Drive System Inverter Modeling Using Symbolic Regression

Round 1

Reviewer 1 Report

The study is comprehensive, and its content is valuable. I find the topic to be very fascinating. Despite this, I believe the following corrections are necessary to improve the article's scientific level.

1- What is the motivation for this project? The authors must demonstrate a scientific interest in the objectives and results of their paper.

2- In the abstract, there are no clear indications of the article's scientific merit or novelty.

3- Can the authors explain clearly in the abstract the originality and value of the proposed system?

4- Compare the proposed systems with similar ones.

5- Would it be possible to share the code of the proposed method on GitHub?

Author Response

Respected Reviewer,

we thank you for your time spent in reviewing our manuscript. Please find the answers to comments posed below. Changes in the manuscript made due to your comments have been marked green.

The study is comprehensive, and its content is valuable. I find the topic to be very fascinating. Despite this, I believe the following corrections are necessary to improve the article's scientific level.

1- What is the motivation for this project? The authors must demonstrate a scientific interest in the objectives and results of their paper.

The goal of this research is to develop an artificial intelligence model based on genetic programming - symbolic regression. Many electrical drives do not measure the phase voltage online for the reason that they require expensive equipment (such as sensors ) to measure the current value of a certain parameter of the machine device, etc.

Furthermore, to further clarify the goal and meaning of this research, the following text was added to the manuscript: ”The motivation for this research is to develop an AI model that will be able to compete with its performance, if not replace the current estimation methods with equal reliability and precision. Unlike online methods that estimate model parameters whenever new data is available during the operation of the physical system, this approach only uses data that has been measured over a certain period and stored, for example, in one simple program file.”

The authors reviewed the paper based on the reviewer's comments, and the novelty is visible and defined at the end of the introduction section with the following text: "The main novelty of this paper lies in the application of the GPSR algorithm for the design of symbolic expressions for drive inverter modeling. In difference with a majority of other AI and ML-based modeling techniques, utilization of GPSR can provide a single numerical expression that can be easily implemented as a part of a control system."

However, to clarify and additionally clarifying and emphasizing the novelty of this work, that part was modified and in the new version of the manuscript it was written as: "Based on previous research, there is a visible lack of methods for estimating inverter parameters, which represents a great financial loss because precise and expensive sensors must be purchased. For this reason, the GPSR AI algorithm is applied, which, with its properties and performance, successfully estimates the parameters of the inverter. Additionally, it is important to mention that, unlike most other AI and ML-based modeling techniques, the use of GPSR provides a numerical expression that can be easily implemented as part of a control system. This information is of high importance because by applying this method and procedures defined in this paper, a high-quality AI algorithm can be achieved that will estimate the parameters of the inverter with minimal error. Furthermore, it is important to define that this algorithm is not limited to this type of problem, i.e. with a minor modification and adding new data to the proposed algorithm, the same results can be achieved for another type of problem.”

In this way, the contribution and application of this artificial intelligence algorithm is additionally emphasized, and the authors hope that the contribution is better defined compared to the previous version of the manuscript.

2- In the abstract, there are no clear indications of the article's scientific merit or novelty.

According to this comment, and comments provided by other reviewers, the following changes are made in the abstract:

Abstract: For accurate and efficient control performance of electric motor drives precise values of phase voltages are required. In order to achieve control of the electric drive, the development of mathematical models of the system and its parts is often approached. Data-driven modeling using artificial intelligence can often be unprofitable due to the large amount of computing resources required. To overcome this problem the idea in this paper is to investigate if a genetic programming- symbolic regressor (GPSR) algorithm could be used to obtain simple symbolic expressions which could estimate the mean phase voltages (black-box inverter model) and duty cycles (black-box compensation scheme) with high accuracy using a publicly available dataset. To obtain the best symbolic expressions using GPSR the random hyperparameter search method and 5-fold cross- validation were developed and used. The best symbolic expressions were chosen based on their estimation performance which was measured using the coefficient of determination (R2), mean absolute error (MAE), and root mean squared error (RMSE). The best symbolic expressions for estimation of mean phase voltages achieved R2, MAE, and RMSE values of 0.999, 2.5, and 2.8, respectively. The best symbolic expressions for the estimation of duty cycles achieved R2, MAE, and RMSE values of 0.9999, 0.0027, and 0.003, respectively. The presented procedure in this paper shows that the symbolic expression for accurate estimation of mean phase voltages and duty cycles can be obtained using the GPSR algorithm.

3- Can the authors explain clearly in the abstract the originality and value of the proposed system?

The originality and contribution of this paper is additionally emphasized in the abstract and the added text is as follows:”The originality of this work lies in the application of the GPSR algorithm, which, based on a mathematical equation it generates, can estimate the value of mean phase voltages and duty cycles in a three-phase inverter. Using the obtained model, it is possible to estimate the given aforementioned values, which represents an opportunity to replace expensive online equipment with a cheaper, more precise, and faster approach such as a GPSR -based model.”

In this way, the authors hope that the abstract contains a sufficient amount and quality of information.

4- Compare the proposed systems with similar ones.

At the end of the manuscript, the cost comparison of proposed modeling approach and similar approaches is added:

To additionally compare certain methods with the proposed one, the text in section 3.3 was added:

"To conclude whether the method applies to a problem, it is necessary to compare the proposed method with other similar ones. This was done using a tabular representation, ie using Table 6. Three approaches were chosen: the proposed GPSR method, ML methods, and deterministic methods. ML methods in the last few years and even decades have been solutions for complex problems, whether it is a classification, regression, or any other kind of problem, depending on the desired solution. Deterministic methods provide an exact solution based on defined rules, and it is important to take this into account as a potential solution. Also, five criteria were selected that must be met to select the appropriate algorithm:

• Model complexity - refers to the structure of the model itself, and how many parameters it contains, for example, models like MLP or CNN are black box-models, that is, the user is not aware of what is happening at a certain moment during training and cannot influence it before the results are calculated. Only input and output are known.
• Model performances - indicates what the user wants, which is what kind of performance a particular algorithm showed, that is, how high-quality the obtained results are.
• Model execution time - refers to the time of execution or obtaining results from the moment of starting the model estimation process.
• Modeling procedure complexity - is the complexity when creating an algorithm to perform a certain task.
• Modeling computational complexity - is the hardware requirement, i.e. how many resources each model uses to perform the task

In addition to GPSR, ML methods and conventional deterministic methods were taken into account. ML methods are quite complex methods, and many AI algorithms are black-box models, which means that the user will hardly be able to independently model a certain algorithm and is not familiar with the current actions that are executed in that model. Deterministic modeling methods are of medium difficulty, various mathematical equations describe relationships and potential future values that can be predicted for a given issue, but there are cases that deterministic methods are rejected because the deviation from the real value has too much oscillation. Contrary to the two competitors, the complexity of GPSR is extremely low, the entire structure of the algorithm is visible, and it is much easier to influence the algorithm itself than other compared methods. As far as model performance is concerned, all three approaches can get top results and one against the others does not pose a challenge. Model execution time is one of the key factors for choosing a method to solve a certain problem. In this case, GPSR gives the fastest result from the beginning of the estimation initialization to its completion, compared to ML and deterministic methods. The reason for this is that with GPSR, the final result is a single equation that describes the system almost perfectly, which was confirmed by the results of this research. Regarding the complexity of the procedure, GPSR and ML models have a low complexity rate for algorithm preparation. The reason for this is that for most ML algorithms, there are already publicly available programming libraries that are easy to implement. In the end, it remains to compare the computational complexity of all three approaches. It can be seen that deterministic models have the least computational complexity, while GPSR and ML models have a high one. The reason for this is the complexity of the dataset and the desired performance of the model. For the output results to be reliable and accurate, more demanding hyperparameters must be defined, which results in higher hardware requirements for the execution of the task.”

5 - Would it be possible to share the code of the proposed method on GitHub?

At the reviewer's request, the code with which this research was conducted is attached to the manuscript under the Data Availability Statement.

Additionally, for review purposes, the link to the page is below:

https://github.com/nandelic2022/InverterGP

We hope you will be satisfied with our answers to the questions posed. We have marked the changes made to the manuscript due to your comments using green highlight.

Kindest regards,
The Authors

Author Response File: Author Response.pdf

Reviewer 2 Report

In the paper titled "Drive System Inverter Modeling Using Symbolic Regression", the authors have investigated if a genetic programming-symbolic regressor (GPSR) algorithm could be used to obtain symbolic expressions which could estimate the mean phase voltages (black-box inverter model) and duty cycles (black-box compensation scheme) with high accuracy using a publicly available dataset. To obtain the best symbolic expressions using GPSR the random hyperparameter search method and 5-fold cross-validation were developed and used. The best symbolic expressions were chosen based on their estimation performance which was measured using the coefficient of determination (R 2 ), mean absolute error (MAE), and root mean squared error (RMSE). The best symbolic expressions for estimation of mean phase voltages achieved R 2 , MAE, and RMSE values of 0.999, 2.5, and 2.8, respectively. The best symbolic expressions for the estimation of duty cycles achieved R 2, MAE, and RMSE values of 0.9999, 0.0027, and 0.003, respectively. The presented procedure in this paper shows that the symbolic expression for accurate estimation of 14 mean phase voltages and duty cycles can be obtained using the GPSR algorithm.

I have the following comments:

1. In this paper, the publicly available dataset [] was used. The initial statistical analysis of the dataset is shown in Table 1. The dataset is missing in the reference.

2. More explanation of Pearson’s correlation analysis is required in the paper.

3. Genetic programming-symbolic regression is an evolutionary algorithm that begins its execution by building a naive population that is unfit for a particular task and through the application of genetic operations for a prespecified number of generations makes them fit for a particular task. Here reference is required.

4. Figure 10. The flowchart of GPSR with random hyperparameter search and 5-fold cross-valdiation should be redrawn.

5. A cost comparison between the presented technique and the sensor-based technique should be done and presented in a separate and new section to motivate the engineers and other researchers to use this technique.

Author Response

Respected Reviewer,

thank you very much for your review of our manuscript. We have tried our best to respond to the issues you have noted in your manuscript. Please find our responses below. Changes in the manuscript made due to your comments have been marked blue.

In the paper titled "Drive System Inverter Modeling Using Symbolic Regression", the authors have investigated if a genetic programming-symbolic regressor (GPSR) algorithm could be used to obtain symbolic expressions which could estimate the mean phase voltages (black-box inverter model) and duty cycles (black-box compensation scheme) with high accuracy using a publicly available dataset. To obtain the best symbolic expressions using GPSR the random hyperparameter search method and 5-fold cross-validation were developed and used. The best symbolic expressions were chosen based on their estimation performance which was measured using the coefficient of determination (R 2 ), mean absolute error (MAE), and root mean squared error (RMSE). The best symbolic expressions for estimation of mean phase voltages achieved R 2 , MAE, and RMSE values of 0.999, 2.5, and 2.8, respectively. The best symbolic expressions for the estimation of duty cycles achieved R 2, MAE, and RMSE values of 0.9999, 0.0027, and 0.003, respectively. The presented procedure in this paper shows that the symbolic expression for accurate estimation of 14 mean phase voltages and duty cycles can be obtained using the GPSR algorithm.

I have the following comments:

1. In this paper, the publicly available dataset [] was used. The initial statistical analysis of the dataset is shown in Table 1. The dataset is missing in the reference.

The following reference is added:

• Stender, M., Wallscheid, O., & Böcker, J. (2020). Data set description: Three-phase IGBT two-level inverter for electrical drives.

2. More explanation of Pearson’s correlation analysis is required in the paper.

According to the provided comment, the following text is added:

“As a unit of correlation measure during Pearson’s correlation analysis, Pearson’s correlation coefficient is used. Pearson’s correlation coefficient between two variables (X and Y) can be defined as:

ρX,Y = cov(X, Y)/σ(X)σ(Y) , (1)

where σ(X) and σ(Y) represent standard deviations od variables X and Y. Furthermore,

cov(X, Y) represent covariance between X and Y, defined as:

cov(X, Y) = E((X − X)(Y − Y)), (2)

where X and Y represent mean values of variables X and Y. Furthermore, in this case E represent the expectation.”

3. Genetic programming-symbolic regression is an evolutionary algorithm that begins its execution by building a naive population that is unfit for a particular task and through the application of genetic operations for a prespecified number of generations makes them fit for a particular task. Here reference is required.

The following references are added:

• Huang, Z., Mei, Y., & Zhong, J. (2022). Semantic linear genetic programming for symbolic regression. IEEE Transactions on Cybernetics.
• Nicolau, M., & McDermott, J. (2020). Genetic programming symbolic regression: What is the prior on the prediction?. In Genetic Programming Theory and Practice XVII (pp. 201-225). Springer, Cham.

4. Figure 10. The flowchart of GPSR with random hyperparameter search and 5-fold cross-validation should be redrawn.

The flowchart is redrawn to address GPSR with random hyperparameter search and 5-fold cross-validation more clearly. The collor coding is used to emphasize the main parts of the methodology and to address the repetitiveness of the algorithm.

5. A cost comparison between the presented technique and the sensor-based technique should be done and presented in a separate and new section to motivate the engineers and other researchers to use this technique.

The authors agree with the proposed comment, and subsection 3.3 has been added.

A cost comparison was additionally written and a tabular record of the comparisons was attached, furthermore, the text was marked in green color for the reason that it was also made at the request of the previous reviewer.

The added text that is linked to the cost comparison table is:

"To conclude whether the method applies to a problem, it is necessary to compare the proposed method with other similar ones. This was done using a tabular representation, ie using Table 6. Three approaches were chosen: the proposed GPSR method, ML methods, and deterministic methods. ML methods in the last few years and even decades have been solutions for complex problems, whether it is a classification, regression, or any other kind of problem, depending on the desired solution. Deterministic methods provide an exact solution based on defined rules, and it is important to take this into account as a potential solution. Also, five criteria were selected that must be met to select the appropriate algorithm:

• Model complexity - refers to the structure of the model itself, and how many parameters it contains, for example, models like MLP or CNN are black box-models, that is, the user is not aware of what is happening at a certain moment during training and cannot influence it before the results are calculated. Only input and output are known.
• Model performances - indicates what the user wants, which is what kind of performance a particular algorithm showed, that is, how high-quality the obtained results are.
• Model execution time - refers to the time of execution or obtaining results from the moment of starting the model estimation process.
• Modeling procedure complexity - is the complexity when creating an algorithm to perform a certain task.
• Modeling computational complexity - is the hardware requirement, i.e. how many resources each model uses to perform the task

In addition to GPSR, ML methods and conventional deterministic methods were taken into account. ML methods are quite complex methods, and many AI algorithms are black-box models, which means that the user will hardly be able to independently model a certain algorithm and is not familiar with the current actions that are executed in that model. Deterministic modeling methods are of medium difficulty, various mathematical equations describe relationships and potential future values that can be predicted for a given issue, but there are cases that deterministic methods are rejected because the deviation from the real value has too much oscillation. Contrary to the two competitors, the complexity of GPSR is extremely low, the entire structure of the algorithm is visible, and it is much easier to influence the algorithm itself than other compared methods. As far as model performance is concerned, all three approaches can get top results and one against the others does not pose a challenge. Model execution time is one of the key factors for choosing a method to solve a certain problem. In this case, GPSR gives the fastest result from the beginning of the estimation initialization to its completion, compared to ML and deterministic methods. The reason for this is that with GPSR, the final result is a single equation that describes the system almost perfectly, which was confirmed by the results of this research. Regarding the complexity of the procedure, GPSR and ML models have a low complexity rate for algorithm preparation. The reason for this is that for most ML algorithms, there are already publicly available programming libraries that are easy to implement. In the end, it remains to compare the computational complexity of all three approaches. It can be seen that deterministic models have the least computational complexity, while GPSR and ML models have a high one. The reason for this is the complexity of the dataset and the desired performance of the model. For the output results to be reliable and accurate, more demanding hyperparameters must be defined, which results in higher hardware requirements for the execution of the task.

We have used blue highlight to note the changes made to the manuscript due to your comments. We hope you will be satisfied with our answers to the questions posed.

Author Response File: Author Response.pdf

Reviewer 3 Report

This paper proposes a genetic programming-symbolic regressor (GPSR) algorithm utilization for black-box modeling of an inverter to estimate phase voltages and duty cycles.

Technical comments:

1. Are there any experimental results or verification?

2. How is this method implemented for any random inverter?

3. Line 2: "However, data acquisition of phase voltage values of electric motor drives requires expensive measurement equipment and contains measurement errors." Measuring phase voltage amplitude is not costly. What exactly do you mean. Please elaborate.

4. More details about the dataset used for Table 1 are needed. 

5. Explain the different parameters introduced in Table 1.

6. Figure 4 is for all dataset variables, Figure 5 is for variables in black-box inverter model, and Figure 6 is dataset variables used in black-box compensation. Can you introduce these variables briefly?

7. Can you please reduce the similarity rating of this work and your previous work [11]?

Editorial comment:

- Lines 118 and 123, reference is missing.

- Indentation formatting is not consistent with the template. 

- Some figure are large compared to their importance and details and occupy too much space.

- The chemical reaction picture for GA in Figure 8 seems irrelevant. Maybe a computer or AI would be a better representation.

- Please use standard shape and format for flowchart in Figure 5.

- Line 340, Figure ??

- Among self citations 13, 14, and 18 one can be enough to explain GPSR.

Author Response

Respected Reviewer,

thank you for your detailed review of our submitted manuscript. The response and answers to the questions you posed are below. Changes in the manuscript made due to your comments have been marked red

.

This paper proposes a genetic programming-symbolic regressor (GPSR) algorithm utilization for black-box modeling of an inverter to estimate phase voltages and duty cycles.

Technical comments:

1. Are there any experimental results or verification?
2. How is this method implemented for any random inverter?

According to comments 1. And 2. The authors have added the following text in the conclusion as a part of future work between lines 428 and 431:

Alongside implementation of more complex regression methods, the authors will examine the performances of proposed method on other inverter models. Furthermore, the physical drive system and inverter will be designed and implemented to experimentally verify the performances of the proposed method.

3. Line 2: "However, data acquisition of phase voltage values of electric motor drives requires expensive measurement equipment and contains measurement errors." Measuring phase voltage amplitude is not costly. What exactly do you mean. Please elaborate.

The authors agree with the comment. This sentence does not describe the problem in a proper manner. The abstract is edited as follows:

Abstract: For accurate and efficient control performance of electric motor drives precise values of phase voltages are required. In order to achieve control of the electric drive, the development of mathematical models of the system and its parts is often approached. Data-driven modeling using artificial intelligence can often be unprofitable due to the large amount of computing resources required. To overcome this problem the idea in this paper is to investigate if a genetic programming- symbolic regressor (GPSR) algorithm could be used to obtain simple symbolic expressions which could estimate the mean phase voltages (black-box inverter model) and duty cycles (black-box compensation scheme) with high accuracy using a publicly available dataset. To obtain the best symbolic expressions using GPSR the random hyperparameter search method and 5-fold cross- validation were developed and used. The best symbolic expressions were chosen based on their estimation performance which was measured using the coefficient of determination (R2), mean absolute error (MAE), and root mean squared error (RMSE). The best symbolic expressions for estimation of mean phase voltages achieved R2, MAE, and RMSE values of 0.999, 2.5, and 2.8, respectively. The best symbolic expressions for the estimation of duty cycles achieved R2, MAE, and RMSE values of 0.9999, 0.0027, and 0.003, respectively. The presented procedure in this paper shows that the symbolic expression for accurate estimation of mean phase voltages and duty cycles can be obtained using the GPSR algorithm.

4. More details about the dataset used for Table 1 are needed.

Information about the dataset is as follows:

“The used data set is collected by meassuring signals of a 3-phase inverter used in the induction motor drive system. Mentioned drive system consist of the induction motor LUST ASH-22-20K13-000 with nominal power 1.5 kW, nominal speed 3000 min−1, and rated phase current of 3.9 A. For the control of the induction motor, 3-phase IGBT inverter SEMIKRON Semiteach IGBT is used. The inverter is fed with a 560 V DC link with rated output current of 30 A. “

In addition to that, Table 1 was added, which describes the input and output parameters, and the entire subsection 2.2 "Data set description" was reformulated, the variables in the data set were described, the number of samples was defined, and a list of symbols found in the entire work was attached using Table 2.

Modified text is as follows:”This research is carried out in two stages. The first stage is the Black Box Inverter Model estimation, while the second is the Black Box inverter compensation scheme. Both stages consist of similar identical parameters, namely: duty cycles (D), phase currents (I), direct current link voltage (Udc), and mean phase voltages (U). However, both stages have different input and output parameters. Before clarifying and defining the division into input and output parameters and which parameters belong to which stage, it is necessary to define the subscripts that are next to each variable. Each variable consists of at least two subscripts: k is the sampling step label, while a, b, andc are labels for one of the three possible phases. The maximum previous samples reach three samples less, i.e. k − 3 sample, which means that the previous three samples are taken to form the current parameter. For example, Ic,k is the phase current parameter on phase c for the k sampling step. Given that signals are generally interpreted as continuous values, and this dataset is a sequence of several small recorded sequences, previous signal values are needed for training the inverter model and inverter compensation scheme, ie they must be included as an addition to the signal in the dataset for each generation step. With the previously given data, it is possible to define the parameters that go into each stage of training. What was done and the input and output parameters are shown in Table 1 below.

Statistical data analysis shown in Table 2 is one of the important steps before embarking on the process before even training the artificial intelligence algorithm. Minimum (Min), maximum (Max), mean and standard deviation (Std) for the dataset size of 234.5 thousand sampling steps is shown. Min indicates the minimum numerical value of the given parameter, the same applies to Max, Mean, and Std. The reason for this is an insight into the state and interrelationships of the data, which shows the complexity of the dataset itself. Furthermore, in addition to the defined symbols and their ratios between Min, Max, Mean and Std values, GP variables have been added that facilitate the subsequent definition of the results, that is, it provides a list of variables behind the GPSR equation which is shown and defined in the Section 3.”

5. Explain the different parameters introduced in Table 1.

Table 1 has been changed to Table 2. The reason for this is because Table 1 was previously added, which defines the variables, i.e. what they represent and what the subscripts indicate. In addition, the Input and output values ​​and which group of variables are used for which stage of the research are defined in Table 1. In addition, the section "2.2. Data set description" is completely restructured, and each variable used in the dataset and their subscripts are described. In addition, symbolic variables related to genetic programming were defined, which will be presented in the Results and discussion section.

Furthermore, the text that was changed in Section 2.2 is the following first, the text that was under Table 1 in the previous version of the manuscript was removed:””It is evident from Table 1 that the research is carried out in two stages. The first stage is the Black Box Inverter Model estimation, while the second is the Black Box inverter compensation scheme. Both stages consist of similar variables, namely Duty cycles (D), phase currents (I), Direct current link voltage (Udc) and mean phase voltages (U).”

The following is the text that was added or changed which is located above Table 1 of the new version of the manuscript.:" This research is carried out in two stages. The first stage is the Black Box Inverter Model estimation, while the second is the Black Box inverter compensation scheme. Both stages consist of similar identical parameters, namely: duty cycles (D), phase currents (I), direct current link voltage (Udc), and mean phase voltages (U). However, both stages have different input and output parameters. Before clarifying and defining the division into input and output parameters and which parameters belong to which stage, it is necessary to define the subscripts that are next to each variable. Each variable consists of at least two subscripts: k is the sampling step label, while a, b, andc are labels for one of the three possible phases. The maximum previous samples reach three samples less, i.e. k − 3 sample, which means that the previous three samples are taken to form the current parameter. For example, Ic,k is the phase current parameter on phase c for the k sampling step. Given that signals are generally interpreted as continuous values, and this dataset is a sequence of several small recorded sequences, previous signal values are needed for training the inverter model and inverter compensation scheme, ie they must be included as an addition to the signal in the dataset for each generation step. With the previously given data, it is possible to define the parameters that go into each stage of training. What was done and the input and output parameters are shown in Table 1 below.”

In addition, a short description of the statistical analysis of the data and the symbols found inside was added and defined by the following text:”Statistical data analysis shown in Table 2 is one of the important steps before embarking on the process before even training the artificial intelligence algorithm. Minimum (Min), maximum (Max), mean and standard deviation (Std) for the dataset size of 234.5 thousand sampling steps is shown. Min indicates the minimum numerical value of the given parameter, the same applies to Max, Mean, and Std. The reason for this is an insight into the state and interrelationships of the data, which shows the complexity of the dataset itself. Furthermore, in addition to the defined symbols and their ratios between Min, Max, Mean and Std values, GP variables have been added that facilitate the subsequent definition of the results, that is, it provides a list of variables behind the GPSR equation which is shown and defined in the Section 3.”

6. Figure 4 is for all dataset variables, Figure 5 is for variables in black-box inverter model, and Figure 6 is dataset variables used in black-box compensation. Can you introduce these variables briefly?

Figures 4, 5, and 6 have been moved to the appendix section, further, a brief description of the variables found in the previously mentioned correlation matrix are described in section 2.2 of the new version of the manuscript.

7. Can you please reduce the similarity rating of this work and your previous work [11]?

Citation [11] has been restructured and modified, the authors hope that the similarity rate is now much reduced. The text that was removed is as follows:" utilized complex ML algorithms and ensemble methods for potential estimation of mean phase voltages and duty cycles for a three-phase Insulated-gate bipolar transistor (IGBT) inverter with two-level control. The black-box inverter model for estimating mean phase voltages and the black-box inverter compensation scheme were examined, where the possibility of estimating duty cycles for

the specified circuit was examined. The research results showed that the application of AI for the black-box inverter model achieves results for

" while the new text describing the quotes in the state of the art is as follows:" used complex ML algorithms and methods to estimate mean phase voltages and duty cycles to solve the problem of a three-phase insulated-gate bipolar transistor converter with two-level control . Two models were tested, one of them is a black-box inverter model for estimating mean phase voltages, while the other is a similar black-box inverter compensation scheme. The given research showed that by using AI it is possible to achieve precise results as shown in the article. The values ​​obtained by the authors are as follows:

In this way, the authors hope that the text has now been brought to an appropriate level.

- Lines 118 and 123, reference is missing.
The missing parenthetical citation has now been placed.

- Indentation formatting is not consistent with the template.
The indentations are now uniform and an equal indentation is made after each new line.

- Some figure are large compared to their importance and details and occupy too much space.
According to the comment, the correlation matrices are moved to the appendix.

- The chemical reaction picture for GA in Figure 8 seems irrelevant. Maybe a computer or AI would be a better
representation.
The authors agree with the given comment and the figure has been modified based on the given proposal. Please also refer to the attachment. 

- Please use standard shape and format for flowchart in Figure 5.
Figure 5 does not represent a flowchart. However, if the Reviewer had referenced Figure 10, the figure is edited according to the comments provided by other reviewers.
The flowchart is redrawn to address GPSR with random hyperparameter search and 5-fold cross-validation more clearly. The color coding is used to emphasize the main parts of the methodology and to address the repetitiveness of
the algorithm.

Please also refer to the attachment.

- Line 340, Figure ??
The figure number (Figure 5) is added.

- Among self citations 13, 14, and 18 one can be enough to explain GPSR.
The authors agree and the citations are reduced to only one of the previous three citations.

We have used red highlight to note the changes made to the manuscript due to your comments. We hope you will be satisfied with our answers to the questions posed.

Kindest regards,
The Authors

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have addressed all of my comments.

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 3 Report

Thank you for answering my previous questions and applying changes.

Lack of experimental validation is a major drawback of this work. Authors mention this as a future work.

There are formatting and grammar errors that requires proofreading.

Table 1 parameters need to be explained.

Figure 6 needs to be explained more.

Derivation method for equations 3-8 need to be briefly explained.

All abbreviations need to be explained the first time they are used (e.g. CNN, MLP, GP, AI, ...)

Size of all figures can be scaled to 70-80% of their original size (in my opinion)

Author Response

Please see the attachment

Author Response File: Author Response.pdf