Overall Thoughts

Overall, this manuscript has a clear objective and application. The authors propose a method to estimate the impact of changes in control design variable on the objective function in an optimization study without directly solving a control co-design (CCD) optimization problem. The authors argue that this method would save the user time in identifying a scenario where the objective function is insensitive to the control design variable—making CCD somewhat trivial—without actually going through the effort of solving the more complex problem. This intention to reduce computational time in the design process is well-motivated, but I believe the discussion of this point could be more comprehensive.

A second thread of the manuscript is the application of the proposed method to the optimal design of a wind turbine tower with fatigue load constraints. I believe the details of this case study convolute and overshadow the main objective at times, and a more refined description is necessary.

After addressing these comments and some other minor revisions, I believe this manuscript will be suitable for publication in Wind Energy Science.

Major Comments

Line 171: What is the surrogate model for LCOE and how was it calibrated? The model form of the objective function and its dependence on the design variables is necessary information for the reader to understand the case study.

Line 189: “A gain schedule is created by varying the parameters r and q over the operational range.” I don’t think this phrase is intuitive for a reader unfamiliar with LQR control. Also, a few lines later: “The weight matrix entry associated to the tower top velocity was found to give a good fatigue damage reduction, without affecting the standard deviation of the power production in a significant manner.” This description is vague and not reproducible. Similar to the previous comment, the details of the case study (LQR control law, LCOE minimization, etc.) are extensive compared to the objective function sensitivity estimator. I think they require more explanation to be comprehensible by the reader or should be removed or moved to an appendix.

Line 261: I think the comparison of computational cost between the proposed estimator and the full CCD solution could be clearer. The authors state that “12 [shouldn’t this be 16?] evaluations of the full set of aeroelastic simulations for each configuration” were run for the estimator, while the CCD problem requires “running the full set of simulations 50 times for the soft-soft configuration and 20 times for the standard configuration.” I’m not sure why these specific numbers of simulations were chosen. Furthermore, how could this result generalize outside of the tower design case study?

Minor Comments

Title: I find the question format of the title to be a bit strange and not descriptive of the particular methods introduced by the paper. Perhaps something in the vein of: "Identifying design optimization problems for control co-design approach with objective function sensitivity estimator”?

Line 95: How valid is the simplification to only consider active constraints, and assuming that the active set of constraints does not change with control tuning? If your constraint is a function of control tuning, it seems plausible that a change in control could cause an inactive constraint to become active.

Line 114: There is repeated use of the phrase “diminishing returns” of controller tuning without any elaboration on what the authors mean. I understand that in Figure 5, a marginal increase in control tuning leads to a diminishing change in optimal tower mass. I think this point could be clearly defined earlier in the manuscript if this phrase is to be used.

Line 133: Is resonance avoidance not included in the soft-soft configuration in order to simplify the problem and focus on the objective function sensitivity? This is a valid approach, but I think the authors should state that point clearly. Otherwise, the problem definition for the soft-soft case seems impractical.

Line 147: “On the other hand, the AEP used to calculate the LCOE is only marginally impacted by the control tuning, since it is based on the average power production, which tends to be relatively insensitive to such changes.” Is there a source to justify this statement?

Line 149: I have trouble following the process for solving the optimization problems. For a given tower height, the loads on the tower are simulated and used to optimize the tower mass. When is the height of the tower changed and LCOE minimized? It seems like the solution of the outer optimization problem has not been described. Also, it is stated that “If the change in [tower mass] design is greater than a given threshold, the process is repeated iteratively (Bottasso et al., 2016).” What threshold is used in this paper?

Line 203: I think more details on the aeroelastic simulations could be given. What certification standards were used for fatigue analysis? How is turbulence synthetically generated and modeled?  How long is the transient period?

Line 203: How is the fatigue damage resulting from different wind speeds combined into a single estimate of lifetime fatigue damage? Is there a probability distribution of wind speeds that inform taking a weighted average of the different fatigue values?

Figure 4: Panel ‘b’ contains the Lagrange multipliers for geometric constraints, and ‘c’ for the fatigue damage. The caption appears to have the wrong labels.

Table 1: The caption could be simpler and less confusing. The reference for the table is the optimization solution without CCD for a control input of zero. Then, LCOE is optimized with CCD and with the estimator method and compared relative to that reference value. There are a few points in the manuscript where the change of optimal LCOE is presented without a clear definition of what the reference value is.

Line 269: The authors state that the estimator accurately predicts the optimal objective value, but not the optimal design. I think it could be useful to clarify here that the goal of this method is to identify how much the objective function can be improved by control tuning, and in cases where the estimator signals much potential, a full CCD study would be performed. Otherwise, the reader could jump to the conclusion that the proposed method would not ultimately reach an optimal design.

Line 284: Would there be any advantage to quantifying uncertainty of the estimator? In other words, high uncertainty in the estimator could encourage a user to explore the CCD problem regardless of the compared sensitivity in the objective function.

Typographical Comments

Equation 10: Could be split up into two separately numbered equations, one for the “outer” optimization problem and one for the “inner” problem.

Reviewer summary

The paper considers control co-design (CCD) for wind energy systems, with as specific case study, the design of soft-stiff and soft-soft wind turbine towers. Such integrated design of the system and its controller is typically computationally demanding, due to the computational cost of an analysis for each controller design. Specifically for tower design, load calculations determine this cost.

The paper presents a methodology to create a simplified version of the CCD optimization problem that is less computationally demanding, but still can provide information about the interaction between system and control design. The idea is that first solving the simplified optimization problem, its results can make it clear whether or not it is worth it to solve the CCD optimization problem.

The paper proposes approximations (‘estimators’) for the CCD problem. They are based on decoupling the design of the system and the controller, assuming the latter fixed. So for a fixed controller design, an optimal system design x* is obtained. The CCD problem is then approximated by doing first-order and second-order approximations of the neighborhood of x* as a function of controller parameters.

When applied to the tower design case studies, it is seen that the first-order gives some, but arguably too limited information about the usefulness of doing CCD. The second-order approximation does give sufficient information. The results are that the effort of doing CCD is worth it for soft-soft towers, but not for soft-stiff towers.

General comments

The paper discusses a topic of widespread interest in the wind energy systems design community: approaches to reduce the computational burden of design optimization. Any advances in this area are scientifically relevant. The paper discusses existing literature touching on CCD-type approaches to show specific interest in this area.

The approximations proposed as part of the methodology are theoretically nontrivial. They require careful derivation of gradients and higher-order derivatives, adding assumptions to simplify expressions obtained. A good part of the paper is dedicated to this, including two appendices, one of which contains a very commendable and informative analysis of how their approximations can break. The notation is generally good, but would need to be introduced more carefully in advance, to avoid readers having to deal with too much while trying to understand the derivations. Furthermore, the assumptions made at different locations in the paper should be made more explicit, to make sure readers have a clear view of the approximation's limitations.

These theoretical discussions are performed on an abstract formulation of the CCD optimization problem. The case study's concrete optimization problems are not directly formulated as such. There is also not a clear translation of this concrete problem to the abstract one, to the detriment of the reader's understanding. What complicates matters is that the simplification performed is not limited to just the approximations introduced, but also involves a surrogate model for one aspect of the concrete optimization problem. With the current presentation, it cannot be expected that readers can understand how the concrete and abstract problems are related with reasonable effort.

The goal of the methodology is, effectively, to substantially reduce the computational cost associated to the optimization of a wind energy system. Therefore, it is as important to get a good quantitative view of computational cost (or time, given fixed computational resources) next to the accuracy of the approximations. In the paper, the accuracy is sufficiently described, but the computational cost is not. It is dealt with in one paragraph, which is unclear and in one possible way of reading it may even imply that there is not much difference between solving the simplified problem and the full CCD problem. (In terms of costly load calculations: 50 vs. 16 for soft-soft and 20 vs. 16 for soft-stiff.) Were this reading to be correct, this would severely undermine the significance of this paper.

Overview of specific aspects

My judgments here are based on my current understanding of the work.

• Does the paper address relevant scientific questions within the scope of WES?
  
   Yes. Reducing computational cost of wind energy system design optimization.
• Does the paper present novel concepts, ideas, tools, or data?
  
  Yes. The specifics of the methodology proposed, i.e., the approximations, are new in this context.
• Is the paper of broad international interest?
  
  Yes. All wind energy system designers could benefit from significant advances in this area.
• Are clear objectives and/or hypotheses put forward?
  
  Yes. The reduction of computational cost of wind energy system design.
• Are the scientific methods valid and clear outlined to be reproduced?
  
  Partial. The general overview of the methodology and its application to the case study are clear, but its details are not.
• Are analyses and assumptions valid?
  
  Yes. The analysis is set up well and much care is taken to discuss the assumptions made and their limitations.
• Are the presented results sufficient to support the interpretations and associated discussion?
  
  Partial. The analysis of accuracy of the approximations seems quite solid, but the analysis of computational cost is insufficient.
• Is the discussion relevant and backed up?
  
  Mostly. The discussion is mostly conceptual, but includes at least one statement that would require further explanation (“We can expect … reduced benefits.”).
• Are accurate conclusions reached based on the presented results and discussion?
  
  Partial. Statements about computational cost are insufficiently backed by presented results.
• Do the authors give proper credit to related and relevant work and clearly indicate their own original contribution?
  
  Yes. They seem to do a good job of mentioning related relevant literature.
• Does the title clearly reflect the contents of the paper and is it informative?
  
  Partial. It mentions CCD and design optimization, but not anything about the nature of the approximations or the concrete case study. (It is my opinion that a more informative title would be desirable. For example: “A second order approximation for investigating control co-design relevance applied to wind turbine tower design”)
• Does the abstract provide a concise and complete summary, including quantitative results?
  
  Yes. Computational cost discussion may need to be amended.
• Is the overall presentation well structured?
  
  Partial. See specific comments.
• Is the paper written concisely and to the point?
  
  Partial. See specific comments.
• Is the language fluent, precise, and grammatically correct?
  
  Mostly. Any remaining issues can be easily fixed by the journal's copy-editors.
• Are the figures and tables useful and all necessary?
  
  Yes. More could/should be added; see specific comments. (N.B.: Figure 1 deserves explicit praise.)
• Are mathematical formulae, symbols, abbreviations, and units correctly defined and used according to the author guidelines?
  
  Mostly. Some may need to be introduced earlier/better; see technical corrections.
• Should any parts of the paper (text, formulae, figures, tables) be clarified, reduced, combined, or eliminated?
  
  Yes. See general comments above and specific comments.
• Are the number and quality of references appropriate?
  
  Yes.
• Is the amount and quality of supplementary material appropriate and of added value?
  
  Yes. The appendices with details about the approximations are actually necessary if no external reference makes their content easily accessible.

Specific comments

Paper presentation, focus, and clarity

As mentioned, the paper's current structure can cause confusion with the reader when trying to understand how the concrete optimization problems (10-12) are reduced to the abstract one (1|2). Namely, in Sec. 3.1, 3.3, 3.4 the case study problem is described. In Sec. 3.2, the reduction to the abstract problem is attempted. I think it can be clearer when Sec. 3 only focuses on presenting the concrete optimization problem and then a new section after that focuses on the reduction of the concrete problem to the abstract version. This new section should be far more elaborate than the current Sec. 3.2 and really explicitly make the correspondence between the f and g of the abstract problem and the constraints and objective of the concrete problem, making sure that it is clear how the two-layered optimization structure of the concrete problem is gotten rid of.

(You could even consider moving the concrete problem first and the abstract problem second, but that is a matter of taste.)

Figures and tables

Figure 6 is very illuminating. It would be good to add such a figure as well for the first-order approximation, so that the difference between the two approximations becomes clearer. Furthermore, the full CCD problem will have given rise to a decent number of LCOE-evaluations and would allow to also visualize the actual underlying LCOE-surface for both tower types. It would be of great value if this were done, as it would give a feeling how far or close the approximations (+surrogate) are from the ‘ground truth’.

Technical corrections

• p1l11: control-co design → control co-design
• p1l20: optimal production → optimal energy production
• All units should be typeset according to the standards, i.e., with a space between number and unit. For example, p2l28: 13MW → 13 MW.
• avoid double parentheses around citations by absorbing by properly using citation macros. For example, p2l29: (e.g. Zahle et al. (2016)) → (e.g. Zahle et al., 2019) [check macros such as \citet and \citep]
• p2l37: “A promising problem for CCD applications is likely to be sensitive to control tuning.”: this is a central assumption that requires more justification
• p3l70: at each iteration → at each iteration of the optimization algorithm
• p3l76: “If Problem 2 can benefit from a CCD formulation”: this makes no sense, as Problem 2 does not depend on c (only on a constant c_r); likely you want to reformulate this
• p3l78-84: The mathematical notation used here needs to be introduced more elaborately and in a more structured way, so in advance of its usage. Moreover, there should be some discussion of the meaning of dx*, as the natural thing to do would be to consider x(c_r) and x(cr+dc), but the latter is effectively replaced by x(c_r) + dx* (I am not yet fully convinced that using dx* isn't introducing some implicit assumptions)
• p3l84/p4l90: Define ‘stationarity condition’/‘stationary point of the objective function’ formally/explicitly (I guess it is there where the gradient is zero?).
• p4l97: f(x,c_r,λ): what is the λ doing there?
• p4l102-103: Why not mention assumptions explicitly? The current formulation is vague.
• p4l112: It would be easier to follow if the explanation of the figure is in-text and the caption is just the title of the figure.
• p5l115-116: Why not mention assumptions explicitly? The current formulation is vague. (Likely you mention them after Eq. 9, but then the connection should be made explicit.)
• p5l120: finite → small? infinitesimal?
• p5l121-122: Again ‘validity assumptions’ are not made explicit (it may be as easy as referring back explicitly to some lines above)
• p5Fig1caption: define ‘coupling’ formally/explicitly in the text before this aspect of the figure is discussed
• p6Eq11: δf should be defined before being used
• p6l144: noted m → denoted by m
• p6l147: discussion of marginal effect of control on AEP is vague/informal; can you make it more formal/explicit?
• p7l163-164: “Therefore, the estimator in Eq. (9) is defined using this constraint only and applied to the tower mass minimization problem.”: I think this statement should become more prominent/explicit
• p7l165: ‘As a result’: say explicitly that Eq. 9's first term therefore becomes zero.
• p7l169-170: “because the active set is robust, there is little interaction between constraints, and the objective and constraints tend to be nearly linear around the optimum”: this statement really needs some justification/references
• p7Fig2caption: the information in the caption should be integrated in the text; currently this part is really confusion, also due to the fact that the relevant information is spread over text and caption instead of forming a unified discussion.
• p7l173: ‘surrogate model’: explain better what the role of the true LCOE is, e.g., by expanding Eq. 14 with an extra part ‘= …’ making the connection explicit
• p9l217-218: 1e-… → 10^{-…}
• p10l237-238: “Adding this constraint also reduces the relative importance of fatigue, reducing the potential for CCD, but also showing why the soft-soft tower has lower mass than the standard configuration.”: vague, so make more explicit
• p12Table1: Make it explicit that the second-order estimator is considered here
• p12l260-265: This paragraph really needs to be expanded to its own subsection at least, as computational cost quantification and discussion is severely underrepresented in the paper. Also, ‘12 evaluations’ are mentioned here, but shouldn't that be 16 as 4 times 4?
• p14l295-296: “We can expect that including this feature in the controller design would translate into reduced benefits.”: clarify
• p16EqB3: dc* → dc
• p16EqB4: f(dc) → f(x*+dx*,dc)?
• p17l349-350: “we assume that the constraints that do not depend on x contribute marginally to the change of optimum”: is this a reasonable assumption (justify)
• p17EqC1: I was wondering why there is no cross term in x and c included.
• p18-21: Make it explicit that the titles refer to some assumption whose violation is studied.