the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Mar 2024
Probabilistic surrogate modeling of damage equivalent loads on onshore and offshore wind turbines using mixture density networks
Abstract. The use of load surrogates in offshore wind turbine site assessment has gained attention as a way to speed up the lengthy and costly siting process. We propose a novel probabilistic approach using mixture density networks to map 10-minute average site conditions to the corresponding load statistics. The probabilistic framework allows for the modeling of the uncertainty in the loads as a response to the stochastic inflow conditions. We train the data-driven model on the OpenFAST simulations of the IEA-10MW-RWT and compare the predictions to the widely used Gaussian process regression. We show that mixture density networks can recover the accurate mean response in all load channels with values for the coefficient of determination (R2) greater than 0.95 on the test dataset. Mixture density networks completely outperform Gaussian process regression in predicting the quantiles, showing an excellent agreement with the reference. We compare onshore and offshore sites for training to conclude the need for a more extensive training dataset in offshore cases due to the larger feature space and more noise in the data.
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RC1: 'Comment on wes-2024-20', Anonymous Referee #1, 03 Apr 2024
General Comments
The paper presents a probabilistic surrogate which models the aleatory uncertainty in the model output. The epistemic uncertainty is not considered in the presented method. The overall quality of the paper is in general good with significant contribution. However, the authors should provide more explanations in some parts of the manuscript (see in the specific comments) in order to improve the reader’s comprehension and to ensure the reproducibility of their work.
Specific Comments
Comment 1:
In Section 3.3, it is mentioned that “The 10-minute fatigue is calculated using short-term damage equivalent loads (DELST)”
But, Section 4.1 says “The network is trained on the tower base fore-aft moment standard deviation (TwrBsMyt [kN-m] stddev)”.
It is not clear what are the inputs and outputs of the surrogate. Please formulate the surrogate model formally and mathematically.
Comment 2:
Section 3.5 Accuracy metric should be better explained with more detailed information.
Equation 18: It is not clear if the summation is over the whole test dataset or over the seed repetitions of each test sample. How many samples are drawn from the MDN output distribution to compute the statistics (mean, std, and the quantiles)?
Equation 19: How are the quantile functions computed, by combining closed-form formulations for each Gaussian or through sampling directly from the output mixed distribution?
Comment 3:
The seed repetitions with the same input variables are not necessary since the prediction is inferred from neighboring inputs, however, it also means that many neighborhood points are still required to have a good estimate of aleatory uncertainty. In addition, when modelling in high dimensional input space, the actual distance between the test point and the input points might be far. What is the robustness of the model in such case?
Comment 4:
Line 215 “It pushes the coefficients of uninformative features towards zero, effectively pruning the feature space. “
Additional explanations or references would improve the reader’s comprehension.
Comment 5:
Table 1. What is the sensitivity of mixture component numbers in terms of accuracy and computational resources?
Comment 6:
Please describe what is the specific purpose of having two different test datasets TEST1 and TEST2.
Comment 7:
Figure 3: Why is the marginal distribution of alpha a deterministic value only? Isn’t it a uniform distribution for the ‘train’ and ‘test 1’ according to Table 2?
Comment 8:
Figure 6: Please define the spread of the boxes and whiskers (within which quantiles?).
Technical Corrections
Comment 9:
Line 174 “L-BFGS-B algorithm” Limited memory Broyden–Fletcher–Goldfarb–Shanno algorithm for simple bound constraints is not defined.
Citation: https://doi.org/10.5194/wes-2024-20-RC1 -
RC2: 'Comment on wes-2024-20', Anonymous Referee #2, 27 May 2024
- Clarify the difference between design standards and site specific analysis
- What is being "designed" during site analysis? Towers and foundations? Or just checking stock design of rotor/nacelle. Clarify for the reader a bit more at the start. Also has implications; eg if we find we need to change the tower design, have to rebuild the surrogate (which is expensive!)
- Include references/discussion to older freq domain model to show how long been around/lineage
- Are 10 minutes simulations enough for off-shore turbines? Standards specific length to include important wave impacts
- Around line 35 would be nice to have more discussions of the importance of linearity and nonlinear. Contrast what linearized models miss of the physics vs data driven models
- Is it just aleatoric uncertainty? It's true the physical system itself has embedded uncertainty, but in the context of a surrogate the choice and extent of the feature selection and training dataset represent epistomalogical sources of uncertainty.
- Section 1.1; if we know the pdf of Y, how does that get used for design verification? E.g. what do we to get cumulative lifetime fatigue and extreme ultimate loads extrapolation? Nice to have clarity on model usage there, beyond being able to provide variance and pdf info from the surrogate
- Explain what heteroscedasticity actually is and where is comes from in wind turbine context
- Line 105 add ref to non-heteroscedastic paper. Also later in section talk about Kriging and GPR; clarify these are essentially the same
- Section 2.1; mention GP challenges in terms of passing though (or not) the training points depending on detailed implementation and variants of eqn 2. See also line 344
- Line 221; l1 and l2 variables aren't actually defined in the eqns above? Meaning lambda values?
- Fig 2 implied hidden layers of various widths, but table 1 only defines 2 hidden layers not their widths
- Below eqn 17 mention m=10 for blades but before said not looking at blade loads?
- Line 298; not clear what Test2 is. "alter only wind speed and turb;" but still creating how many new sims?
- Fig 3 redo axis labels. Eg "speed" is windspeed? I'd suggest using the Greek variables directly to avoid confusion
- Fig 5. Say in caption what MDN [ x, x] denote. What is x?
- In the results it's not clear how GPR is being used. Eg fig 5; does that 4 separate GPRs trained independently, one for each predicted quantity? I can't see where this is explicitly in results section
- Around line 370 confused if the r squared value is so dependent on choice of training subset, what is table 4 actually showing us? Is that for an average across multiple training sets?
- Should make mention in table 4 for Sigma of the GPR you have negative values. This is a typical for me statistical definition of r squared so good to remind the reader what this means in the context of training and test data
- Would be nice to see a bit more discussion on how the results changed for the different load channels you're looking at EG tower bottom versus later bending
- Figure 7 and the discussion would indicate some optimum number of samples between 500 and 4500. The GPR uses 500. The conclusions talk about reduced simulation needs with the proposed mdn method, so it would be nice to revisit what practicalities these numbers of samples and plies in terms of creating a database. Back to the use case in terms of how the model might need to be rebuilt when changing tower heights, etc in intended use cases for the tool.
Citation: https://doi.org/10.5194/wes-2024-20-RC2 - RC3: 'Comment on wes-2024-20', Anonymous Referee #3, 31 May 2024
Data sets
Training and validation datasets for training probabilistic machine learning models on NREL's 10-MW reference wind turbine Deepali Singh https://doi.org/10.4121/21939995
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