Algorithm for Propeller Optimization Based on Differential Evolution

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This article describes the choice of an optimization algorithm for solving problems and the optimal design of an unmanned aerial vehicle propeller. To solve the problem using evolutionary algorithms, it was transformed into an unconstrained optimization problem using a penal function. The paper needs some improvements. My detailed comments are as follows:

 

1.      Are the optimization algorithms described in the paper applicable to all types of air propeller designs? Or is it only applicable to specific propeller types or sizes?

2.      In describing the optimization algorithm, the paper mentions the use of a genetic algorithm for optimization. However, there is no detailed explanation as to why the genetic algorithm was chosen and no comparison with other optimization algorithms.

3.      When describing the air propeller design, the paper mentions the use of Bezier curves to describe the shape of the blades. However, not enough details are provided to explain how these curves are used for shape description.

4.      Do the experimental results mentioned in the paper adequately verify the validity of the proposed design methodology? More experiments are recommended, including performance tests under different conditions and comparative analysis.

5.      The paper does not mention anything about propeller materials, manufacturing processes, etc. Should these factors, which have a significant impact on both propeller performance and cost, be considered in the optimization process?

6.      In the introduction section, some of the following relevant literature studies are missing:

[1] Bocii, L. S., Di Noia, L. P., & Rizzo, R. (2019). Optimization of the Energy Storage of Series-Hybrid Propelled Aircraft by Means of Integer Differential Evolution. Aerospace, 6(5), 59.

[2] Muratoglu, A., Tekin, R., & Ertuğrul, Ö. F. (2021). Hydrodynamic optimization of high-performance blade sections for stall regulated hydrokinetic turbines using Differential Evolution Algorithm. Ocean Engineering, 220, 108389.

 

[3] Lin, J., Yao, H. D., Wang, C., Su, Y., & Yang, C. (2023). Hydrodynamic performance of a rim-driven thruster improved with gap geometry adjustment. Engineering Applications of Computational Fluid Mechanics, 17(1), 2183902.

Author Response

Thank you for your valuable comments that have improved the quality of the paper.

1. Are the optimization algorithms described in the paper applicable to all types of air propeller designs? Or is it only applicable to specific propeller types or sizes?

The article presents two case studies: in one of the cases a tractor propeller is optimized for an electric motor of a fixed-wing aircraft; in the second case a propeller is optimized for a helicopter.In addition, the algorithm can be used to optimize the shape of any propeller as long as transonic speeds are not reached.

2. In describing the optimization algorithm, the paper mentions the use of a genetic algortihm for optimization. However, there is no detailed explanation as to why the genetic algorithm was chosen and no comparison with other optimization algorithms

In point 2.8, it is made clear that an algorithm based on Differential Evolution was used, the explanation of why the SHADE evolutionary algorithm was used was extended.

3. When describing the air propeller design, the paper mentions the use of Bezier curves to describe the shape of the blades. How, not enough details are provided to explain how these curves  are used for shape description.

The explanation of how the propeller geometries are generated was extended in points 2.3 and 2.4.

4. Do the experimental results mentioned in the paper adequately verify the validity of the proposed design methodology? More experiments are recommended, including performance test under different conditions and comparative analysis.

In the Results section, two different cases were exposed in order to evaluate the performance of the algorithm. In one of the cases, the size of the initial population was evaluated. While in another case, the obtained result was evaluated by comparing it with experimental tests.

5. The paper does not mention anything about propeller materials, manufacturing process, etc. Should these factors, which have a significant impact on both propeller performance and cost, be considered in the optimization process?

Subsection 3.3 was added where details of the experimental tests carried out are given.

6. In the introduction section, some ot the following relevant literature studies are missing:

The suggested references have been added.

Reviewer 2 Report

Comments and Suggestions for Authors

The article proposes the algorithm (optimization based) for design of the highly efficient unmanned aerial vehicle propeller.

 

 

Below are my observations:

1)     The novelty of the work must be clearly addressed and discussed, compare your research with existing research findings and highlight novelty.

2)     Figure 2:  The quality of this figure should be improved.

3)     The article is generally well structured and the algorithm is well described, but validation of this algorithm and comparison is lacking. How exactly does unmanned aerial vehicle propeller performance increase using this algorithm. There is no quantification of the advantages of using this algorithm. In the discussion part, it is mentioned that the methodology was proven on a real propeller of unmanned aerial vehicles; in the paper itself, there is a missing part where more data would be given about that real propeller that was constructed on the basis of the proposed algorithm.

Comments on the Quality of English Language

Minor editing of English language required

Author Response

Thank you for your valuable comments that have improved the quality of the paper.

1. The novelty of the work must be clearly addressed and discussed, compare your research with existing research findings and highlight novelty

 

We clearly addressed in Introduction and Conclusion that the novelty of the paper is in the use of Bézier curves to optimize the shape of the screws and the associated flexibility of the resulting solutions:

 

In Introduction: “The issue of applying Bézier curves to propeller shape optimization problems is not sufficiently covered in existing studies and is the novelty of the current work.”

In Conclusion: “The methodology of air propellers optimization using Bézier curves is developed. Compared to existing solutions, the proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span.”

 

 

2. Figure 2: The quality of this figure should be improve.

All the figures were re-created to improve the quality of these.

3. The article is generally well structured and the algorithm is well described, but the validation of this algorithm and comparison is lacking. How exactly does unmanned aerial vehicle propeller performance increase using this algorithm. There is no quantification of the advantages of using this algorithm. In discussion part, it is mentioned that the methodology was proven on a real propeller of unmanned aerial vehicles; in the paper itself, there is a missing part where more data would be given about that real propeller that was constructed on the basis of the proposed algorithm.

The section 3 of Results was extended to expose the comparison that was made between a propeller created with our algorithm and a commercial propeller used in unmanned aerial vehicles. The main characteristic that was evaluated was to compare the required power of the propellers to achieve certain thrusts. The propeller created with our algorithm requires less power compared to the commercial propeller.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have conducted a very interesting and thorough study of a propeller optimization algorithm. I do not have any major remarks on the methodology and results presentation but I have some remarks that should not be too much trouble for the authors to address:

 

1. Line 79, please elaborate on how did you conclude to the penalty function.

 

2. Line 189, please elaborate on the CFD methodology. The analysis at this point is extremely limited.

Comments on the Quality of English Language

I would recommend a spelling check, as the manuscript has some typos that must be corrected before proceeding with journal publication.

 

E.g.,

line 89 "a continuous and continuous" must be rephrased

line 119 "infinite" should probably be "finite"

line 127, please check the legend in the figure, it does not correspond to the curves illustrated (none of the legends has markers)

 

 

Author Response

Thank you for your valuable comments that have improved the quality of the paper.

1. Line 79, please elaborate on how did you conclude to the penalty function.

This part is explained in section 2.1

2. Line 189, please elaborate on the CFD methodology, The analysis at this point is extremely limited.

Section 2.7 was created to describe this step.

Reviewer 4 Report

Comments and Suggestions for Authors

1) The Abstract should be rewritten. At the moment, the Abstract is too general and overall, I did not believe that it presents this paper and performed research in concise and most appropriate way. In the Abstract the Authors should add some of the most important results obtained in this research (their exact values), which are currently completely missing. These results will surely help to highlight novelty of the preformed research already in the Abstract.

2) The English is readable and understandable, but it should be notably improved in many sentences or whole paper parts. For example, in the sentence: “This paper describes step-by-step the selection of methods of parametrization of geometric model, design calculation of propeller aerodynamics, and the use of metaheuristic algorithms of parametric optimization.” – there is no necessity to use the word “of” so many times in one sentence – the whole sentence should be clearer written. This is just one example, the corrections, modifications and improvements related to the English are required throughout the paper text.

3) Due to the high number of abbreviations, symbols and markings used throughout the paper text, in the paper should be added a Nomenclature inside which will be listed and explained in one place all of them. The Nomenclature will notably improve paper readability.

4) Figure 3 - Aerodynamic coefficients – it is completely unknown how the Experimental and Open Foam coefficients presented in this Figure are obtained (or from where these results are taken) – it should be clearly highlighted and explained.

Aerodynamic coefficients obtained by Xfoil have a much higher deviation in relation to the Experimental values than coefficients obtained by Open Foam. The Authors should discuss, highlight and present which the acceptable deviations in relation to the Experimental values are? From this Figure 3, no one cannot be sure that the deviations obtained by Xfoil, in relation to the Experimental values, are acceptable and that they can be effectively used in the calculations. The explanations related to this matter are highly required.

5) Figure 5 – the same mentioned in my previous comment related to Figure 3 is also valid for this Figure (the second part of previous comment related to Figure 3). The Authors should present some kind of evidence that the results obtained by CFX and Cross-section are sufficiently accurate and precise and that they can be proper representatives of Experimental results. In any Figure, a curve which is visually close to the experimental one can result with unacceptably high deviations during the further calculations, so it is not sufficient to state that the obtained results are close to Experimental ones – the acceptable accuracy and precision ranges are required.

In addition, Figure 5 I consider as validation. It is advisable that the Authors add more validation results to this Figure and involve one or two more parameters for which will be performed the same comparison. Validation based on the two parameters only, in some cases, may not be sufficient – so it is advisable to add more validation parameters (of course, the Authors should select which exact additional parameters will be the most appropriate ones).

6) Figure 8 – this Figure is very interesting, but it is not fully clear. The Authors should select better and more intuitive representation of this Figure.

7) The scientific novelty and contribution of this paper to the specific research field surely exists, but they should be much better explained and described. In the literature can be found many different methods for propeller optimization – so the Authors should better and more precisely describe the benefits (and possible disadvantages) of this method in comparison to all others currently known from the literature.

8) As the Abstract, the Conclusions section should also be improved with the most important obtained results (their exact values). Also the Conclusions seem to be too descriptive and general at the moment, without any exact details and results obtained in the presented analysis.

9) The references used in the List of References are dominantly older than 5 or 10 years. I can agree that for some base literature the year of publication is irrelevant, but the majority of the References should be recent ones. Otherwise, the paper’s relevance nowadays can be questionable. The Authors should involve much more recent literature from this research field in the paper text (along with proper explanations).

 

Final remarks: This paper is interesting, but it should be properly corrected and improved (according to the comments above) during the revision process.

Comments on the Quality of English Language

Extensive editing of English language required.

Author Response

Thank you for your valuable comments that have improved the quality of the paper.

1. The Abstract should be rewritten. At the moment, the Abstract is too general an overall. I did not believe that it presents this paper and performed research in concise and most appropriate way. In the Abstract the authors should add some of the most important results obtained in this research (their exact values), which are currently completely missing. These results will surely help to highlight novelty of performed research already in the Abstract.

The Abstract was rewritten, your comments were taken into account.

2. The English is readable and understandable, but it should be notably improved in many sentences or whole paper parts.

 

English has been corrected throughout the article.

 

3. Due to the high number of abbreviations, symbols and markings used throughout the paper text, in the paper should be added a Nomenclature inside which will be listed and explained in one place all of them. The Nomenclature will notably improve paper readability.

We decoded many of the acronyms in the text and we found the article to be more readable this way than by adding a separate Nomenclature section. If we still need to add Nomenclature, we can add it quickly.

4. Figure 3 – Aerodynamic coefficients – it is completely unknown how the Experimental and OpenFOAM coefficients presented in this Fugure are obtained – it should be clearly highlighted and explained.

Section 2.6 was created to explain this point.

5. Figure 5 – the same mentioned in my previous comment related to Figure 3 is also valid for this Figure.

Section 2.7 was created to explain this point.

6. Figure 8 – this Figure is very interesting, but it is not fully clear. The Authors should select better and more intuitive representation of this Figure.

New figures were proposed to make a qualitative comparison of the geometries of the obtained propellers.

7. The scientific novelty and contribution of this paper to the specific research field surely exist, but they should be much better explained and described. In the literature can be found many different methods for propeller optimization – so the authors should better and more precisely describe the benefits of this method in comparison to all others currently known from literature.

 

We clearly addressed in Introduction and Conclusion that the novelty of the paper is in the use of Bézier curves to optimize the shape of the screws and the associated flexibility of the resulting solutions:

 

In Introduction: “The issue of applying Bézier curves to propeller shape optimization problems is not sufficiently covered in existing studies and is the novelty of the current work.”

In Conclusion: “The methodology of air propellers optimization using Bézier curves is developed. Compared to existing solutions, the proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span.”

 

8. As the Abstract, the Conclusions section should also be improved with the most important obtained results. Also the Conclusions seem to be too descriptive and general at the moment, without any exact details and results obtained in presented analysis.

The conclusions were rewritten, taking into account the new results presented in section 3.

9. The references used in the List of References are dominantly older than 5 o 10 years. I can agree the for some base literature the year of publication is irrelevant, but the majority of the References should be recent ones.

New references were added (no older than 5 years old), to support our work.

Reviewer 5 Report

Comments and Suggestions for Authors

This paper is well organized, the methods used seem to be proper and the results seem to be reliable. 

It is suggested to accept this paper for publication if the following questions are clarified, 

1. How the quality of grids is conserved if the shape of propeller was changed? 

2. specify the numerical methods used, will the turbulence model influence the results?, is it a 2D or 3D simulation?

3. many abbreviations were used but not explained.

4. what's the cost for the computation?

Author Response

Thank you for your valuable comments that have improved the quality of the paper.

1. How the quality of grids is conserved if the shape of propeller was changed?

The number of stations used in the Isolated Sections Method was the same to optimize different propeller types, the most notable change is in the design space, the design intervals change according to the propeller type, at the moment these intervals are empirically defined.

2. Specify the numerical methods used, will the turbulence model influence the results?, is it a 2d or 3d simulation?

Sections 2.6 and 2.7 were created to explain these points.

3. Many abbreviations were used but no explained.

We’ve decoded abbreviations in the text. If this is not enough, we are ready to supplement explanation the required acronyms.

4. What is the cost for the computation?

The characteristics of the computer equipment used to perform the tests are set out in Section 3. In addition, the calculation time of each of the tests is mentioned.

Reviewer 6 Report

Comments and Suggestions for Authors

The aim of the work was defined in the first sentence of the abstract: "this article describes the choice of an optimization algorithm for solving problems and the optimal design of an unmanned aerial vehicle propeller". However, it is not really clear what this is about. For optimization, an objective function is necessary. What is the function in this case? What does "this article describes the choice of an optimization algorithm" mean? Have the authors developed their own new algorithm? What are the results of this work? All this is not in the abstract.

 

I see a similar problem in the conclusions. What is the purpose of this work? Is it about the aerodynamic optimization of the propeller or maybe about the presentation of the algorithm. These are two different issues in my opinion.

 

Page 1, row 27 "An optimized air propeller can significantly reduce emissions, improve acoustic response, and enhance performance." Exactly. These may be different optimization goals.

 

The first paragraph of the introduction section only shows that many researchers have optimized various aspects of this issue. Moreover, these researchers used different tools. However, there are no conclusions in the context of the presented manuscript.

 

I think that the presented algorithms do not increase the value of this paper. I think that the most important thing is the method itself.

 

Figures 7 and 8 show the optimal (in what sense?) propeller geometry. How do you know that it is optimal? There is no comparison with other methods. Sketch 8 shows 30 different profile shapes. How is the reader supposed to read the coordinates of these profiles from such small images? How to technically implement 30 different chords on one small propeller? I'm not a technologist, but I think this is not an easy issue. Moreover, do I understand correctly that the characteristics of this propeller (fig. 8) are in figure 5?

Author Response

Thank you for your valuable comments that have improved the quality of the paper.

1. The aim of the work was defined in the first sentence of the abstract: “this article describes the choice of an optimization algorithm for solving problems and the optimal design of an unmanned aerial vehicle propeller”. However, it is not really clear what this is about. For optimization, an objective function is necessary. What is the function in this case? What does “this article describes the choice of an optimization algorithm” mean? Have the authors developed their own new algorithm? What are the results of this work? All this is not in the abstract.

The abstract and the main objective were restated to give a better understanding of the work carried out.

2. I see a similar problem in the conclusions. What is the purpose of this work? Is it about the aerodynamic optimization of the propeller or maybe about the presentation of the algorithm. These are two different issues in my opinion.

The conclusions were also rewritten.

3. Page 1, row 27 “An optimized air propeller can significantly reduce emissions, improve acoustic response, and enhance performance” Exactly. These may be different optimization goals.

It was made clear in the text that the optimization of the propeller is merely aerodynamic, specifically it is to reduce the required power of the propeller to achieve a predetermined thrust.

4. The first paragraph of the introduction section only shows that many researchers have optimized various aspects of this issue. Moreover, these researchers used different tools. However, there are no conclusions in the context of the presented manuscript.

 

We have indicated both in the introduction and in the conclusion that the value of the proposed methodology lies in the flexibility of the geometric description of the propeller blades, obtained through the use of Bézier curves. And noted that this approach distinguishes us from known works.

 

In Introduction: “The issue of applying Bézier curves to propeller shape optimization problems is not sufficiently covered in existing studies and is the novelty of the current work.”

In Conclusion: “The methodology of air propellers optimization using Bézier curves is developed. Compared to existing solutions, the proposed approach allows for more flexibility in setting the propeller shape, for example, using a variable airfoil over the blade span.”

 

 

5. I think the the presented algorithms do not increase the value of this paper. I think that most important thing is the method itself.

 

Thank you, we agree that the method is the most important thing. We clearly addressed in Introduction and Conclusion that the novelty of the paper is in the use of Bézier curves to optimize the shape of the screws and the associated flexibility of the resulting solutions.

 

 

6. Figures 7 and 8  show the optimal propeller geometry. How do you know that is optimal? There is no comparison with other methods. Sketch 8 shows 30 different profile shapes. How is the reader supposed to read the coordinates of these profiles from small images? How to technically implement 30 different chords on one small propeller? I am not a technologist, but I think this is not an easy issue. Moreover, do I understand correctly that the characteristics of this propeller are in the figure 5?

The images were improved so that the reader can better understand the geometries of the obtained propellers.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

1.      The study did not demonstrate significant innovations in topic selection, research methodology, or conclusions and lacked a breakthrough from existing research.

2.      The experimental design of the study failed to adequately account for a variety of possible confounding factors, leading to the questionable credibility of the results. A rigorous experimental design is a prerequisite for drawing reliable conclusions, which is exactly what the manuscript lacks.

3.      There are incomplete and inaccurate references to relevant literature cited in the text. This not only affects the completeness of the study but may also lead to a misinterpretation of the study by the readers.

4.      There are instances of unclear language and poor logic in the manuscript. This makes it possible for readers to be confused or misinterpreted during the reading process.

 

In summary, I believe that the manuscript has more problems and do not recommend it for publication. In order to improve the quality and academic value of the manuscript, I suggest that the authors carefully revise and improve the manuscript to address the above issues.

Author Response

We thank the reviewer for his careful attitude to our work! We have revised the manuscript, taking into account the received comments:

1. The study did not demonstrate significant innovations in topic selection, research methodology, or conclusions and lacked a breakthrough from existing research.

We have highlighted a feature of our technique, which is the transition from twist angle to angle of attack in design variables in the optimization problem, which eliminates the need for an internal cycle for the angle of attack searching and significantly reduces the number of calculations when solving the optimization problem.

We’ve added in Introduction detailed description of our approach:

“BEMT and IMS methods for calculating a given shape propeller aerodynamic performance require an airfoil angle of attack selection cycle, causing the need for an airfoil drag polar calculation database [40] and drag curve extrapolation when the geometric twist exceeds the limits on angles of attack. Replacing the twist angle in the design variables with the angle of attack allows the optimization process to immediately search for the desired airflow field and eliminate an additional cycle of searching for the angle of attack for each propeller individual. In the current work a modification of ISM consists in the airfoil effective angle usage instead of the geometrical twist angle is presented. The geometric twist is calculated for the optimum propeller when the optimum distribution of the angle of attack over the span is found. This approach also improves the robustness of the optimization process by directly controlling the constraints imposed on the angle of attack.

The parametric geometric model of the propeller used in the current work implements Bezier curves to specify the dependencies of the parameters of thickness, curvature, and positions of the maximum thickness and maximum curvature of the airfoil along the blade span. This makes it possible to use different thicknesses and curvatures of airfoils along the blade span, which allows the aerodynamic twist of the blade to be varied.

In addition, the article considers the use of interpolation of the calculation of the aerodynamic characteristics of the airfoils on the blade span, which also allows to accelerate the process.“

 

2. The experimental design of the study failed to adequately account for a variety of possible confounding factors, leading to the questionable credibility of the results. A rigorous experimental design is a prerequisite for drawing reliable conclusions, which is exactly what the manuscript lacks.

We have revised the experimental section of the manuscript, adding plots with the original data (Figure 18), adding details for constructing averaged curves (“The dependences of thrust on required power for each test were interpolated by smooth univariable spline of 3 degree and then lines of average values and scatter equal to the standard deviation”), and also adding the scatter between the experimental results (Figure 19).

Statistics have been added to the experimental section - the characteristics of each propeller are built based on seven series of measurements, which allows us to obtain reliable data and build scatter for analyzing measurements.

3. There are incomplete and inaccurate references to relevant literature cited in the text. This not only affects the completeness of the study but may also lead to a misinterpretation of the study by the readers.

We have significantly revised the literature review, increasing it to 57 sources, adding a review of Bezier curves to describe propeller geometry and a review of methods for calculating aerodynamic characteristics.

We also supplemented all the references proposed in the previous review round.

4. There are instances of unclear language and poor logic in the manuscript. This makes it possible for readers to be confused or misinterpreted during the reading process.

We worked on the language of the manuscript, especially completely updating the Introduction and Conclusion. We hope that the manuscript has become clearer for understanding.

Reviewer 2 Report

Comments and Suggestions for Authors

I would not fully agree with the sentence in the Introduction part: „The issue of applying Bézier curves to propeller shape optimization problems is not sufficiently covered in existing studies and is the novelty of the current work.”

The use of Bezier curves for shape (geometry) optimization is by no means new. Bezier curves are very often used for geometry (shape) parameterization and optimization of propellers and all other turbomachines, therefore it is not entirely correct to state that this is the contribution of this work. I recomend you to study a literature in the field of turbomachinery design, more precisely propellers, and state exactly what the advantage of your approach is.

I believe that the literature review in the Introduction part should still be improved and that it should include relevant literature in the field of turbomachinery design or at least in the propeller area.

Author Response

We thank the Reviewer for his attention to our article.

The second reviewer's recommendation is that a more comprehensive review of the literature on Bezier curves in the propeller area is needed, as well as exactly statement what the advantage of authors approach is.

We have added the literature review citing 9 sources of the use of Bezier curves to describe the shape of both airfoils and blades in general:

“The methodology of propeller geometry parametrization is part of its the aerodynamic optimization problem. Bezier surfaces and curves have the possibility of generating complex shapes using some control points [29]. This feature makes the use of Bezier curves and surfaces suitable for describing propeller cross-sections (airfoils [30, 31, 32]) and the propeller blades in general, including chord and twist [1, 33]. Due to the reduced number of control points, these curves and surfaces are used in propeller shape optimization processes [34], especially when used in conjunction with evolutionary algorithms [35, 36] and artificial neural networks [1, 33].”

We also add more detail statement the advantage of presented approach:

“The parametric geometric model of the propeller used in the current work implements Bezier curves to specify the dependencies of the parameters of thickness, curvature, and positions of the maximum thickness and maximum curvature of the airfoil along the blade span. This makes it possible to use different thicknesses and curvatures of airfoils along the blade span, which allows the aerodynamic twist of the blade to be varied.”

Reviewer 4 Report

Comments and Suggestions for Authors

The Authors have performed all mentioned elements in my review. Also, an experimental validation is also performed and added in the paper.

After proper and detail paper revision, I have no more concerns related to this paper.

The paper should be accepted and published in a presented (revised) form. 

My congratulations to the Authors.

Author Response

We thank the reviewer for his careful attitude to our work!

Reviewer 6 Report

Comments and Suggestions for Authors

In my opinion, this paper looks much better than the previous version. The purpose of the work was clearly presented. I also see a lot of effort these authors put into improving the rest of this paper. That is why I recommend this paper for publication in this form.

Author Response

We thank the reviewer for his careful attitude to our work!