A Novel Analytical Tool to Capture Wind Profile Variability for Wind Energy Assessment: Fast, Simple, and Beyond State-of-the-Art in Complex Terrain

van Schaik, Brandon Jacobus Adrianus; Baruzier, Alice Juliette; Loyer, Clément; Huwald, Hendrik; Lehning, Michael

doi:10.5194/wes-2025-271

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https://doi.org/10.5194/wes-2025-271

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30 Dec 2025

| 30 Dec 2025

Status: this preprint is currently under review for the journal WES.

A Novel Analytical Tool to Capture Wind Profile Variability for Wind Energy Assessment: Fast, Simple, and Beyond State-of-the-Art in Complex Terrain

Brandon Jacobus Adrianus van Schaik, Alice Juliette Baruzier, Clément Loyer, Hendrik Huwald, and Michael Lehning

Abstract. Reliable wind energy assessment is often limited by reliance on single-point wind measurements at nacelle height, as commonly used in international standards. Here, we present a closed-form analytical rotor-averaging model for turbine power that integrates the vertical variation of horizontal wind speed – the changes in wind with height – using a Taylor expansion of the inflow velocity over the rotor disk. We assess idealised wind profiles to understand the isolated effects of typical shear, veer profiles, and turbulence on the power production of the turbine. This analytical model is then validated against power measurements from five Enercon E92 turbines on the Gotthard Pass, a complex-terrain wind park in the Swiss Alps, across 10 distinct wind events. The analytical model consistently outperforms both OpenFAST simulations and the International Electrotechnical Commission (IEC) standards, providing smoother and more accurate power estimates, reducing RMSE by 8.8% and bias by 25.4% relative to IEC-standard single-point extrapolations. This equates to an annual energy production estimate that is 23.7 MWh closer to reality, for a 2.35 MW turbine at this specific site. Additionally, it enables effective filtering of wake-affected LiDAR measurements, demonstrating flexibility, robustness, and applicability for pre-construction assessment and wind park expansion in complex terrain.

Received: 04 Dec 2025 – Discussion started: 30 Dec 2025

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Brandon Jacobus Adrianus van Schaik, Alice Juliette Baruzier, Clément Loyer, Hendrik Huwald, and Michael Lehning

Status: final response (author comments only)

RC1:
'Comment on wes-2025-271', Anonymous Referee #1, 17 Mar 2026
Summary:
The paper presents a new analytical model accounting for the vertical variations in horizontal wind speed to assess wind turbine power production. The approach relies on the Taylor expansion of wind profile measurements in the rotor swept area obtained from Doppler lidar observations. Results from the analytical model are validated against OpenFAST simulations and measurements from wind turbines operating in the Gotthard Pass wind farm in the Swiss Alps. The topic of the manuscript is of interest since this method provides an estimate of turbine power production at a relatively low computational cost compared with other approaches. Moreover, the detailed knowledge of the turbine mechanical characteristics is not required, and the wake effects between turbines can be addressed by this new method. However, the manuscript lacks clarity, organisation, and rigour, which hinder its readability and reproducibility. More work is needed to improve the manuscript by addressing the following comments:
Major comments:
Key explanations are missing or not clearly formulated. While the Taylor series decomposition of the wind profile is well detailed, the complete methodology is not fully described. In particular, the extension of this decomposition to the wind variance profile is not explicitly explained. Furthermore, it remains unclear how this variance decomposition is used to predict turbine power output and how the power curves are obtained. All these points should be addressed in the revised manuscript.

Several points raised by the editor were not addressed in the manuscript. In particular, the suggested revision to the opening sentence of the abstract was not applied. Furthermore, while a few references were added, the proportion of self-citations remains high, and the paper still does not introduce or discuss previous studies focusing on alternative rotor equivalent wind speed formulations.

The current structure of the methodology section (section 2) is confusing. The analytical model is introduced (section 2.1) before the input data is presented. The OpenFAST configuration is split between sections 2.2 and 2.5, making it difficult to get an overall view of the numerical modelling approach. Moreover, the current section titles (e.g., "Analytical method", "Numerical method") are imprecise and should be more descriptive. Here is a suggested restructuring:

2.1 Wind profiles
    2.1.1 Idealized wind profiles
    2.1.2 Gotthard Pass wind profiles
2.2 Power prediction through analytical method
    2.2.1 Rotor-averaged cubic wind using Taylor expansion of the wind profile (which corresponds to the current paragraph before section 2.1.1)
    2.2.2 Turbulence modelling
    2.2.3 Power curve fitting (current section 2.1.1)
    2.2.4 Application to idealized and real wind profiles
2.3 Power prediction through numerical modelling (OpenFAST, current section 2.2 and a part of the 2.5)
2.4 Analytical model validation strategy (current section 2.5.1, to merge with the current introduction of the results and discussion part (section 3))

Minor Comments:
General comments on the manuscript (please refer to detailed comments below for details):
Units are written in italics throughout the manuscript, but should be written in normal font.

Units are sometimes specified in brackets in the text (e.g., line 83) but omitted elsewhere (e.g., line 267). It should be consistent throughout the manuscript. I recommend removing the brackets as this improves readability.

Some formatting issues create large blank spaces in the manuscript (e.g., pages 4, 6, 10, etc), which should be corrected.

The reference to equations, figures, tables, sections, and appendices in the text does not follow the WES guidelines, as for the format of the tables.

There are several punctuation and capitalization issues throughout the manuscript. For example, there are unnecessary commas (e.g., line 216), incorrect capitalization where it should not appear (e.g., caption of Fig. 2), and missing capitalization where it should be used (e.g., line 651).

Some words extend beyond the page margins on several lines and should be corrected.

Some parts of the methodology appear in the results section, and conversely, some results are presented in the methodology section.

Detailed comments:
Line 84-85: “ρ [kg m−3] is the air density which is assumed to be constant throughout the swept area of the blade”. The specific value of ρ should be specified here, or a reference to Table 2 should be made.

Line 94: the sentence “we integrate over the rotor-swept area…” should specify which quantity is being integrated. This holds for the caption of Fig. 1, which should be detailed.

Figure 1: a unique notation should be used for the vertical coordinate (z, h, or Z) and should remain the same in the following equations.

Equation 3: Variable N is not defined in the text. It can be specified in the sentence above by adding “at the order N”.

Line 104: “The full derivation is given in A…”. Here, the authors refer to the appendix, but A corresponds to the rotor swept area.

Line 106: “expansion coefficients i, j & k.” Please replace “&” with “and”. Same remark for line 113.

Table 1: this table is present two times in the paper (in the main text and in the appendix). It should be removed from the main text.

Line 114: “recommended at N = 3”. Is there a reason for this specific value?

Line 125: “. It has been shown that a linear scaling factor of 1.81 applied to σv effectively accounts for the heavier tails of the Student-t distribution in the case of the Gotthard Transect Experiment for all wind directions and speeds.” A reference is missing for this statement.

Equation 6: some terms are not explained. Please detail what vin, vout and P(v) are here. Same remark for Eq. 7, where Pr is not defined. In this equation it is not clear whether v refers to v or v . Moreover, the P(v) in Eq. 6 seems to be different from the one in Eq. 4. If this is the case, another notation should be used.

Line 135: “showing good agreement with the analytical solution (Fig. 6b, further detailed in the Results).” This sentence should be in the results section.

Line 178: “When comparing to the measured power output of the turbine (Ptrue), setting the shaft tilt to zero introduces a small bias of approximately 8kW for the 2.35MW turbine.” Here, the discussion refers to a turbine model that is not yet introduced in the paper.

Table 2: how was this specific density value selected?

Line 208: “constant and uniform wind profile with no turbulence, which is detailed in section 3.2.” The reference case is actually defined in the same section, from lines 212 to 215.

Line 211: What is [a)]?

Line 219: “and z = 0m at the nacelle of the turbine”. This sentence should be moved and included in the description of Fig. 1.

Line 232: please correct “Von Kárman” to “von Kármán”. On the same line, how did the authors select the value of z0 = 0.1 m?

Line 239: “indicated orange in Figure 3a”. There is no a) or b) on this figure.

Line 242: “In 2023, the site hosted a Doppler wind LiDAR campaign in which the instrument was deployed at 10 locations near the turbines over a three-month period (Figure 3).” The lidar model and the dates of the field campaign should be briefly indicated.

Line 295: “we have selected a four hour window of 10-minute data”. It is not clear whether the analysis is based on the mean profile over this period or on the 24 individual profiles available.

Line 324: “For the analytical model, we align with the Enercon E92 2.35MW wind turbines installed at the Gotthard Wind Park. Power curve data is taken from the Enercon E92 specification sheet and is fitted (Equation 7), resulting in a smooth analytical expression (Fig. 5a).” These sentences, except the reference to Fig. 5a, describe the methodology and should be moved to the methodology section.

Line 330: The term “power curve” is incorrectly written as “powercurve” several times in the text.

Figure 5: the text indicates that a scaling between the Enercon E92 and the WindPACT turbines is applied to allow comparison of the results. This figure should therefore be plotted using this scaling. Moreover, including additional points on the curve in Fig. 5b (for example, between 9 and 14 m s⁻¹) would result in a smoother curve. The wind speed unit should be written as ms-1 instead of m/s; this should be corrected in the x-axis label.

Line 359: “The OpenFAST model consistently increases with shear…”. I think the word “power” is missing here.

Line 376: the description here refers to Fig. 6c and not Fig. 6b.

Line 376-377: “a characteristic shape which has previously been reported in similar works such as Gasser et al. (2025)”. This reference does not seem relevant here, as no curve (or result) similar to the one presented in Fig. 6c is presented in Gasser et al. (2025).

Line 391: brackets are missing around the references.

Line 392: the authors likely mean Fig. 6d.

Line 395: “from the uniform inflow amounting to only about 15 kW.” This value is right for the analytical model, but the deviation reaches 20 kW in the case of the OpenFAST simulations.

Figure 6: The same remark as above applies here for the x-label unit. As explained before in the text, different values of a, max, and TI are considered, however, the impact of these different values is not discussed. The meaning of these parameters should also be briefly indicated in the caption.

Line 413-414: “Compared to the turbulence-free case (Fig. 7b), this model version achieves much better agreement with the measured power production, particularly at high wind speeds.” This conclusion is not obvious from the figure alone. Although there is indeed a slight improvement when turbulence is included in the model, this only becomes clear when considering the bias values reported in Table 3, since the order of magnitude of the biases is much smaller than the scale of the figure.

Figure 7: the caption does not describe what is shown in the figure. Given that the inclusion of turbulence in the model has a limited impact, it would be interesting to present the power curves obtained with the analytical model and/or OpenFAST in comparison with the one derived from the standard approach. This would confirm the statement in lines 432-433.

Line 423: “the third subfigure (Fig. 7)” corresponds to Fig. 7c.

Table 3: it seems that this table is included in the manuscript as an image. If this is the case, this should be modified, and the table format should follow the WES guidelines.

Line 434-435: it is not clear how the values of 8.6% and 25.4% were obtained. They should therefore be verified.

Line 454: “In Figure 8a, the original LiDAR wind measurements are shown in orange”. One could understand that the authors refer to the measurements from turbine 1 under southerly winds, but this should be stated in the text.

Line 457: “the wake from Turbine 2 intersects the LiDAR beam but consistently bypasses Turbine 1”. Please clarify how this was established.

Line 490: the link between the discussion in this paragraph and the analytical model is not clear. This should be better explained in a revised version of the manuscript.

Line 510: “such as shear, veer”. Please specify linear shear and linear veer.

Equation A4: the lower limit of the summation is missing. The same applies to the remaining summations in the appendix.

Equation A5: this equation is basically the same as Eq. A3 and should not be repeated.

Equation A6: standard functions as sin, cos, and arcsin should be written in normal font rather than italics. The same holds for the differential in the integrals.

Equation A11: it should be specified that the function Q depends on the indices i, j, and k. Qm can then be used in the following equations.

Line 580-585: the demonstration can be simplified by not splitting the integral into two parts, as it can be directly evaluated using the Beta function with x=s+1/2 and y=3/2.

Line 591: it should be specified that the second term is included.

Line 596: another variable should be used since z already corresponds to the vertical coordinate in the main text.

Equations A25 to A27: these equations should not be detailed here since the formula given in Eq. A28 is well known.

Line 634: I recommend introducing a notation for this fraction (e.g., F(m)) and defining it explicitly, since it is currently unclear whether the factor 1/2 is included or not.

Line 649: a detailed calculation of the integral should be provided to confirm that statement. The same applies to section A3 of the appendix.

Equation A9: since the term O(h4) is only used once in the manuscript, I suggest removing it from the equation, replacing “=” with “≈” and specifying “Taylor expansion at the order 2” in the sentence above. Moreover, the choice for the vertical coordinate notation should be consistent in the manuscript (h or z).
Citation: https://doi.org/10.5194/wes-2025-271-RC1
RC2: 'Comment on wes-2025-271', Anonymous Referee #2, 03 Apr 2026

Paper presents a new method for estimating wind measurements, improving on single point measurements, for use in power curve assessments. The paper is well written and provides clear explanation of methods and experimental setup. The proposed methods are demonstrated in a clear way in studies using collected field data. All in all a well-written paper and a good contribution.
General comments:
Good that the paper makes frequent comparison to the standard approach in EIC 61400. Is there an intention to submit this work for consideration to a future standard?
Specific comments:
Page 2 Line 50: "These discrepancies scale to wind-farm level and can lead to significant instantaneous and cumulative power prediction errors" -- This depends a little if the error per-turbine is biased in one direction, it could also cancel out.
Page 3 Line 65: "developed by NREL" (could note now NLR per name change)
Page 3 Line 69: "2106m a.m.s.l." please define
Page 3 line 70: "while modelling a virtual wind turbine with open-source blade geometry", do you mean modeling the Enercon E92s as this turbine, or seperately modeling this turbine?
Page 4: "where horizontal variations across the rotor disk are usually minor compared to the strong vertical gradient in wind speed." -- perhaps unless wakes are present?
Page 7: "given arrays of measured wind speeds v and heights z, t", are the wind speeds v instantaneous or averaged measurements?
Page 7: "We refer the reader to the Data Availability section of this manuscript for more details." Thank you for this contribution to open-source and data availability.
Page 8: Equation 7, I think a figure comparing the data-based, or binned power curve against the functional approximation would be helpful to see. Does the equation have a range of wind speeds for which it is valid or does it work cut-in to cut-out?
Page 8: "it naturally enables the removal of anomalous or clearly unphysical measurements before constructing the analytical profile", a more standard approach might be to only include data from wind sectors where the turbine is not waked right?
Page 11: "resulting in 25 simulations per power curve." how many power curves in total? A standard approach might include 4-6 random seeds per wind speed but that could be unnecessary here and I'm not saying you should do it. Just noting.
Page 21: Line 387: , "so the controller cannot hold the blades exactly at the optimum." Since this is above rated operation, the controller would not be trying to keep the blade at optimum but instead be pitching toward feather to keep the turbine from exceeding rated power. If the torque controller is a simplified constant power (versus constant torque) operation at and above rated speed, then the explanation is more probably that the variations induce rotor speed excursions above and below the rated speed, but the power is clipped above but still falls below, leading to a downward bias.
Still I'm a little surprised these above rated power deviations can be on par with the deviations happening near rated. If you take a look at a standard power curve adjustment for TI you often see the pattern you see with an increase in power away from rated, a decrease near and slightly above rated (like a rounding of a sharp corner) and then minimal effect above rated. There could also be a bug in the controller.
page 25: "OpenFAST generates power earlier because it lacks a defined cut-in wind speed – a known unrealistic feature of the software", this could be defined in the controller. But it's not a major point worth the effort to fix.

Citation: https://doi.org/10.5194/wes-2025-271-RC2

Brandon Jacobus Adrianus van Schaik, Alice Juliette Baruzier, Clément Loyer, Hendrik Huwald, and Michael Lehning

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https://doi.org/10.5194/wes-2025-271-supplement

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A Novel Analytical Tool to Capture Wind Profile Variability for Wind Energy Assessment: Fast, Simple, and Beyond State-of-the-Art in Complex Terrain Brandon van Schaik et al. https://doi.org/10.5281/zenodo.17804408

Brandon Jacobus Adrianus van Schaik, Alice Juliette Baruzier, Clément Loyer, Hendrik Huwald, and Michael Lehning

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Short summary

We created a new way to estimate how much energy a wind turbine can produce by using the full change of wind with height instead of a single measurement. Testing it on turbines in the Swiss Alps showed that it predicts energy production more accurately and smoothly than current industry methods. It performs as well as advanced models but is much faster, helping guide future wind projects in complex landscapes.


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