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: open (extended)
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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
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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
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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
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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
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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
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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.
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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: