A Multi-Objective Optimization Method and System for Energy Internet Topology Based on Self-Adaptive-NSGA-III

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper proposes an improved NSGA-III algorithm (SA-NSGA-III) for the multi-objective optimization of Energy Internet (EI) topology. The authors introduce a "free edge" transformation strategy to maintain scale-free characteristics and propose an adaptive reference point generation method to enhance the algorithm's search capability. The topic is relevant to the scope of energy systems and complex network optimization. The proposed "free edge" method for maintaining node degree distribution during optimization is an interesting constraint-handling technique. However, the manuscript suffers from several clarity issues, questionable data visualization choices, and minor typographical errors that must be addressed.

 

1. Several challenges for the construction and management of EI are discussed. However, the most important and practical challenge is the renewable energy uncertainty, which requires robust optimization methods to handle, such as “fortifying renewable-dominant hybrid microgrids: a bi-directional converter based interconnection planning approach”. It is suggested to cite and discuss this study in the introduction to enhance the quality of the review.
2. The authors propose a "biased Dirichlet sampling". Please clarify why the bias direction w = [0.6, 1.5, 0.2] was chosen11. Is this specific to the user's preference for Robustness? If so, this should be explicitly stated as a "preference-based" modification, as standard NSGA-III aims for a uniform spread across the Pareto front.
3. The text states: "In four nodes, there are fewer than two nodes connected by a single edge". This phrasing is confusing and mathematically imprecise. Does this mean the subgraph induced by these four nodes must not have isolated nodes, or does it refer to specific connectivity constraints? The authors should provide a clearer graph-theory explanation of the conditions required for a valid free edge exchange. Figure 2 helps, but the text definition is weak.
4. Algebraic connectivity is widely accepted as a spectral measure of graph robustness/connectivity. The metric R (based on high-degree attacks) is also a robustness metric. The authors should justify optimizing both simultaneously. Is there a scenario where lambda_2 increases but R decreases? If they are highly positively correlated, treating them as separate conflicting objectives in a Pareto optimization might be redundant. A correlation analysis between lambda_2 and R on the generated topologies would strengthen the justification.
5. The conclusion mentions that the algorithm faces "memory resource limitations". There is no experimental data regarding the running time (execution time) or memory consumption of SA-NSGA-III compared to the baselines (MOPSO, MOEA/D).

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript presents a multi-objective optimization approach for Energy Internet (EI) topology design using a self-adaptive variant of NSGA-III (SA-NSGA-III). The work addresses an important and timely problem—balancing connectivity, robustness, and operational efficiency in EI networks—and proposes algorithmic enhancements to improve convergence and adaptability. The paper is generally well-structured, the methodology is sound, and the experimental validation is comprehensive. While the contributions are meaningful, there are areas where clarity, justification, and depth could be improved to strengthen the manuscript for publication in a high-impact journal.

 

 

Comments to authors:

 

• Please revise the abstract to include specific numerical achievements from your experimental results. Currently, the abstract describes the proposed method and states that it outperforms other algorithms, but it lacks concrete, quantitative evidence of its performance. Including key metrics such as percentage improvements in fitness, convergence iteration counts, and enhancements in objective values (e.g., algebraic connectivity, robustness, average path length) will make the abstract more impactful and informative for readers.The “free edge” concept and its three types (Figure 2) are briefly introduced but not fully explained in the main text. A more detailed description or a reference to a clearer definition would help.
• The fitness function (Eq. 4) uses fixed weights (α, β, γ). How were these weights determined? Were they tuned empirically? A sensitivity analysis or justification would strengthen the methodology.
• All experiments are conducted in simulated environments (MATLAB). While acceptable, the authors should discuss the applicability to real EI topologies, which may have additional constraints (e.g., geographical, economic, regulatory).
• The parameter settings (Table 1) are provided, but no parameter sensitivity analysis is included. This is important for reproducibility and robustness.
• The discussion focuses largely on fitness values. It would be valuable to analyze the Pareto fronts qualitatively—e.g., how the trade-offs among the three objectives manifest in different topologies.
• The claim that the algorithm avoids local optima is supported by convergence curves, but a more direct analysis (e.g., diversity metrics over iterations) would strengthen the argument.
• Figures 6 and 7, parts (a) and (b): The visual difference between the initial and optimized topologies is not clearly discernible in the current layout. To improve readability and impact, consider the following:
  • Use contrasting visual styles—for example, differentiate the optimized topology with thicker, colored edges or distinct node shapes.
  • Add insets or zoomed panels highlighting regions where edge rewiring is most pronounced.
  • Include a small adjacency matrix or degree distribution plot alongside each topology to quantitatively illustrate structural changes.
  • If space permits, overlay both topologies with transparency to allow direct visual comparison of edge shifts.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors 1. What is the main question addressed by the research? The question to be addressed in this paper is, "How can we optimize an Energy Internet (EI) topology system using a collaborative optimization method to achieve multiple objectives of network connectivity, robustness, and operating efficiency whilst still maintaining all the properties associated with scale-free network topology?" To answer this question, the authors propose a multi-objective walking optimization algorithm, SA-NSGA-III, which utilizes self-adaptive approaches to solve the collaborative multi-objective optimization problem associated with the EI topology.   2. Do you consider the topic original or relevant to the field? Does it address a specific gap in the field? Please also explain why this is/is not
the case.   The answer is yes to all of the above. As far as complex networks, energy systems, and optimization go, network topology plays a pivotal role in determining how well an energy internet (EI) performs, how much it consumes, and its overall viability. The paper fills the void created by the bulk of previous studies focusing on single-objective optimization. The Energy Internet is far more complex than previous studies have recognized, and thus, many real-world EIs require optimization of multiple competing factors at once: connectivity of the network, robustness of the network, and operating efficiency of the network. Moreover, the proposed approach directly addresses the issues associated with optimizing network topology based on existing node degree distributions without modifying the existing properties of that distribution; hence, it will maintain a scale-free property of EI networks.   3. What does it add to the subject area compared with other published material? In comparison to the existing literature, the manuscript presents multiple contributions that are new ideas and concepts through the integration of adaptive capabilities. For example:
• Self-adaptive NSGA-III (SA-NSGA-III) represents a novel multi-objective evolutionary algorithm developed for explicitly addressing topology optimization using non-dominating sorting and self-adaptive methods.
• The Adaptive Dynamic Reference Point Generation strategy describes a technique that continuously adapts the reference point density based on the diversity of the population as measured by a diversity indicator, which provides a better balance between exploration of the global search space and refinement of the local search.
• The Cosine Similarity Penalty Mechanism is an adaptive penalty that calculates the cosine similarity between all the reference points that are selected. As a result, it filters out two similar reference points, which improves solution set diversity and prevents clustering of the solution sets.
• Annealing Strategy for Genetic Operations describes how using annealing provides an example of how a genetic operation may change dynamically over time. Providing a high degree of mutation in the early stage of the optimization process improves the chances for successful convergence to the global optimum and minimizes the chances of premature convergence.
• Free Edge-Based Optimization is a strategy that allows for the ability to perform exchanges of edges using a method based on free edges of the current tree structure of the network. By using this method, the degree of each of the nodes in the network does not change as a result of the exchange and helps to maintain the scale-free structure of the network throughout the optimization process.
4. What specific improvements should the authors consider regarding the methodology? The methodology is sound; however, the following changes could help clarify and validate the methodology further.
- A better justification for the weights applied to the three objectives within the fitness function is needed. It would be helpful if the authors included some form of simple sensitivity analyses or rationale based on the known vulnerabilities of engineering infrastructure.
- Quantitative measures or charts need to be provided that show the degree distribution of the nodes both prior to and following optimization in Section 5.3 (Figures 6 or 7) to better substantiate that the power law nature of the results has indeed been preserved.
- The scalability of the methodology should be assessed through the proposed computational time complexity for each iteration of optimization run on various network sizes to clarify and support how the algorithm will perform in relation to larger networks (500 nodes) and what limitations exist related to memory usage when designing for thousands of nodes.
5. Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed? Please also explain why this is/is not the case.   The results reported in this study support the validity of the conclusions made based on the evidence provided. In addition, the data supports the answer to the primary research question as follows:
• The algorithm is proven to optimize the three objectives (algebraic connectivity, robustness, average shortest path), thus achieving multi-objective success.
• The results presented in Figures 8, 9, & 10 demonstrate that SA-NSGA-III consistently achieves superior fitness and converges faster than NSGA-III; MOEA/D; and MOPSO for multiple scenarios.
• Figure 11 confirms that the proposed adaptive reference point generation method has outperformed alternative methods for generating reference points throughout the entire iterative process.   6. Are the references appropriate? Yes, all cited sources are suitable; they incorporated major contributions in the area of complex network theory (including scale-free models), worked in evolutionary multi-objective algorithms, and new case studies showing how network topology optimization has been applied in Energy Internet & related systems.

7. Any additional comments on the tables and figures.
• Figure 1: The diagram that illustrates how the crossover between the upper triangle of an adjacency matrix is converted to a chromosome is an excellent example of a memory-saving encoding scheme.
• Figure 4: This plot proves that the initial topology produced is consistent with a power-law distribution, which supports the scale-free process employed in this study.
• Figures 6d and 7d: The 3D surface graphs present an effective way of depicting the trade-offs for three objectives and overall improvement in three-dimensional objective space. This is vital for any study on multi-objective optimization.
• Figures 8 through 11: The quantitative information provided by these figures strongly substantiates the statements made about the superior performance and efficiency of the adaptive components of SA-NSGA-III and validates other assertions included in this article.

Author Response

Please see the attachment.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors have fully addressed all my concerns. It is suggested to accept for publications.

Reviewer 2 Report

Comments and Suggestions for Authors

Good improvements have been made.