Optimal Control of Degrading Units through Threshold-Based Control Policies

Round 1

Reviewer 1 Report

The authors of this article consider two degradation models to analyze the behavior of a degrading system. The optimal control problem is solved in steady state. A Markovian model is considered for numerical calculations.

In section 2 the authors include the variable V. The variable U_{x(V)} is defined as 'the unit is restored to a new state in a random period of time', what is this state?

Page 4. Formulas (1) and (2) should be revised.

The variable Y is used throughout the document since (2), was it previously defined?

One thing I would like to see clarified is that the state space given in section 2.3. From Figure 3 we can see that after a signal state, the system could occupy states less than m.

The algorithm given in (10) needs to be improved.

Author Response

Answer to Reviewer 1.

 

Dear Reviewer 1,

I would like to thank you for your useful and appropriate remarks and comments. The paper was considerably revised. Changes in the article are marked in red for ease of reference   Below I give some explanations to the given remarks.

1. The sentence has been reworded. What is meant here is the fact that after repair the unit is as good as new one and the degradation process starts again from state 0.
2. The formulas (1) and (2) were revised.  
3. Inaccuracy in cycle length notation has been eliminated. The random variable Y_{m,n} denotes for any model the random duration of the regeneration cycle. 
4. At the beginning of Section 2.3, the distribution function for the time to absorption for an auxiliary absorbing Markov chain with a set of states E_Y is given. This general result is used further for the original degradation process X(t) with the set of states E.

 

In the revised version I have tried to improve the content and logic of the paper with respect to your remarks and comments. I hope that now the paper will satisfy the requirements of the journal and you could support me with the publication. Thank you once more.

 

With best regards,

Dmitry Efrosinin

 

Reviewer 2 Report

This paper focuses on degradation systems operating under a threshold-based policy. However, some descriptions are not clear. Some revisions are necessary in the manuscript.

1. There are too many parameters. It is recommended to add a nomenclature.

2. The title says some aspects are too general. Please be more precise.

3. FIG. 6 has no corresponding value for the horizontal and vertical coordinates.

4. Please explain the robustness, accuracy, and stability of the proposed method.

5. In the review, authors have focused on optimal control problems. The comparisons of different optimal control methods are suggested to supply to indicate advantages of your work, which can refer to：[a] Journal of Modern Power Systems and Clean Energy, vol. 9, no. 4, pp. 919-929, July 2021; [b] Journal of Modern Power Systems and Clean Energy, vol. 10, no. 2, pp. 286-299, 2022

Author Response

Answer to Reviewer 2.

 

Dear Reviewer 2,

I would like to thank you for the time you spent reviewing my paper. The paper was considerably revised.  Changes in the article are marked in red for ease of reference   Below I give some explanations to the given remarks.

1. A table of the main terms used in the article has been added at the end of the Introduction section.
2. The title of the article has been changed to a simpler and more accurate version “Optimal control of degrading units through threshold-based control policies”.
3. Figure 6 has been changed in accordance with the comment.
4. Some of the arguments for the proposed approach are added in the introduction. The main advantage is the relative simplicity of the model, where performance and reliability characteristics are presented explicitly. Optimization of these characteristics is carried out using only two parameters, which simplifies the computational process.
5. Thank you very much, very interesting articles. But I think the issues covered in the articles are quite different from what is suggested in this article.

 

In a new version I did my best to improve content and logic as well as grammar and style of the text. I hope that the paper will satisfy now the requirements of the journal Mathematics and you could support me with the publication. Thank you once more.

 

With best regards

 

 

Round 2

Reviewer 1 Report

I think that this new version is approppriate to be published by Mathematics