Optimization with Time and Frequency Constraints Using Automatic Differentiation: Application to an Aircraft Electrical Power Channel

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper presents a hybrid ODE optimization method incorporating automatic differentiation and applies it to aircraft power channel optimization. However, the following aspects need improvement:

1. The paper states that the proposed solver can handle hybrid ODEs (including discontinuous events), but it does not clearly compare its advantages and disadvantages with existing methods. For example, compared to conventional event-driven simulation methods, how does this solver improve computational efficiency, stability, or accuracy? Is there numerical evidence to support these claims? A more detailed explanation is needed to clarify these points.
2. The paper mentions the use of the Routh criterion to ensure system stability. However, this criterion is typically used for stability analysis of continuous systems rather than as an optimization constraint. How does the optimization process ensure that this criterion does not introduce non-convexity or convergence issues?
3. The study applies the SQP (Sequential Quadratic Programming) method with exact gradient computation but does not discuss potential alternatives or hybrid approaches, such as gradient descent or evolutionary algorithms. Including such discussions would enhance the completeness of the optimization analysis.
4. The computational cost of DE (Differential Evolution) in this study is significantly higher than that of SQP (210 hours vs. 6 hours). However, as a gradient-free stochastic optimization method, DE inherently requires more iterations. There are many improved versions of DE, such as those incorporating advanced evolutionary strategies (e.g., CMA-ES or L-BFGS hybridized with DE). Additionally, SQP requires a multi-start approach to find the global optimum, whereas DE is inherently a global optimization method. Is this comparison truly fair?
5. How does the proposed method compare with existing standard tools such as MATLAB Simulink or other ODE solvers? A direct comparison with widely used tools would help demonstrate its advantages.

Formatting Issues:

1. Parameters should be italicized, and vectors should be bold to follow mathematical notation standards (e.g., lines 198–205 on page 6).
2. The phrase "Error! Reference source not found." appears on line 245 of page 8. Please check and correct this issue.
3. Flowcharts have inconsistent formatting—some have fill colors while others do not. Additionally, arrows appear in different styles, and some are misaligned. Please standardize the formatting across Figures 2, 3, 4, and 5.
4. The term multi-start appears in different quotation styles (“” and 《》) throughout the text. Please ensure consistency.
5. Many references are outdated (some over 40 years old). Consider removing or replacing them with more recent works.

 

Author Response

Thank you for your comments on our paper. It helped us to deepen some aspects of the paper. We hope our improvements will answer to your questions.

1. The paper states that the proposed solver can handle hybrid ODEs (including discontinuous events), but it does not clearly compare its advantages and disadvantages with existing methods. For example, compared to conventional event-driven simulation methods, how does this solver improve computational efficiency, stability, or accuracy? Is there numerical evidence to support these claims? A more detailed explanation is needed to clarify these points.

The Runge-Kutta 45 (ODE 45 in Matlab) solver was used in this optimization problem because:

• It is a common used one for electrical drives problems (containing power electronics models) in many software tools such as Matlab/Simulink.
• Its variable step size can be adapted, especially for discrete events and FFT computation.

This solver is currently used in our research in Matlab-simulink with a specific management of the time step that allows to manage the possible stiff behaviours without using a stiff ODE method. It allows to show the feasibility of the approach. Practically, it consists on implementing C S-Function using the method mdlGetTimeOfNextVarHit(SimStruct *S) to manage events and specifically dynamic changes.

Therefore, there is no need to have a ‘stiff’ solver, because for each switch commutation, the time step is strongly reduced, to avoid numerical issues and to manage the apparition of a faster or slower dynamic. Note after a commutation, even if the time step is strongly reduced, if the dynamic is slow, this time step increases rapidly to be adapted to the simulated dynamic.

Moreover, we needed a solver that can be stopped automatically after the sizing criteria have been computed :

• After detecting the steady-state and running one over operating period.
• Or after detecting the crossing of a threshold that justifies to no simulate more (this is also the case for the detection of a time response).

This functionality is not available in main state-of-the-art solvers.

Finally, one of our constraints when programming was to have access to the code of the solver, in order to couple it to the model. Thus, a compromise was done between computational performances and the fact to have a “white-box” solver.

It is important to know that the ODE solver is not derivated, the derivation deals only with the ODE equation (creating an extended ODE). Therefore, the ODE solver can be changed without questioning the approach.

This explanation was added in a new sub-section 3.2 of the paper.

2. The paper mentions the use of the Routh criterion to ensure system stability. However, this criterion is typically used for stability analysis of continuous systems rather than as an optimization constraint. How does the optimization process ensure that this criterion does not introduce non-convexity or convergence issues?

Yes, it is true that Routh Criterion is only for continuous system. In fact, here to compute it, a global equivalent circuit of the aircraft power channel (based on a RL-serie circuit, cf. reference 11 of the paper) was established. It was done by bringing the resistances and inductances of all components to the continuous side. Therefore, the Routh criterion is applied on a global (average) behaviour of the system.

These explanations were added in section 4.2.2 to explicit this point.

 

3. The study applies the SQP (Sequential Quadratic Programming) method with exact gradient computation but does not discuss potential alternatives or hybrid approaches, such as gradient descent or evolutionary algorithms. Including such discussions would enhance the completeness of the optimization analysis.

 

4. The computational cost of DE (Differential Evolution) in this study is significantly higher than that of SQP (210 hours vs. 6 hours). However, as a gradient-free stochastic optimization method, DE inherently requires more iterations. There are many improved versions of DE, such as those incorporating advanced evolutionary strategies (e.g., CMA-ES or L-BFGS hybridized with DE). Additionally, SQP requires a multi-start approach to find the global optimum, whereas DE is inherently a global optimization method. Is this comparison truly fair?

Answer to questions 3 and 4 :

SQP and DE methods have their own advantages and disadvantages, so it is possible to compare them :

• Indeed, DE is an optimization method often used by designers, because it avoid being trapped in a local minimum.
• However, we showed in the paper that, despite it can find local optimal solution, SQP is better to deal with tens of constraints (DE is generally limited to 10 ones), and the possibility to have gradients improves the running time of the optimization.

CMA-ES is similar to the differential evolution (DE) algorithm used in the paper. L-BFGS is a Quasi-Newton method, which is the same type of SQP, but it was not natively designed to deal with constraints. In the optimization problem of the paper, there are several constraints, and SQP was specifically designed to solve optimization problems with equality and inequality constraints, using KKT method.

Regarding hybridized methods (combining gradients and global method, such as SQP with DE), yes it could have been applied to our optimization problem, to avoid the “multi-start” procedure used and can provide a good strategy to face with global optimum.

This last point was added to the perspectives of the methodology.

5. How does the proposed method compare with existing standard tools such as MATLAB Simulink or other ODE solvers? A direct comparison with widely used tools would help demonstrate its advantages.

We wanted to have an open-source and non-commercial software tool, which includes the ODE solver, and it was the not case with Matlab. Our objective was not to compare the performances of our ODE solver to other ones, but the performances of the different optimization approaches (stochastic without gradients : DE, deterministic with gradients : SQP) we implemented. The same ODE solver was used for every optimization algorithm we run.

This point was added to the new sub-section 3.2 of the paper.

 

Formatting issues

1. Parameters should be italicized, and vectors should be bold to follow mathematical notation standards (e.g., lines 198–205 on page 6).

The text was modified to check this.

2. The phrase "Error! Reference source not found."appears on line 245 of page 8. Please check and correct this issue.

The text was modified to check this (it’s a reference of Figure 6).

3. Flowcharts have inconsistent formatting—some have fill colors while others do not. Additionally, arrows appear in different styles, and some are misaligned. Please standardize the formatting across Figures 2, 3, 4, and 5.

Arrows misalignment was checked, such as their style. The white blocs in Figure 4 means these blocs are not derivated (this point was added in the legend of the Figure 4).

4. The term multi-startappears in different quotation styles (“” and 《》) throughout the text. Please ensure consistency.

The 《》quotation style was retained for all multi-start occurrences.

 

5. Many references are outdated (some over 40 years old). Consider removing or replacing them with more recent works.

References outdated over 40 years old were replaced by more recent ones, especially :

• Reference 33 by :
  • Oberst, Ulrich. "The fast Fourier transform." SIAM journal on control and optimization 46.2 (2007): 496-540.
• References 51 to 55 by :
  • Allain, L., Merdassi, A., Gerbaud, L., & Bacha, S. (2009, September). Automatic modelling of Power Electronic Converter, Average model construction and Modelica model generation. In Proceddings of the Modelica Conference.
  • Cappuzzo, F., Broca, O., & Allain, L. (2015, July). Methodologies and processes to achieve earlier virtual integration of aircraft systems. In 6th European Conference for Aerospace Sciences.
  • Allain, L., Gerbaud, L., & Van DerSchaeghe, C. (2002, November). Object Oriented Modelling for model capitalisation and simulation of electromechanical systems. In IEEE 2002 28th Annual Conference of the Industrial Electronics Society. IECON 02(Vol. 1, pp. 431-436). IEEE.
  • Allain, L., L. Gerbaud, and Ch Van Der Schaeghe. "Capitalisation and treatment of models for the optimization of electric drives." Optimization and Inverse Problems in Electromagnetism (2003): 205-212.
  • Merdassi, Asma, Laurent Gerbaud, and Seddik Bacha. "A new automatic average modelling tool for power electronics systems." 2008 IEEE Power Electronics Specialists Conference. IEEE,

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents a methodology and tool to address the problem of optimal sizing in the context of the domain of power electronics in the presence of time-domain and frequency-domain constraints and objectives. The article also presents an efficient optimizer that is coupled with a differential equation solver that is capable of leveraging automatic differentiation and symbolic derivation. The authors claim that the developed solver automatically detects the steady state and extracts time-domain and frequency-domain features from simulations to be optimized or constrained. The methodology is illustrated through its application to the optimal design of an aircraft electrical power channel.

The paper is interesting and well-written, and the illustrations are quite motivational. However, some readers would appreciate the possibility to take a look at their codes in Python, whether that is via a repository on the cloud, or through a reasonable request to the authors. Also, the originality of the paper could improve if the applications went beyond signal processing and electric signals, and at least mentioned a reference on pricing financial intruments.

There is a high level of similarity with reference 17. I recommend to rephrase these parts.

Author Response

Thank you for your comments on our paper. It helped us to deepen some aspects of the paper. We hope our improvements will answer to your questions.

The paper is interesting and well-written, and the illustrations are quite motivational. However, some readers would appreciate the possibility to take a look at their codes in Python, whether that is via a repository on the cloud, or through a reasonable request to the authors.

Sure, the code is available by a simple request to the authors. This statement has been added at the end of the paper.

Also, the originality of the paper could improve if the applications went beyond signal processing and electric signals, and at least mentioned a reference on pricing financial intruments.

The methodology was tested on power electronics application, but it can be used in other fields (e.g. mechanical engineering). In particular, economical criteria could be added to the optimization problem.

This point was explained in the second paragraph of the section 5 of the paper (Discussion).

There is a high level of similarity with reference 17. I recommend to rephrase these parts.

These parts were rephrased. However, the sub-section 2.6.1.2 (Average and Root Mean Square (RMS) values) was removed, since it was not used in the optimization problem of the paper, and it has already been detailed in reference 17.

Reviewer 3 Report

Comments and Suggestions for Authors

Review on the article “Optimization with time and frequency constraints using Automatic Differentiation: Application to an aircraft electrical power channel” by Lucas Agobert, Laurent Gerbaud  and Benoit Delinchant

In the presented article, the authors proposed an original methodology for using automatic differentiation along with mathematical properties for calculating derivatives. An exact calculation method was developed that can be used both for time modeling and for fast Fourier transform calculations, which requires access to the code of the computational program implementation. The developed method is illustrated by a practical application for modeling the aircraft power supply channel, which is discussed in the article.

The results of the paper are interesting, based on well-known approaches that the authors originally used to create their own effective methodology.

Comments to the text.

1.Lines 207-208. Remove repetitions in sentences.

2.Line 245. Correct the typo - incorrect reference.

3.Line 334. Correct the typo x ̂

4. Does the * sign mean multiplication? Add an explanation.

5.Line 490. For what value of f is the value k=37? More detailed explanations are needed.

6.Line 500. Additional explanations are needed for the notation in Table 2: iL2?

Generally speaking, the text of the article requires careful correction of formatting flaws. The authors revealed carelessness in writing the text (absence of commas, other punctuation marks, etc., shortcomings in the design of the list of references).

I suggest that the authors carefully correct the text, taking into account the above.

Author Response

Thank you for your comments on our paper. It helped us to deepen some aspects of the paper. We hope our improvements will answer to your questions.

 The results of the paper are interesting, based on well-known approaches that the authors originally used to create their own effective methodology.

Comments to the text.

1. Lines 207-208. Remove repetitions in sentences.

The repetition was removed.

2. Line 245. Correct the typo - incorrect reference.

The text was modified to check this (it’s a reference to Figure 6).

3. Line 334. Correct the typo x ̂

The text was modified to check this.

4. Does the * sign mean multiplication? Add an explanation.

The * sign means multiplication. It was replaced by  to avoid confusion.

5. Line 490. For what value of f is the value k=37? More detailed explanations are needed.

k=37 represents a frequency f=14800 Hz (=37*400, where 400 Hz is the fundamental frequency). In this sizing problem, the frequency-domain constraints have to be respected for low frequencies (up to 14800 Hz).

This precision was added in sub-section 4.2.2.

6. Line 500. Additional explanations are needed for the notation in Table 2: iL2?

iL2 is the channel input current, linked to R2 resistor. Table 2 gives the upper constraint that the Individual Harmonic Rate (IHR) of iL2 has to respect. A line was added after Table 2 to explicit that statement.

Generally speaking, the text of the article requires careful correction of formatting flaws. The authors revealed carelessness in writing the text (absence of commas, other punctuation marks, etc., shortcomings in the design of the list of references).

The text was modified to check this.

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript presents an optimization methodology that integrates automatic differentiation to handle time-domain and frequency-domain constraints in dynamic systems efficiently. The approach is applied to the optimal design of an aircraft electrical power channel, demonstrating significant improvements in computational efficiency. 

 

The text is well-written, clear, and presents the technical aspects with property. However, because of this emphasis, the paper lacks intuition about the problem of sizing it is approaching. In summary, the problem is taken for granted, and this could be a problem for non-expertise readers. One paragraph in the introduction illustrating a simple example of what should be sized would correct this deficiency. There is also a missing reference in line 245. Under such modifications, I recommend the publication of the work.

   

 

Author Response

Thank you for your comments on our paper. It helped us to deepen some aspects of the paper. We hope our improvements will answer to your questions.

The text is well-written, clear, and presents the technical aspects with property. However, because of this emphasis, the paper lacks intuition about the problem of sizing it is approaching. In summary, the problem is taken for granted, and this could be a problem for non-expertise readers. One paragraph in the introduction illustrating a simple example of what should be sized would correct this deficiency.

A paragraph has been added in the introduction, explicating the sizing of a ballistic shooting (a simple optimization problem) to present the problematic of sizing dynamic models.

There is also a missing reference in line 245. Under such modifications, I recommend the publication of the work.

The missing reference was corrected.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

All have been revised.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have updated their draft according to my suggestions. I thank them for this. I have no further recommendations.