SCADA-based calibration of analytical wake models: uncertainty-aware generalisation across offshore wind farms and the role of atmospheric stability

van Binsbergen, Diederik; Daems, Pieter-Jan; Verstraeten, Timothy; Vratsinis, Konstantinos; Nejad, Amir; Helsen, Jan

doi:10.5194/wes-2026-120

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https://doi.org/10.5194/wes-2026-120

Preprints

16 Jul 2026

| 16 Jul 2026

Status: this preprint was under review for the journal WES. A revision for further review has not been submitted.

SCADA-based calibration of analytical wake models: uncertainty-aware generalisation across offshore wind farms and the role of atmospheric stability

Diederik van Binsbergen, Pieter-Jan Daems, Timothy Verstraeten, Konstantinos Vratsinis, Amir Nejad, and Jan Helsen

Abstract. This study presents a Bayesian framework for generalising SCADA-calibrated wake-model tuning parameters across offshore wind farms. The framework infers cluster-wide, farm-specific, and new-farm parameter distributions while accounting for calibration uncertainty, intra-farm variability, inter-farm variability, and residual model-data mismatch. It is applied to the Jensen and Gaussian TurbOPark wake models and extended to condition the central tuning parameter on atmospheric stability, represented by the bulk Richardson number and the inverse Monin-Obukhov length. For both wake models, unstable conditions are associated with larger tuning parameters, indicating faster wake expansion and recovery, whereas stable conditions yield smaller tuning parameters and more persistent wakes. The results show that the globally pooled stability-independent parameter should not be interpreted as a neutral-stability parameter because it reflects the stability mix at the case-study offshore site. Propagating representative stability-class-specific parameters to wake-loss estimates shows that atmospheric stability substantially affects predicted wake losses, with model differences becoming more pronounced when external wakes from surrounding wind farms are included.

Received: 07 Jul 2026 – Discussion started: 16 Jul 2026

Competing interests: At least one of the (co-)authors is a member of the editorial board of Wind Energy Science.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.

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Diederik van Binsbergen, Pieter-Jan Daems, Timothy Verstraeten, Konstantinos Vratsinis, Amir Nejad, and Jan Helsen

Status: closed (peer review stopped)

RC1:
'Comment on wes-2026-120', Anonymous Referee #1, 12 Aug 2026
The authors obtain two datasets of the wake expansion rate in two engineering wake models. These datasets are constructed by fitting the wake expansion rate on the mean absolute turbine power error, for selected wind farms in the BE-NL concession zone and at each 10-min time interval of 2-years of SCADA data. To analyze the distribution of the wake expansion rate, the authors develop a Bayesian hierarchical model. As such, the mean and standard deviation of the calibrated (L1 optimal) wake expansion rate are obtained for every wind farm, while pooling information across the wind farms for the mean wake expansion rate and assuming that its standard deviation σ_intra is the same for each wind farm. In the Bayesian hierarchical model, the pooling is done by assuming that the mean wake expansion rate of every farm is drawn from a cluster-wide wake expansion distribution, for which the mean and standard deviation are also estimated from the dataset of calibrated wake expansion rates. The uncertainty-aware generalization then rests on the assumption that the L1 optimal wake expansion rate for a new wind farm in the same cluster will have a distribution with a mean that follows the cluster-wide distribution and the same standard deviation as for the other wind farms. The authors also examine two extensions of the baseline hierarchical model: (1) associating larger parameter uncertainty with larger mean absolute turbine power errors, and (2) conditioning the wake expansion rate on a stability proxy.
Overall, the authors present a novel approach to quantify the parameter uncertainty after calibration, by developing a Bayesian hierarchical model with two extensions. However, the current manuscript does not validate the Bayesian model(s), which is typically done through a posterior predictive check. The clarity and conciseness of exposition should also be improved, mainly regarding the definition of Bayesian hierarchical model. Below I have listed my comments.
General comments:
Before analyzing the posteriors of parameters in a Bayesian (hierarchical) model, it is common to perform a posterior predictive check to validate whether the Bayesian formulation adequately describes the data distribution (see for example Bayesian Data Analysis by Gelman et al.). Such posterior predictive checks should still be performed for the baseline Bayesian model, the inclusion of the ‘residual mismatch model’, and the inclusion of conditioning on a stability proxy. By doing so, the authors can establish the validity of the posterior analysis, and demonstrate the value of the two extensions. If these extensions do not improve the agreement between the posterior predictive distribution and the data distribution, it would be useful to discuss whether the additional model complexity is warranted, or whether the main benefit lies in providing insight into the sources of uncertainty.

Currently the mean wake expansion rate per farm w_p is pooled in a Bayesian hierarchical framework. The same can be done for the standard deviation of the wake expansion rate per farm sigma_intra. Could the authors comment on why they did assume the same uncertainty sigma_intra, rather than pooling it?

How large is the spread in the results for the repeated calibration per timestep? Is this an artefact of the Optuna framework, in the sense that too many model evaluations are needed to reach convergence? Before convergence, the error on the turbine power will be larger. Is the ‘residual mismatch model’ then mainly useful in situations where the optimization method is not yet converged?

Major comments:
L134: Could the authors comment on the total number of wake model evaluations required for their three-stage calibration procedure, with R=5 and time-series based calibration for 10-min intervals and 2 years of SCADA data?

L150: What is meant by ‘simplifies the cost-function definition’? A generic turbine based L2 error is also ‘simpler’ than an error with weighting factors, but L1 errors are more robust to outliers. Please provide additional motivation for this choice. Also note that the same reference is displayed three times on line 152, or explicit identifiers (a,b,c) should be used.

L155: How is the turbulence intensity used in the considered wake models? It is common to let the wake expansion rate depend on the turbulence intensity. Does removing the uncertainty on the turbulence intensity also remove uncertainty on the calibrated wake expansion rate parameter?

L220: Should the sample standard deviation be used here or rather an estimate for the standard error on the sample mean? It seems that the authors want to quantify the uncertainty on the sample mean of the R calibration runs.

L323: According to Eq. 14, the farm-level tuning parameter can be negative. Is there a reason why the non-negativeness of the parameter is no longer enforced? A similar comment holds for Eq.13 & 15.

L325: The specification of w_ref from the same data used for posterior inference appears somewhat unconventional from a Bayesian perspective, where priors are generally defined independently of the observed data. Could the authors provide additional justification for this choice? As alternative, standard literature values for w_ref could be used. Since the prior is weakly informative, this should not noticeably change the results.

L346: Please define the posterior distributions for the parameters. This is typically done through Bayes’ theorem by explicitly defining the likelihood and prior.

L349 Please state the number of chains, warm-up samples and posterior samples, as well as which MCMC diagnostics were used (including references for those diagnostics).

L350-356: The terms 'physical separation' and 'components may overlap' may be vague. Are the authors referring to confounding of the sigmas? Is this mentioned because the authors observed posterior correlations between alpha and sigma_intra? If so, that should be mentioned in the manuscript. Currently, only summary statistics of the marginal posterior distributions are provided in Table 1, while samples of the joint posterior are available. If other posterior correlations are found, those should also be mentioned in the manuscript.

L431: Why are the two calibration datasets not identical? This should be mentioned in the data filtering section. Which calibration dataset is then displayed in Figure 2?

Figure 6: The authors should demonstrate whether the ‘residual mismatch model’ improves the probabilistic description of the uncertainties in the data {w_{p,i}} through a posterior predictive check, before analyzing how it may do so.

L508: Supposing that the ‘residual mismatch model’ improves the posterior predictive check, which value of epsilon should be used when constructing the posterior predictive for a new farm?

Section 6.1: It is unclear how the authors define the prior and likelihood, which jointly define the posterior discussed in 6.2. For instance, it is unclear whether Eq. 34-35 describe the prior for w_b, or parametrize its posterior in a hierarchical fashion. Also, posterior distributions for sigma_stab and sigma_park,stab are obtained, but no prior distribution is specified for these parameters. Please provide an expression for the joint posterior distribution by specifying the likelihood and the prior. Please also mention the total amount of parameters for which the posterior is obtained.

L634: How/why does the stability bin b(p,i) depend on the wind farm?

L667: Please mention how many chains, burn-in and posterior samples were used to sample this posterior, and whether the same metrics were used to verify that the chains converged to the posterior.

L715: By letting the cluster-based mean wake expansion rate depend on the stability proxy, large variations in the mean are captured (Fig. 21). Since the total uncertainty in the data remains the same, one would expect to observe a reduction of the quantified intra-farm uncertainty. A posterior predictive check would allow identifying whether the total uncertainty in the data is adequately captured by both Bayesian models. Currently, one of them may over/underestimate the total uncertainty.

Figure 24: To what extent do these results depend on the considered wind farms (e.g. large vs. small farm) within the BE-NL concession zone?

Minor comments:
L6: Since there is only one tuning parameter per model, and that tuning parameter is directly related to the wake expansion rate, it may help to explicitly state that. The authors may consider changing ‘larger tuning parameters’ by ‘larger wake expansion rate’. A similar comment holds for the conclusion.

L99: Could the authors mention how the turbulence intensity is obtained from the SCADA data?

L116: Which quantitative criterion is used for ‘nearly constant active power’ in the filtering procedure?

Section 3.2: Is it correct that the wake expansion calibration parameter (k or A) is calibrated separately for each of the four wind farms at each accepted time stamp, but not for every turbine separately? Mentioning this explicitly would improve the clarity.

L149: Since mathematical description of the wake models is not given, the authors may need to add references so that k and A uniquely defined.

L160: ‘earlier work’ Are the authors referring to the calibration procedure of section 3?

Section 4 is rather lengthy (page 7-21) and at times difficult to follow. The manuscript may benefit from making this section more concise, for example, by focusing on the mathematical exposition of the Bayesian hierarchical model, and motivating/interpreting it on the go.

L274: A good model should both have small bias and small error variance. Consequently, variance-sensitivity seems related to the representativeness of a calibration parameter.

L280: The normalization mainly seems to get rid of the power-curve scaling, but the effect of the free stream wind speed is not mentioned. It is also not entirely clear how such a scaling would reduce the dependence of the spread.

L326: The specification of a prior for z_p implies that a posterior distribution is obtained for z_p. Is that the case? Otherwise, z_p only serves as a reparameterization variable for the (hierarchical) prior w_p \sim N(w_global, sigma_park^2). The authors may consider reorganizing this to improve the clarity.

L443: What is meant by ‘how well the parameter is constrained’?

Figure 5: Is there only SCADA data available from 4 out of the 12 farms in the BE-NL concession zone? If so, this should be mentioned in section 2.

Figure 7: What do the dots represent?

Section 5 currently seems to interrupt the flow of the Bayesian analyses in Section 4 and 6. The manuscript may benefit from transferring the parts of Section 5 required to understand Section 6 to Section 2, and the rest to the appendix. However, this remains at the authors’ discretion.

L571: Does 10-100% mean 10-100% of the peak value of the PDF?

Table 5 & L750: ‘summarised’ How are the posteriors over the 25 quantile bins translated to the stability classes (e.g. closest bin, linear interpolation, average, …)?

L761: Please define the total, internal, and external wake losses mathematically or give a reference which defines them.
Citation: https://doi.org/10.5194/wes-2026-120-RC1
RC2: 'Comment on wes-2026-120', Anonymous Referee #2, 13 Sep 2026

Please see the attached for the referee report.

Citation: https://doi.org/10.5194/wes-2026-120-RC2

Status: closed (peer review stopped)

RC1:
'Comment on wes-2026-120', Anonymous Referee #1, 12 Aug 2026
The authors obtain two datasets of the wake expansion rate in two engineering wake models. These datasets are constructed by fitting the wake expansion rate on the mean absolute turbine power error, for selected wind farms in the BE-NL concession zone and at each 10-min time interval of 2-years of SCADA data. To analyze the distribution of the wake expansion rate, the authors develop a Bayesian hierarchical model. As such, the mean and standard deviation of the calibrated (L1 optimal) wake expansion rate are obtained for every wind farm, while pooling information across the wind farms for the mean wake expansion rate and assuming that its standard deviation σ_intra is the same for each wind farm. In the Bayesian hierarchical model, the pooling is done by assuming that the mean wake expansion rate of every farm is drawn from a cluster-wide wake expansion distribution, for which the mean and standard deviation are also estimated from the dataset of calibrated wake expansion rates. The uncertainty-aware generalization then rests on the assumption that the L1 optimal wake expansion rate for a new wind farm in the same cluster will have a distribution with a mean that follows the cluster-wide distribution and the same standard deviation as for the other wind farms. The authors also examine two extensions of the baseline hierarchical model: (1) associating larger parameter uncertainty with larger mean absolute turbine power errors, and (2) conditioning the wake expansion rate on a stability proxy.
Overall, the authors present a novel approach to quantify the parameter uncertainty after calibration, by developing a Bayesian hierarchical model with two extensions. However, the current manuscript does not validate the Bayesian model(s), which is typically done through a posterior predictive check. The clarity and conciseness of exposition should also be improved, mainly regarding the definition of Bayesian hierarchical model. Below I have listed my comments.
General comments:
Before analyzing the posteriors of parameters in a Bayesian (hierarchical) model, it is common to perform a posterior predictive check to validate whether the Bayesian formulation adequately describes the data distribution (see for example Bayesian Data Analysis by Gelman et al.). Such posterior predictive checks should still be performed for the baseline Bayesian model, the inclusion of the ‘residual mismatch model’, and the inclusion of conditioning on a stability proxy. By doing so, the authors can establish the validity of the posterior analysis, and demonstrate the value of the two extensions. If these extensions do not improve the agreement between the posterior predictive distribution and the data distribution, it would be useful to discuss whether the additional model complexity is warranted, or whether the main benefit lies in providing insight into the sources of uncertainty.

Currently the mean wake expansion rate per farm w_p is pooled in a Bayesian hierarchical framework. The same can be done for the standard deviation of the wake expansion rate per farm sigma_intra. Could the authors comment on why they did assume the same uncertainty sigma_intra, rather than pooling it?

How large is the spread in the results for the repeated calibration per timestep? Is this an artefact of the Optuna framework, in the sense that too many model evaluations are needed to reach convergence? Before convergence, the error on the turbine power will be larger. Is the ‘residual mismatch model’ then mainly useful in situations where the optimization method is not yet converged?

Major comments:
L134: Could the authors comment on the total number of wake model evaluations required for their three-stage calibration procedure, with R=5 and time-series based calibration for 10-min intervals and 2 years of SCADA data?

L150: What is meant by ‘simplifies the cost-function definition’? A generic turbine based L2 error is also ‘simpler’ than an error with weighting factors, but L1 errors are more robust to outliers. Please provide additional motivation for this choice. Also note that the same reference is displayed three times on line 152, or explicit identifiers (a,b,c) should be used.

L155: How is the turbulence intensity used in the considered wake models? It is common to let the wake expansion rate depend on the turbulence intensity. Does removing the uncertainty on the turbulence intensity also remove uncertainty on the calibrated wake expansion rate parameter?

L220: Should the sample standard deviation be used here or rather an estimate for the standard error on the sample mean? It seems that the authors want to quantify the uncertainty on the sample mean of the R calibration runs.

L323: According to Eq. 14, the farm-level tuning parameter can be negative. Is there a reason why the non-negativeness of the parameter is no longer enforced? A similar comment holds for Eq.13 & 15.

L325: The specification of w_ref from the same data used for posterior inference appears somewhat unconventional from a Bayesian perspective, where priors are generally defined independently of the observed data. Could the authors provide additional justification for this choice? As alternative, standard literature values for w_ref could be used. Since the prior is weakly informative, this should not noticeably change the results.

L346: Please define the posterior distributions for the parameters. This is typically done through Bayes’ theorem by explicitly defining the likelihood and prior.

L349 Please state the number of chains, warm-up samples and posterior samples, as well as which MCMC diagnostics were used (including references for those diagnostics).

L350-356: The terms 'physical separation' and 'components may overlap' may be vague. Are the authors referring to confounding of the sigmas? Is this mentioned because the authors observed posterior correlations between alpha and sigma_intra? If so, that should be mentioned in the manuscript. Currently, only summary statistics of the marginal posterior distributions are provided in Table 1, while samples of the joint posterior are available. If other posterior correlations are found, those should also be mentioned in the manuscript.

L431: Why are the two calibration datasets not identical? This should be mentioned in the data filtering section. Which calibration dataset is then displayed in Figure 2?

Figure 6: The authors should demonstrate whether the ‘residual mismatch model’ improves the probabilistic description of the uncertainties in the data {w_{p,i}} through a posterior predictive check, before analyzing how it may do so.

L508: Supposing that the ‘residual mismatch model’ improves the posterior predictive check, which value of epsilon should be used when constructing the posterior predictive for a new farm?

Section 6.1: It is unclear how the authors define the prior and likelihood, which jointly define the posterior discussed in 6.2. For instance, it is unclear whether Eq. 34-35 describe the prior for w_b, or parametrize its posterior in a hierarchical fashion. Also, posterior distributions for sigma_stab and sigma_park,stab are obtained, but no prior distribution is specified for these parameters. Please provide an expression for the joint posterior distribution by specifying the likelihood and the prior. Please also mention the total amount of parameters for which the posterior is obtained.

L634: How/why does the stability bin b(p,i) depend on the wind farm?

L667: Please mention how many chains, burn-in and posterior samples were used to sample this posterior, and whether the same metrics were used to verify that the chains converged to the posterior.

L715: By letting the cluster-based mean wake expansion rate depend on the stability proxy, large variations in the mean are captured (Fig. 21). Since the total uncertainty in the data remains the same, one would expect to observe a reduction of the quantified intra-farm uncertainty. A posterior predictive check would allow identifying whether the total uncertainty in the data is adequately captured by both Bayesian models. Currently, one of them may over/underestimate the total uncertainty.

Figure 24: To what extent do these results depend on the considered wind farms (e.g. large vs. small farm) within the BE-NL concession zone?

Minor comments:
L6: Since there is only one tuning parameter per model, and that tuning parameter is directly related to the wake expansion rate, it may help to explicitly state that. The authors may consider changing ‘larger tuning parameters’ by ‘larger wake expansion rate’. A similar comment holds for the conclusion.

L99: Could the authors mention how the turbulence intensity is obtained from the SCADA data?

L116: Which quantitative criterion is used for ‘nearly constant active power’ in the filtering procedure?

Section 3.2: Is it correct that the wake expansion calibration parameter (k or A) is calibrated separately for each of the four wind farms at each accepted time stamp, but not for every turbine separately? Mentioning this explicitly would improve the clarity.

L149: Since mathematical description of the wake models is not given, the authors may need to add references so that k and A uniquely defined.

L160: ‘earlier work’ Are the authors referring to the calibration procedure of section 3?

Section 4 is rather lengthy (page 7-21) and at times difficult to follow. The manuscript may benefit from making this section more concise, for example, by focusing on the mathematical exposition of the Bayesian hierarchical model, and motivating/interpreting it on the go.

L274: A good model should both have small bias and small error variance. Consequently, variance-sensitivity seems related to the representativeness of a calibration parameter.

L280: The normalization mainly seems to get rid of the power-curve scaling, but the effect of the free stream wind speed is not mentioned. It is also not entirely clear how such a scaling would reduce the dependence of the spread.

L326: The specification of a prior for z_p implies that a posterior distribution is obtained for z_p. Is that the case? Otherwise, z_p only serves as a reparameterization variable for the (hierarchical) prior w_p \sim N(w_global, sigma_park^2). The authors may consider reorganizing this to improve the clarity.

L443: What is meant by ‘how well the parameter is constrained’?

Figure 5: Is there only SCADA data available from 4 out of the 12 farms in the BE-NL concession zone? If so, this should be mentioned in section 2.

Figure 7: What do the dots represent?

Section 5 currently seems to interrupt the flow of the Bayesian analyses in Section 4 and 6. The manuscript may benefit from transferring the parts of Section 5 required to understand Section 6 to Section 2, and the rest to the appendix. However, this remains at the authors’ discretion.

L571: Does 10-100% mean 10-100% of the peak value of the PDF?

Table 5 & L750: ‘summarised’ How are the posteriors over the 25 quantile bins translated to the stability classes (e.g. closest bin, linear interpolation, average, …)?

L761: Please define the total, internal, and external wake losses mathematically or give a reference which defines them.
Citation: https://doi.org/10.5194/wes-2026-120-RC1
RC2: 'Comment on wes-2026-120', Anonymous Referee #2, 13 Sep 2026

Please see the attached for the referee report.

Citation: https://doi.org/10.5194/wes-2026-120-RC2

Diederik van Binsbergen, Pieter-Jan Daems, Timothy Verstraeten, Konstantinos Vratsinis, Amir Nejad, and Jan Helsen

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Short summary

A key challenge for offshore wind farms is to quantify wakes reliably from turbine measurements. This study develops a statistical approach that learns from measured turbine data how these wakes vary between wind farms and transfers that knowledge to new sites with quantified uncertainty. It also assesses how atmospheric conditions affect wake recovery. The results show that stable conditions lead to slower recovery and larger predicted losses, while unstable conditions support faster recovery.


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