge the SLS framework. Critical analysis and independent reproduction attempts are essential parts of the scientific process. However, when auditing the vdLW implementation, code, and reproduced results, we identified several implementation errors and undocumented modifications to the original SLS methodology that materially affected the reproduced results. Importantly, however, once the most significant implementation errors are corrected, the results of the vdLW methodology are found to be in very good agreement with those of SLS.
The extent and impact of the implementation issues identified during the audit made the analysis extensive. Hence, to fully go through all points, it has been required to formulate a rather lengthy reply to the comment, which is attached to the present text. In the main body of the reply, we focus on the principal scientific points and key reproducibility results, while the detailed audit of the vdLW implementation and responses to specific comments are presented in Appendices.
Jens N. Sørensen, Carlos Simão Ferreira and Gunner Chr. Larsen
May 16, 2026Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer. -
CC4: 'Comment on wes-2026-59 - Executable Reproducibility Notebook for the Rebuttal', Carlos Simao Ferreira, 22 May 2026
WES Discussion Comment — Executable Reproducibility Notebook for the Rebuttal to “Comment on ‘A theoretical upper limit for offshore wind energy extraction’ by Simão Ferreira et al. (2026)”
Following the submission of our rebuttal discussion comment in a previous comment, there were subsequent requests from the authors of the wes-2026-59 comment paper (vdLW) for additional reproducibility material and clarification regarding the generated results presented in the rebuttal. Therefore, we make available the complete executable reproducibility notebook and supporting material used to generate the analyses and figures discussed in the manuscript.
The contention raised by vdLW is that their implementation could not produce the results presented in the rebuttal. Our position is that, once the identified implementation inconsistencies are corrected, the vdLW methodology can in fact and did produce the results shown in the rebuttal and remains in substantial agreement with the conclusions of the original SLS framework. The executable notebook provided here documents this process explicitly.
The repository contains the full reproducibility workflow associated with the rebuttal, including:
- the executable Jupyter notebook,
- intermediate analyses,
- figure-generation routines,
- supporting visualizations,
- and the progressive correction analyses associated with the identified Type 1 and Type 2 implementation and methodology inconsistencies by vdLW.
The material is provided in the interest of transparency, reproducibility, and open scientific discussion.
As discussed in the rebuttal manuscript itself, the analyses further demonstrate that, once the identified Type 1 implementation inconsistencies are corrected and selected Type 2 preprocessing inconsistencies are mitigated, the alternative finite wind-farm methodology proposed by vdLW remains in substantial agreement with the SLS framework. In particular, the corrected vdLW methodology continues to reproduce the central result that operational offshore wind-farm data cluster below the SLS theoretical limit curve and close to the 90% theoretical limit curve.
The visualization of the progressive correction of the vdLW implementation inconsistencies, together with the representation of highly complex interacting wind-farm clusters that the methodology can still approximate, further reinforces the soundness and robustness of the SLS framework. Once again, we thank vdLW for continuing to challenge the methodology and stimulate further refinement of the framework. During the present audit process, we also identified and corrected a minor data-point inconsistency associated with one split wind farm, resulting in a marginally improved agreement relative to the originally published regression.
The executable notebook therefore also serves to illustrate that the discussion raised in the original comment paper ultimately concerns refinement of finite wind-farm correction methodologies rather than invalidation of the underlying asymptotic theoretical framework proposed by SLS.
Resources for viewing analysis and downloading code
- Jupyter notebook through NBViewer:
https://nbviewer.org/github/csimaoferreira/offshore-wind-limit-reproducibility-audit/blob/main/Generation_figures_1_2_3_5_and_6.ipynb - GitHub repository of the code:
https://github.com/csimaoferreira/offshore-wind-limit-reproducibility-audit - Jupyter notebook viewer at GitHub repository:
https://github.com/csimaoferreira/offshore-wind-limit-reproducibility-audit/blob/main/Generation_figures_1_2_3_5_and_6.ipynb
In addition, an HTML export and PDF of the executable notebook is attached to this comment for archival purposes and ease of inspection without requiring a Jupyter environment.
The notebook reproduces the analyses associated with Figures 1–3 and Figures 5–6 of the rebuttal manuscript and documents the corresponding reproducibility workflow discussed in the paper.
Disclaimer: this community comment is written by an individual and does not necessarily reflect the opinion of their employer. -
RC3: 'Comment on wes-2026-59', Anonymous Referee #3, 17 Jun 2026
The manuscript under review presents a comment on a publication by Simao Ferreira et al. published earlier in 2026 that raised some attention also outside the academic community. Van der Laan and Watson – i.e., the authors of the comment – raise and elaborate on a number of concerns that may highly impact the interpretation of the findings from the earlier study. Overall, I believe, it is very relevant to raise these concerns and discuss them at a proper level in the community. The public discussion in Wind Energy Science confirms this but also shows that the comment may not be complete (and without shortcomings and misinterpretations) as commented by one of the authors of the original work. Considering this and several more structural shortcomings (see below), I am of the opinion that the “comment” under review should not be published in its current form and therefor suggest a rejection. This is not to stop or hinder the critical discussion of the previous study but rather to ask the authors to find a more comprehensive and diligent way of doing so.
Specific review comments:
[rather a formal aspect] Checking the acceptable manuscript types for WES (https://www.wind-energy-science.net/about/manuscript_types.html), I would expect a peer-reviewed comment only as a comment to a paper published in the same journal. As the two journals in our case seem to differ quite a lot in various aspects, I believe another manuscript type might be more appropriate here (e.g., a research article repeating more of the original work and describing the made amendments in event more detail, or a brief communication focusing on the impact of the previous study it is suggested shortcomings.)
[abstract] The abstract is very short and mentions “a number of concerns” and the single conclusion on the previously proposed “theoretical limit” only being “a model limit”. I’d suggest that the authors detail this a little further – e.g., if this is the conclusion, what is their specific suggestion / next step? Is the previous study shown to be wrong or the results just associated with a higher uncertainty? The authors need to be more specific here.
[l 16] Interviews regarding the previous findings are mentioned, and it seems that the publicity around the original study has been the driver for the authors to draft their comment. Though this seems very obvious, it is not mentioned explicitly in the manuscript. I’d suggest that the authors are more specific on this aspect.
[l 19] “complex” (in “complex models”) is not specified properly here, but it seems that the authors refer to mesoscale models like WRF as a better truth. This should be further detailed here – not least because also mesoscale models / simulations depend on specific model assumptions and wind farm parametrizations that should not be considered as a (high-level) standard solution.
[p 2] The specific concerns are introduced in detail before introducing the basic details of the model and its application. This is not optimal as it requires the reader to have read the original study in great detail and depth beforehand. I recommend to the authors to find a better way of introducing their list of concerns.#
[l 63] k=2.4 is mentioned here, but it is only stated in l 72 that Simao Feirrera et al. used this single value for k in their study. This detail should already be mentioned at the first location.
[Figure 3 / 4] For both figures, “model results” should be specified further – which model here?
[ll 192] The sentence lists four wind farms but states them in pairs (which becomes clear when reading the sentence twice) – this is a bit confusing. I’d suggest to first list all four wind farm, and then explain the issue about the pairs.
[Figure 5] It is very hard to differentiate between red and pink symbols – please use another colour.
[l 204] “possibly due to differences in model implementation” is a very vague formulation – the authors should be more specific about their assumption / suspicion here.
[l 214] The statement “makes it impossible to draw strong conclusions about a limit for finite wind farms” seems key to the comment and requires further elaboration. I understand this is linked to the observed scatter – but where should we draw a line between possible / impossible? I do not say that I disagree with the authors here, but I strongly suggest to be more explicitly here and give more reasoning.
[ll 224] The statement in the conclusions “it is not trivial to use ..” is gain a hint that needs further elaboration. Generally, I feel, the authors must find a better balance between criticizing the original study for their approach in the interpretation of the results and correcting the technical approach behind that.
Citation: https://doi.org/10.5194/wes-2026-59-RC3
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Review of “Comment on "A theoretical upper limit for offshore wind energy extraction" by Simão Ferreira et al. (2026)” by van der Laan and Watson.
Overview:
van der Laan and Watson have provided a careful and technically rigorous critique of Simão Ferreira et al. (2026) by clarifying the interpretation and the methodological robustness of the proposed “theoretical limit” for offshore wind farm capacity factors. Van der Laan and Watson raise substantive concerns that significantly affect how the original results should be understood and applied.
Most importantly, van der Laan & Watson (vdLW) argue that Simão Ferreira et al.’s (SFLS’s) work cannot be considered a theoretical or universal bound analogous to the Betz limit. I strongly agree with vdLW that “the proposed limit … is not a theoretical limit … but a model limit ….” The SFLS derived capacity factor limit depends explicitly on modeling assumptions (turbine spacing, the treatment of finite wind farm effects). The language used by SFLS risks overstating the generality and physical inevitability of their result. More precise language would be “a model dependent upper bound under specific simplifying assumptions” rather than a “theoretical limit” which implies a fundamental constraint on wind energy extraction.
vdLW carefully examine the finite wind farm correction of SFLS and show that the correction method is ad hoc and insufficiently documented, that it is not reproducible without subjective interpretation. vdLW give a good-faith effort to develop (and document, and provide the python code via zenodo) an effort to provide an automated method to replicate the SFLS approach and demonstrate the insufficiencies of that method, further emphasizing that the SFLS approach is not reproducible. Further, they show clearly in Figs 5 and 6 that the SFLS model is highly sensitive to this parameter. I recommend this important comment for publication in Wind Energy Science.
Major comments:
Minor comments: