Introduction of the virtual center of wind pressure for correlating large-scale turbulent structures and wind turbine loads

Schubert, Carsten; Moreno, Daniela; Schwarte, Jörg; Friedrich, Jan; Wächter, Matthias; Pokriefke, Gritt; Radons, Günter; Peinke, Joachim

doi:10.5194/wes-11-1267-2026

Articles | Volume 11, issue 4

https://doi.org/10.5194/wes-11-1267-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/wes-11-1267-2026

© Author(s) 2026. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 11, issue 4

Research article

|

17 Apr 2026

Research article |

| 17 Apr 2026

Introduction of the virtual center of wind pressure for correlating large-scale turbulent structures and wind turbine loads

Carsten Schubert, Daniela Moreno, Jörg Schwarte, Jan Friedrich, Matthias Wächter, Gritt Pokriefke, Günter Radons, and Joachim Peinke

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Final revised paper (published on 17 Apr 2026)
Preprint (discussion started on 04 Mar 2025)

Interactive discussion

Status: closed

CC1:
'Comment on wes-2025-28', Alex Rybchuk, 28 Mar 2025

Hi Carsten, I stumbled across your pre-print today. I'm not a reviewer so I won't give detailed feedback, but I wanted to chime in and say I really enjoyed this paper!

Disclaimer: 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-2025-28-CC1
- AC1: 'Reply on CC1', Carsten Schubert, 04 Sep 2025
  
  Dear Alex, thank you very much for your comment! The manuscript took a long way as we wanted to tell a consistent and interesting story. We appreciate the positive feedback.
  
  Citation: https://doi.org/10.5194/wes-2025-28-AC1
RC1: 'Comment on wes-2025-28', Anonymous Referee #1, 06 Apr 2025

The present work proposes an additional parameter for describing wind fields in order to better understand differences between measured and simulated loads. The concept of the virtual center of pressure - which in some ways complements the rotor-averaged wind speed by providing spatial information - could be a useful tool for comparing i.e. measured wind fields, LES wind fields, and synthetic wind fields (as exemplified here). The inclusion of LES wind fields would have been a very useful addition to the work, as this would also provide insight into whether such wind fields could be used to improve design analysis.
The argumentation for the virtual center of pressure is based on assuming a constant thrust coefficient over the rotor, which is not a very convincing argument. I don't see any real reason for introducing the thrust at all, as the formulation is really just a spatial averaging of the longitudinal wind speed squared.
The use of DEL10 to identify large load cycles also seems a bit contrived to me. The components of interest are generally made of steel, and damage would typically be calculated with a much lower exponent. For the bearings, different damage formulations are typically used (i.e. load distribution duration). The argument for using DEL10 is that this metric emphasizes large load ranges. Why not simply examine the distribution of load ranges? If using DEL, there should also be more discussion of the counting algorithm that is used, particularly how unclosed cycles are addressed (since large load cycles can remain unclosed at the end of the time series). Why is it not possible to show a distribution based on simulations in Figure 2? This would be better to compare against measurements.
For rotor-level bending moments, it seems rather obvious that these should be dependent on the spatial variation of the wind field rather than the magnitude of the thrust itself. Other authors have proposed an asymmetry index (i.e. Lavely's PhD thesis) that could also be a useful metric for comparison to the proposed metric.

Citation: https://doi.org/10.5194/wes-2025-28-RC1
RC2: 'Comment on wes-2025-28', Anonymous Referee #2, 27 Jun 2025

The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes-2025-28/wes-2025-28-RC2-supplement.pdf

Citation: https://doi.org/10.5194/wes-2025-28-RC2
AC2: 'Reply on RC1 and RC2', Carsten Schubert, 04 Sep 2025

The comment was uploaded in the form of a supplement: https://wes.copernicus.org/preprints/wes-2025-28/wes-2025-28-AC2-supplement.pdf

Citation: https://doi.org/10.5194/wes-2025-28-AC2

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Carsten Schubert on behalf of the Authors (04 Sep 2025) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (04 Oct 2025) by Claudia Brunner

RR by Anonymous Referee #1 (16 Oct 2025)

RR by Anonymous Referee #3 (07 Nov 2025)

Suggestions for revision or reasons for rejection

Overall, this is a well-written paper that proposes a new variable for quantifying wind field properties that are highly correlated with large bending moment loads on the main shaft of a wind turbine. The two main contributions of the paper are to 1) use a combination of field measurements and aeroelastic simulations to show that traditional stochastic wind fields used in the wind turbine design process underpredict the magnitude of main shaft yaw and tilt fatigue loads observed in the field, and 2) propose a new variable called the virtual center of wind pressure that predicts the occurrence of large main shaft tilt and yaw loads and suggesting that this variable could be incorporated into the design process.

Although most of the research presented in the paper is clear, a few high-level comments for how the paper could be improved are as follows. First, the study focuses on damage equivalent loads with a Wöhler exponent of 10. As the authors acknowledge, an exponent of 10 is not commonly used for the main shaft/main bearing, but this exponent is used "to give predominance to such large amplitude events within the calculation of the DELs." Following this logic, why are DELs used at all? If it is large amplitude events that are of interest, why not use a simpler metric such as the difference between the maximum and minimum value of the load signal or the difference between the maximum and the mean value of the load signal (or similar) to compare simulated and measured wind fields? Or if DELS are used, it would be helpful to understand the sensitivity of the results to the choice of exponent. For example, in Sections 3 and 4 can results using the more common exponent of 4 be compared to the DEL10 results? Would the same trends be observed? This would make the results more impactful since most engineers will use an exponent less than 10 when calculating loads on the main shaft.

Second, the paper discusses both yaw and tilt loads at the main shaft, but only results for tilt loads and corresponding centers of wind pressure along the vertical direction are presented. It is therefore difficult to tell how well this work generalizes to yaw moments. It would be helpful to include some yaw moment distributions similar to Fig. 2 as well as an example case comparing simulated and virtual yaw moments and CoWP_y similar to Figs. 6 and 7 to better understand if the same trends exist.

Lastly, consider moving the content of Appendix B, which is very relevant to the main study, to its own section or subsection in the main body of the paper. Perhaps it would be better as an appendix if the paper was already very long, but I think the paper is concise enough that a separate appendix in not necessary. At the very least, I believe Fig. B2 should be included in Section 2 to clarify how the wind field is measured in the GROWIAN campaign.

Specific Comments:

1. Lines 23-28: The description of the IEC turbulence model parameters and coherence is relevant to the Kaimal turbulence model, but the Mann model is fundamentally different in that it describes a three-dimensional velocity spectral tensor that is parameterized by a shear distortion parameter, isotropic wind speed variance, and length scale. I recommend discussing the differences in the models briefly.

2. Line 37: Although most readers will be familiar, a reference Blade Element Momentum theory would be helpful when introduced here.

3. Line 80: "typically assumed as the number of cycles to failure": Can you clarify this statement? I'm not sure if this is correct. In the IEC 61400-13 (2015) standard, in Section 10.8: Damage equivalent load, n_ref is defined as the reference number of cycles for which the DEL would produce the same damage as the measured time series. For example, the standard uses a value of 600 to represent the 1 Hz equivalent DEL for a 10-minute time series. This is different than saying it is the number of cycles to failure, which would depend on the amplitude of the load.

4. Line 96: Can you mention how many modes are included in the linearized tower model, i.e., 1st and 2nd fore-aft and side-side modes, or just 1st modes?

5. Line 97: Is it possible to provide a reference for "generalized-alpha method"?

6. Line 101: It would be relevant to discuss how far from the turbine the met mast is located.

7. Line 116: The description of the measurements form the two GROWIAN met masts is not clear. Measurements are obtained at five heights at the two met tower locations, but it isn’t clear if the measurement locations have different horizontal positions aside from the two discrete met mast positions. Later, a very clear picture is shown in Appendix B illustrating the configuration. I think this figure belongs in Section 2 of the paper to avoid confusion.

8. Line 117: "a total of 334 collections of time series": Are these 10-minute time series? If so, I recommend adding that detail here.

9. Line 143: The std. dev. and the mean wind speed in the TI equation should be swapped.

10. Line 163: "At this point, we have demonstrated significant differences between the simulated and measured loads under comparable standard wind conditions": Could the larger DELs observed in the measured data possibly be caused by strain gauge calibration errors, or perhaps the strain gauges being located at different locations along the main shaft than in the simulations? I imagine these potential issues have been addressed, but it would be worth discussing this in the paper.

11. Line 229: "The center of pressure is defined as the point where the total sum of a pressure field acts on a body, causing a force to act on that point." It seems like something might be missing in this definition, such as "…causing a force to act on that point that results in the same moment on the body as the original pressure field."

12. Line 244: "We assume C_T constant over the rotor disk and over time." How realistic is this assumption? In addition to the radial dependence of C_T, in the presence of vertical wind shear, the tip speed ratio could be lower at the bottom of the rotor than the top, changing the local C_T.

13. Line 270: What is the triaxial asymmetry index?

14. Line 279: Please discuss how the double met mast array measurements were reconstructed as wind fields. Does the grid of measurement points completely encompass the rotor disk area, or do the measurements need to be extrapolated to determine wind speeds near the edge of the rotor? How do you determine the wind speed for points on the rotor disk in between measurement points from the met mast array?

15. Line 341: "As the two quantities are independent of the turbine…" I don’t think this is true, because the virtual center of wind pressure and pressure-induced moments depend on C_T, which is a turbine property.

16. Appendix A: Only the turbulence model for the "u" component is discussed. Do you also simulate the "v" and "w" turbulence?

17. Line 383: Please define D_{i,j}

18. Figure B3: This figure shows conditions of 12.9 m/s and 6% turbulence intensity, but Fig. B1 shows a case with 12.6 m/s and TI = 7%. Did you mean to show the same 10-minute period in these two figures. If so, why are the wind statistics different?

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ED: Reconsider after major revisions (10 Nov 2025) by Claudia Brunner

AR by Carsten Schubert on behalf of the Authors (27 Jan 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (29 Jan 2026) by Claudia Brunner

ED: Publish as is (01 Feb 2026) by Julia Gottschall (Chief editor)

AR by Carsten Schubert on behalf of the Authors (11 Feb 2026) Manuscript

Short summary

For modern wind turbines, the effects of inflow wind fluctuations on loads are becoming increasingly critical. Based on field measurements and simulations, we identify “bump” events responsible for high damage equivalent loads. In this article, we introduce a new characteristic of a wind field: the virtual center of wind pressure, which highly correlates to the identified load events observed in the operational measured data.