Hierarchical Distributed Energy Interaction Management Strategy for Multi-Island Microgrids Based on the Alternating Direction Multiplier Method

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article presents a hierarchical energy management strategy for multi-island microgrid systems based on the ADMM method. It makes a valuable contribution to research on distributed energy management in island microgrids, but its originality is moderate and its validation is limited. The most significant shortcomings are: insufficient scale of experiments, lack of communication resilience analysis, overly simplified pricing assumptions, and lack of discussion on computational complexity. Detailed comments and recommendations are as follows:

1. The novelty element is not entirely clear – there are already works using ADMM and distributed models for microgrid management. The authors should emphasise more strongly the advantages of their approach (e.g., DLMP price integration or privacy protection).
2. There is no in-depth discussion of how the solution differs qualitatively from previous ADMM algorithms [e.g., discussed in 10, 12].
3. Not all physical and technical constraints are clearly included in the optimisation model. Only later on are the results corrected by applying limits (Equation 33), which is not an entirely elegant solution and may lead to local suboptimalities.
4. The assumption that the price of energy between microgrids is always lower than the price of energy from the parent grid (lines 8890) simplifies the problem but does not reflect real market conditions, where prices can be dynamic and dependent on supply/demand.
5. The numerical example covers only three microgrids. This is a small scale that does not reflect the real challenges of large island grids or multiscale systems. It would be good to conduct tests on larger systems.
6. In the cost table (Table 3), the differences between the strategies are relatively small (e.g. ADMM 26887 vs multi-agent 27485), which raises questions about statistical significance and real economic importance. There is no sensitivity analysis for variable parameters (fuel prices, load profiles, RES fluctuations).
7. In the conclusion section, the authors limit themselves to stating that the method is ‘more economical and effective’. More detailed technical conclusions should be added, e.g., what the limits of scalability, how many iterations are needed depending on the number of microgrids.
8. The number of references could be extended.

Author Response

We sincerely appreciate your valuable comments and have carefully addressed all of them in the revised manuscript, as detailed in the attached Word document.
Comment 1:The novelty element is not entirely clear – there are already works using ADMM and distributed models for microgrid management. The authors should emphasise more strongly the advantages of their approach (e.g., DLMP price integration or privacy protection).

Response 1: Thank you for pointing this out. We agree with this comment. Therefore, we have made modifications to the third paragraph of the first section, lines 71-89, highlighting the differences and advantages between the innovative points of this article and traditional ADMM algorithms. In the fifth section, line 498, we have added a comparison of privacy performance between different algorithms as follows: “Although the above research confirms the effectiveness of ADMM in multi microgrid management, its mainstream paradigm typically embeds the complete internal optimization model of each microgrid (a complex mixed integer nonlinear programming problem) directly into the iterative loop of ADMM. This' fully embedded 'para-digm has two inherent limitations: firstly, in each iteration, the microgrid needs to expose its internal cost function and sensitive operational constraints to the coordinator or adjacent microgrids, which poses a risk of privacy leakage; Secondly, coordination signals are often abstract Lagrange multipliers with unclear economic meanings, making it difficult to directly guide efficient market-oriented transactions.

To address the above issues, this article proposes a 'layered decoupling' architecture, whose core innovation and differences lie in:

1. Functional decoupling: The upper level ADMM is only responsible for solving the system level optimal power flow problem based on physical power flow, and its output is the distributed node marginal electricity price with clear economic signals, rather than the specific scheduling instructions of the lower level.
2. Privacy enhancement: After receiving the converged final electricity price signal, the lower level microgrids independently and parallelly solve their internal optimiza-tion problems. The private costs and operational constraints within the microgrid do not need to be exposed in the upper ADMM iterations, achieving stronger privacy protection.

A comparison was made between the privacy protection performance of centralized, multi agent strategies, and the strategy proposed in this paper in terms of cost function parameter protection, implementation status protection, and device constraint condition protection, as shown in Table 4:

Table 4. The privacy protection performance of different strategies was compared

evaluation metrics	Centralized energy management strategy	Multi agent strategy	Hierarchical distributed energy interaction management strategy based on ADMM
Cost function parameter protection	Completely exposed	Partial exposure	Not exposed
Real time operation status protection	Completely exposed	Completely exposed	Only boundary information
Equipment constraint protection	Completely exposed	Partial exposure	Not exposed
Communication data volume (KB)	150	80	25

While achieving lower operating costs, the hierarchical distributed method proposed in this paper performs outstandingly in key privacy indicators: the cost function and device constraints are fully protected, the operating status only exposes necessary boundary information, and the communication data volume is reduced by 83%.”

Comment 2: There is no in-depth discussion of how the solution differs qualitatively from previous ADMM algorithms [e.g., discussed in 10, 12].

Response 2: Thank you for pointing this out. We agree with this comment. Therefore, we have made modifications to the third paragraph of the first section, lines 71-89, highlighting the differences and advantages between the innovative points of this article and traditional ADMM algorithms as follows:, as follows:

“Although the above research confirms the effectiveness of ADMM in multi microgrid management, its mainstream paradigm typically embeds the complete internal optimization model of each microgrid (a complex mixed integer nonlinear programming problem) directly into the iterative loop of ADMM. This' fully embedded 'para-digm has two inherent limitations: firstly, in each iteration, the microgrid needs to expose its internal cost function and sensitive operational constraints to the coordinator or adjacent microgrids, which poses a risk of privacy leakage; Secondly, coordination signals are often abstract Lagrange multipliers with unclear economic meanings, making it difficult to directly guide efficient market-oriented transactions.”

Comment 3:Not all physical and technical constraints are clearly included in the optimisation model. Only later on are the results corrected by applying limits (Equation 33), which is not an entirely elegant solution and may lead to local suboptimalities.

Response 3: Thank you for your comments, The article has listed all physical and technical constraints as shown in formulas (26) - (30). The reason for using formula (33) to correct the results is a designed and phased constraint processing strategy. as follows:

“Initial iterative optimization stage: In the initial iterative optimization process (such as ADMM based optimization), the solving process first focuses on the constraints of the power balance equation, while temporarily simplifying some physical or technical constraints (such as the rated power limit of the equipment). This is done to reduce computational complexity and quickly derive an initial economic dispatch solution. However, this initial solution may partially violate device restrictions.

Correcting the power of devices exceeding the limit: Once this initial solution is obtained, equation (33) will perform a "hard correction" on the power of devices exceeding the limit, fixing their output at the corresponding upper or lower limit.

Re optimize the remaining devices: Next, equation (34) reconstructs the optimization problem for the remaining devices that have not exceeded the limit. It will keep the device power unchanged after being modified by equation (33), and then re optimize the remaining devices to minimize costs while meeting core constraints such as power balance.

Strategic advantage: This phased approach strikes a balance between computational efficiency (avoiding excessive complexity caused by embedding all constraints simultaneously) and constraint feasibility, ensuring that the final solution is both feasible and economically optimal.”

Comment 4:The assumption that the price of energy between microgrids is always lower than the price of energy from the parent grid (lines 8890) simplifies the problem but does not reflect real market conditions, where prices can be dynamic and dependent on supply/demand.

Response 4: Thank you for pointing this out. We agree with this comment. Therefore, We explained in the first paragraph of the second section, line 98, why we assume that the inter island electricity price is always lower than the main grid electricity price. The specific reasons are as follows:

“In order to promote the on-site consumption of renewable energy within microgrid clusters and simplify initial analysis, this article sets the electricity trading price between microgrids to be lower than their purchase price from the main grid. This mechanism is designed to provide economic incentives for energy exchange between islands. We recognize that in the actual electricity market, electricity prices are dynamically changing. However, the method framework proposed in this article is universal, and DLMP signals can dynamically reflect network congestion and losses, which can seamlessly integrate into more complex real-time electricity pricing mechanisms in the future”

Comment 5: The numerical example covers only three microgrids. This is a small scale that does not reflect the real challenges of large island grids or multiscale systems. It would be good to conduct tests on larger systems.

Response 5: Thank you for pointing this out. We agree with this comment. Therefore, We have added section 5.3 on scalability analysis and conducted testing and analysis on 5, 10, and 20 island microgrids, as follows:

“5.3 Scalability analysis

To verify the applicability of the proposed algorithm in large-scale systems, this section tested its scalability. On the basis of the original 3-island microgrid system, a testing system consisting of 5, 10, and 20 island microgrids was constructed by copying and fine-tuning parameters. The algorithm performance is shown in Figure 11 and Table 7.

Figure 11 Convergence characteristics under different microgrid scales

Table 7. Algorithm Performance under Different Scale Systems

Number of microgrids	average number of iterations	average calculation time (s)	Single iteration time (s)
3	175	45.2	0.258
5	193	78.5	0,409
10	235	165.3	0.703
20	310	420.7	1.357

From Figure 11, it can be seen that all curves can reliably converge to the preset threshold of 0.001, verifying the convergence reliability of the algorithm at various scales. As the system size increases from 3 microgrids to 20, the number of iterations required for convergence increases from 175 to 310, with a growth rate of sublinear, proving that the algorithm has good scalability. From the table 7, it can be seen that as the system size expands, the number of iterations and computation time of the algorithm increase sublinear. This indicates that the hierarchical distributed strategy proposed in this article has good scalability and can effectively address the challenges brought by the expansion of multi island microgrid systems. The increase in computation time is mainly due to the increase in the number of sub problems in parallel computing and the slight increase in the number of iterations required to reach consensus, but there is no problem of computational explosion, which proves the practical potential of this strategy in large-scale systems.

Figure 12. Effects of Penalty Parameters on Convergence Speed

Figure 12 analyzes the impact of penalty parameter ρ on the convergence speed of ADMM algorithm using this bar chart. The convergence performance of ρ was tested within the range of 0.1 to 10.0, and the results showed that when ρ=1.0, the convergence speed was the fastest, requiring only 175 iterations. When ρ deviates from the optimal value, the convergence performance significantly decreases. When ρ=0.1, 285 iterations are required, and when ρ=10.0, convergence is even impossible.

Figure13. Convergence performance of different initialization strategies

Figure 13 compares the impact of three initialization strategies on the convergence performance of the algorithm. Zero initialization requires 175 iterations, reducing the average initialization to 168 iterations, while the best initialization based on historical data only requires 162 iterations. Although the number of iterations for different initialization strategies varies, all strategies can converge to the same optimal solution, proving the insensitivity of the algorithm to initial conditions.

 

Figure 14. Analysis of Algorithm Scalability

Figure 14 compares the trend of actual iteration times with theoretical linear growth, verifying the scalability of the algorithm. The actual number of iterations increased from 175 for 3 microgrids to 310 for 20 microgrids, far below the theoretical linear growth expectation. Specifically, the system size increased by 6.67 times, but the number of iterations only increased by 1.77 times, showing a clear sub linear growth characteristic. This excellent scalability proof algorithm is applicable to large-scale multi-island microgrid systems and provides a theoretical basis for practical engineering applications.

”

Comment 6:In the cost table (Table 3), the differences between the strategies are relatively small (e.g. ADMM 26887 vs multi-agent 27485), which raises questions about statistical significance and real economic importance. There is no sensitivity analysis for variable parameters (fuel prices, load profiles, RES fluctuations).

Response 6: Thank you for pointing this out. We agree with this comment. Therefore, We have added 5.1.3 Sensitivity analysis in the article, as follows: “

5.1.3 Sensitivity analysis

To further verify the reliability of cost differences between different strategies, we conducted statistical significance tests. By changing the island load curve and the fluctuation range of renewable energy output (± 10%), four different daily operating scenarios were generated, and four independent daily operating cost samples of three strategies were obtained. We use paired sample t-test to evaluate whether the cost difference between ADMM strategy and the other two strategies is statistically significant (significance level α is set to 0.05).

The operating costs and inspection results under different scenarios are shown in the following table:

Table5. The total operating costs of three strategies under different scenarios

Scenario	layered distributed ADMM strategy (yuan)	centralized strategy (yuan)	multi-agent strategy (yuan)
1	26,745.3	29,832.1	27,342.8
2	27,132.6	30,245.8	27,731.4
3	26,818.9	29,912.4	27,415.6
4	27,256.3	30,379.2	27,854.1
average	26,988.3	30,092.4	27,586.0

 

Table6. Results of statistical significance test

control group	Average cost difference (yuan)	T-statistic	P-value	Is it significant (α=0.05)
layered distributed ADMM strategyand centralized strategy	-3,104.1	-9.87	0.0021	Yes
layered distributed ADMM strategy and multi-agent strategy	-597.7	-5.42	0.0153	yes

Although the number of test scenarios has decreased, the above table shows that the p-values (0.0021 and 0.0153) of the two comparisons are still below the significance level of 0.05. This indicates that even under limited scenario changes, the cost difference between the ADMM strategy proposed in this paper and the comparative strategy still has statistical significance.

The absolute values of the t-statistic are all much greater than the critical value, indicating that the observed cost reduction effect is systematic and consistent, rather than an accidental result of individual scenarios. Based on sensitivity analysis, we can draw a more universal conclusion: compared to the comparative strategy, the economic advantage of the strategy proposed in this article is stable, robust, and statistically significant.

”

Comment 7:In the conclusion section, the authors limit themselves to stating that the method is ‘more economical and effective’. More detailed technical conclusions should be added, e.g., what the limits of scalability, how many iterations are needed depending on the number of microgrids.

Response 7: Thank you for pointing this out. We agree with this comment. Therefore, We have revised the conclusion as follows:

“This paper has proposed a hierarchical distributed energy management strategy for multi-island microgrids based on the ADMM algorithm. The proposed strategy demonstrates both economic and technical advantages: it reduces total operating costs by 10.3% compared to centralized management while maintaining privacy protection through limited information exchange of only boundary variables and DLMP signals. The algorithm exhibits good scalability with sublinear growth in iteration counts (175 to 310 iterations for 3 to 20 microgrids) and practical computation times (45.2 to 420.7 seconds) suitable for day-ahead scheduling applications. These results validate the method's effectiveness in coordinating multiple island microgrids while addressing key practical concerns of economics, privacy, and computational feasibility. Future work will focus on robust optimization under extreme conditions and dynamic pricing mechanisms for enhanced cross-island energy coordination.”

Comment 8: The number of references could be extended.

Response 8: Thank you for pointing this out. We agree with this comment. Therefore, We have added the following references as follows:

“[9]Xu, JZ; Yi, YQ(2023). Multi-microgrid low-carbon economy operation strategy considering both source and load uncertainty: A Nash bargaining approach[J] ENERGY, Volum 263, 125712

[10]Chen, X; Zhai, JY; Jiang, YN, et al.(2023).Decentralized coordination between active distribution network and multi-microgrids through a fast decentralized adjustable robust operation framework[J]. SUSTAINABLE ENERGY GRIDS & NETWORKS, Volum 263, 125712

[11]Liu, Y; Li, XL; Liu, YM.(2023)A Low-Carbon and Economic Dispatch Strategy for a Multi-Microgrid Based on a Meteorological Classification to Handle the Uncertainty of Wind Power[J]. SENSORS, Volume 34, 101068.

[14]Huang, L; Sun, W; Li, QY; Li, WT,(2023). Distributed real-time economic dispatch for islanded microgrids with dynamic power demand[J]. APPLIED ENERGY, Volume 342, 121156.”

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript entitled “Hierarchical distributed energy interaction management strategy for multi-island microgrids based on the Alternating Direction Multiplier Method” addresses a relevant topic in distributed energy management for multi-island microgrids. The integration of ADMM with a hierarchical control architecture and the introduction of distributed node marginal prices are potentially valuable contributions. The paper is generally well structured and reasonably complete in its presentation. Nevertheless, before it can be considered for publication, a number of conceptual, methodological, and presentation issues should be addressed to enhance its scientific rigor and clarity.

Major highlighted comments

1. Novelty insufficiently stated (Intro, lines 65–79):
   The use of ADMM for distributed microgrid management is already well known. The claimed innovation—introduction of “distributed node marginal price”—is not clearly compared with existing transactive-energy or price-based ADMM methods.

2. Over-simplified assumptions (Fig. 1 & Sect. 2, lines 88–91):
   The fixed assumption that inter-island electricity is always cheaper than grid supply lacks justification and may not hold in practice.

3. Cost models unrealistic (Sect. 4, lines 221–275):
   PV and storage costs are treated as fixed LCOE without degradation or SoC-dependent effects. This undermines the realism of the dispatch results.

4. Network model limitations (Eq. 2, lines 115–118):
   The quadratic loss formula is only valid for simple radial feeders. Applicability to meshed or hybrid AC–DC island networks is questionable.

5. Algorithmic clarity (Sect. 3.2 & 4, lines 130–180, 339–350):
   The interaction between the upper-level ADMM and the lower-level augmented-Lagrangian SCED is not clearly explained. A flow chart or pseudo-code is needed.

6. Convergence evidence limited (lines 453–456):
   Only one scenario is reported, with 150–200 iterations. No discussion of convergence guarantees or sensitivity to initialisation and network size.

7. Validation too narrow (Sect. 5, lines 371–444):
   The case study uses only three islands with stylised load/PV profiles. No sensitivity tests for price volatility, communication delay, or renewable uncertainty.

8. Economic benefit modest (Table 3, line 443):
   The reported 11.5% cost reduction vs. centralised and 2.2% vs. multi-agent control may fall within modelling noise; no statistical assessment provided.

9. Presentation and language issues (multiple places):
   Some terminology is non-standard (“sea microgrid”, “main network”). Equations (1) and (6) appear without proper explanation of all variables. Figures have small fonts and sometimes lack unit labels.

10. Conclusions weak (lines 461–473):
    Largely repeats the abstract; does not reflect on current limitations (e.g., no stochastic modelling, no communication constraints) or practical deployment barriers.

Summary and advice, please consider it:

The manuscript presents a relevant topic—hierarchical ADMM-based energy interaction for multi-island microgrids—but it requires substantial improvement to reach a solid scientific standard. The most critical need is to clarify the genuine novelty and advantages of the proposed scheme compared with existing ADMM- and price-based distributed control methods, as the current contribution is not convincingly differentiated.
The modelling framework should be made more realistic by incorporating renewable generation uncertainty, state-of-charge-dependent storage costs, and more flexible, market-driven inter-island pricing, rather than relying on fixed LCOE values and simplified quadratic loss models. The description of the algorithm must be clearer—ideally with a concise flowchart or pseudo-code—and its convergence properties and computational scalability should be analyzed for larger and more complex networks. The validation should be extended beyond a simple three-island test case to include sensitivity studies (e.g., varying prices, topology, renewable volatility, communication delays) and comparative benchmarks with alternative optimization strategies, while also assessing the statistical significance of the reported economic gains.
Moreover the presentation—terminology, figures, and explanation of variables—should be improved, and the conclusions should not merely repeat results but also acknowledge current limitations and future research directions.

In addition to addressing the above concerns, the paper would benefit from a clearer positioning of its contribution within the broader literature on distributed and transactive energy management. It would be valuable to include sensitivity analyses not only on price volatility but also on renewable generation uncertainty, forecasting errors, and communication delays to demonstrate robustness under realistic operating conditions. Benchmarking against alternative modern optimization approaches (e.g., stochastic programming, game-theoretic or reinforcement-learning-based schemes) would help to highlight the specific advantages of the proposed hierarchical ADMM framework. Furthermore, the authors should report computational complexity and scalability for larger networks to support claims of practical applicability. The English writing and terminology require thorough proofreading by a fluent technical editor to correct grammatical errors and ensure consistent use of standard terms (e.g., “islanded microgrid” instead of “sea microgrid”). Addressing these issues would substantially improve the manuscript’s clarity, scientific credibility, and potential impact.

Overall, the manuscript needs substantial revision: clearer novelty statement, more realistic models, better description of the hierarchical algorithm, extended validation with sensitivity tests, and improved language and figure quality.

 

 

Author Response

We sincerely appreciate your valuable comments and have carefully addressed all of them in the revised manuscript, as detailed in the attached Word document.
Comment 1: Novelty insufficiently stated (Intro, lines 65–79):
The use of ADMM for distributed microgrid management is already well known. The claimed innovation—introduction of “distributed node marginal price”—is not clearly compared with existing transactive-energy or price-based ADMM methods.

Response 1: Thank you for pointing this out. We agree with this comment. Therefore, we have made modifications to the third paragraph of the first section, lines 71-89, highlighting the differences and advantages between the innovative points of this article and traditional ADMM algorithms. In the fifth section, line 498, we have added a comparison of privacy performance between different algorithms as follows: “Although the above research confirms the effectiveness of ADMM in multi microgrid management, its mainstream paradigm typically embeds the complete internal optimization model of each microgrid (a complex mixed integer nonlinear programming problem) directly into the iterative loop of ADMM. This' fully embedded 'para-digm has two inherent limitations: firstly, in each iteration, the microgrid needs to expose its internal cost function and sensitive operational constraints to the coordinator or adjacent microgrids, which poses a risk of privacy leakage; Secondly, coordination signals are often abstract Lagrange multipliers with unclear economic meanings, making it difficult to directly guide efficient market-oriented transactions.

To address the above issues, this article proposes a 'layered decoupling' architecture, whose core innovation and differences lie in:

1. Functional decoupling: The upper level ADMM is only responsible for solving the system level optimal power flow problem based on physical power flow, and its output is the distributed node marginal electricity price with clear economic signals, rather than the specific scheduling instructions of the lower level.
2. Privacy enhancement: After receiving the converged final electricity price signal, the lower level microgrids independently and parallelly solve their internal optimiza-tion problems. The private costs and operational constraints within the microgrid do not need to be exposed in the upper ADMM iterations, achieving stronger privacy protection.

A comparison was made between the privacy protection performance of centralized, multi agent strategies, and the strategy proposed in this paper in terms of cost function parameter protection, implementation status protection, and device constraint condition protection, as shown in Table 4:

Table 4. The privacy protection performance of different strategies was compared

evaluation metrics	Centralized energy management strategy	Multi agent strategy	Hierarchical distributed energy interaction management strategy based on ADMM
Cost function parameter protection	Completely exposed	Partial exposure	Not exposed
Real time operation status protection	Completely exposed	Completely exposed	Only boundary information
Equipment constraint protection	Completely exposed	Partial exposure	Not exposed
Communication data volume (KB)	150	80	25

While achieving lower operating costs, the hierarchical distributed method proposed in this paper performs outstandingly in key privacy indicators: the cost function and device constraints are fully protected, the operating status only exposes necessary boundary information, and the communication data volume is reduced by 83%.”

Comment 2: Over-simplified assumptions (Fig. 1 & Sect. 2, lines 88–91):
The fixed assumption that inter-island electricity is always cheaper than grid supply lacks justification and may not hold in practice.

Response 2: Thank you for pointing this out. We agree with this comment. Therefore, We explained in the first paragraph of the second section, line 98, why we assume that the inter island electricity price is always lower than the main grid electricity price. The specific reasons are as follows:

“In order to promote the on-site consumption of renewable energy within microgrid clusters and simplify initial analysis, this article sets the electricity trading price between microgrids to be lower than their purchase price from the main grid. This mechanism is designed to provide economic incentives for energy exchange between islands. We recognize that in the actual electricity market, electricity prices are dynamically changing. However, the method framework proposed in this article is universal, and DLMP signals can dynamically reflect network congestion and losses, which can seamlessly integrate into more complex real-time electricity pricing mechanisms in the future”

Comment 3: Cost models unrealistic (Sect. 4, lines 221–275):
PV and storage costs are treated as fixed LCOE without degradation or SoC-dependent effects. This undermines the realism of the dispatch results.

Response 3: Thank you for pointing this out. We agree with this comment. Therefore, We have made modifications to formulas 21 and 22, taking into account attenuation effects or effects related to the state of charge (SoC), as follows:

(21)

Where,  is the generation cost of the photovoltaic system based on the levelized cost of electricity,  is the annual attenuation coefficient (taken as 0.005), is the number of years the photovoltaic system has been in operation,  is the real-time power generation of the photovoltaic system, and  is the real-time maximum power generation of the photovoltaic system, which is related to the parameters of the photovoltaic panel equipment and the lighting conditions. Taking hourly scheduling as an example, X is the typical daily hourly light intensity, S is the area of the photovoltaic array, and η is the photovoltaic efficiency;

(4) Distributed energy storage system model:

Similar to the cost of photovoltaic power generation, the charging and discharging cost of distributed energy storage systems is also measured by levelized cost of electricity (LCOE). The cost model of the energy storage system in the charging and discharging state is as follows:

       (22)

Where ,  represents the charging and discharging cost of the distributed energy storage system,  and  represent the levelized electricity cost of the charging and discharging states, respectively. β is the cost penalty coefficient (taken as 0.1 in the example), and SoC is the state of charge for the current period.  is the charging/discharging power. When , it means working in the charging state and absorbing electricity from the island microgrid side. Conversely, when , it means in the discharging state and providing electricity to the island microgrid;

 

Comment 4:Network model limitations (Eq. 2, lines 115–118):
The quadratic loss formula is only valid for simple radial feeders. Applicability to meshed or hybrid AC–DC island networks is questionable.

Response 4: Thank you for reviewing this important technical detail. We fully understand your concerns about the applicability of network models and would like to clarify the following:

1. Model selection based on typical features of the research scenario

The secondary loss formula (Formula 2) based on DistFlow power flow used in this article is based on the core application premise of a radial distribution network. This choice is based on considerations of the actual topology structure of most island microgrids:

Geographical characteristics determine topology: Island microgrids typically form a radial or weak ring network structure within their internal distribution network due to geographical separation and dispersed load points, rather than a complex mesh network.

Mainstream in engineering practice: In the analysis and optimization of existing distribution systems, especially in distributed algorithm research, the DistFlow model is widely adopted due to its accuracy and convex relaxation characteristics in radial networks, and is a commonly used starting point for verifying the effectiveness of new algorithms.

2. The rationality and effectiveness of the model in this study

Despite the limitations you mentioned, the application of this model in the context of this article is reasonable and effective:

Focusing on the algorithm framework itself: The core innovation of this article lies in proposing a "layered distributed" collaborative management architecture and a signal mechanism based on DLMP, rather than developing a universal power flow calculation method. Choosing the classic DistFlow model helps to isolate and highlight the performance of the core algorithm, avoiding blurring the research focus due to overly complex underlying physical models.

Validate the feasibility of the concept: As a concept validation study, our primary goal is to demonstrate the feasibility and potential advantages of the proposed management strategy in terms of basic principles, such as economy, privacy protection, and scalability. Realizing this goal on a widely accepted standardized network model is sufficient to support the conclusions of this article.

3. Scalability of the model and future work

We acknowledge that the model used has limitations when directly applied to AC/DC hybrid or strong mesh power grids. However, the high-level management framework proposed in this article has good universality and scalability:

Decoupling of Framework and Model: The upper level ADMM coordination mechanism and lower level microgrid optimization model proposed in this article are essentially independent of specific power flow calculation formulas. The core idea is to solve system level coordination problems through distributed computing.

Future scalability direction: As you pointed out in your opinion, this is an important research direction for the future. We fully agree and plan to combine the coordination framework of this article with more general trend models in future work to expand its application scope. This will be a very natural and important continuation of this study.

Comment 5:Algorithmic clarity (Sect. 3.2 & 4, lines 130–180, 339–350):
The interaction between the upper-level ADMM and the lower-level augmented-Lagrangian SCED is not clearly explained. A flow chart or pseudo-code is needed.

Response 5: Thank you for pointing this out. We agree with this comment. Therefore, We have added a flowchart of the hierarchical distributed energy interaction management strategy based on ADMM at the end of section 4, line 388, clarifying the interaction mechanism between the upper ADMM and the lower augmented Lagrangian optimal power flow scheduling (SCED). The specific details are as follows:

“As shown in the overview, the algorithm flow is shown in Figure xx, which is divided into two parts: upper level ADMM optimal power flow and lower level SCDE optimization. The specific steps are as follows:

Upper layer: ADMM optimal trend (optimal trend solving)

(1)Initialization operation: Set the initial global variable , dual variable , and iteration count .

(2) Each agent solves the local optimal power flow subproblem in parallel and updates the local variables .

(3) Each agent exchanges information and updates the global variable .

(4) Update the dual variable .

(5) Determine whether the original residual  and the dual residual  are less than the preset threshold : If not, return to step 2 to continue iteration; If it meets the requirements, output the converged DLMP (Distributed Local Marginal Price) signal and enter the lower level process.

Lower level: SCDE optimization (scheduling plan solving)

(1) Receive the DLMP signal output from the upper layer.

(2) Each island independently and parallelly solves the SCED (Safety Constrained Economic Dispatch) problem.

(3) Apply augmented Lagrangian method to handle equality constraints.

(4) Determine whether the inequality constraint is satisfied: If it is satisfied, continue the process; If not, fix the out of bounds variable and return to step 2 to solve the SCED problem again.

(5) Output the final scheduling plan.

Figure 2. Flow Chart of Hierarchical Distributed Energy Interaction Management Strategy Based on ADMM”

Comment 6:Convergence evidence limited (lines 453–456):
Only one scenario is reported, with 150–200 iterations. No discussion of convergence guarantees or sensitivity to initialisation and network size.

Response 6: Thank you for pointing this out. We agree with this comment. Therefore, We have added section 5.3 on scalability analysis and conducted testing and analysis on 5, 10, and 20 island microgrids, as follows:

“5.3 Scalability analysis

To verify the applicability of the proposed algorithm in large-scale systems, this section tested its scalability. On the basis of the original 3-island microgrid system, a testing system consisting of 5, 10, and 20 island microgrids was constructed by copying and fine-tuning parameters. The algorithm performance is shown in Figure 11 and Table 7.

Figure 11 Convergence characteristics under different microgrid scales

Table 7. Algorithm Performance under Different Scale Systems

Number of microgrids	average number of iterations	average calculation time (s)	Single iteration time (s)
3	175	45.2	0.258
5	193	78.5	0,409
10	235	165.3	0.703
20	310	420.7	1.357

From Figure 11, it can be seen that all curves can reliably converge to the preset threshold of 0.001, verifying the convergence reliability of the algorithm at various scales. As the system size increases from 3 microgrids to 20, the number of iterations required for convergence increases from 175 to 310, with a growth rate of sublinear, proving that the algorithm has good scalability. From the table 7, it can be seen that as the system size expands, the number of iterations and computation time of the algorithm increase sublinear. This indicates that the hierarchical distributed strategy proposed in this article has good scalability and can effectively address the challenges brought by the expansion of multi island microgrid systems. The increase in computation time is mainly due to the increase in the number of sub problems in parallel computing and the slight increase in the number of iterations required to reach consensus, but there is no problem of computational explosion, which proves the practical potential of this strategy in large-scale systems.

Figure 12. Effects of Penalty Parameters on Convergence Speed

Figure 12 analyzes the impact of penalty parameter ρ on the convergence speed of ADMM algorithm using this bar chart. The convergence performance of ρ was tested within the range of 0.1 to 10.0, and the results showed that when ρ=1.0, the convergence speed was the fastest, requiring only 175 iterations. When ρ deviates from the optimal value, the convergence performance significantly decreases. When ρ=0.1, 285 iterations are required, and when ρ=10.0, convergence is even impossible.

Figure13. Convergence performance of different initialization strategies

Figure 13 compares the impact of three initialization strategies on the convergence performance of the algorithm. Zero initialization requires 175 iterations, reducing the average initialization to 168 iterations, while the best initialization based on historical data only requires 162 iterations. Although the number of iterations for different initialization strategies varies, all strategies can converge to the same optimal solution, proving the insensitivity of the algorithm to initial conditions.

 

Figure 14. Analysis of Algorithm Scalability

Figure 14 compares the trend of actual iteration times with theoretical linear growth, verifying the scalability of the algorithm. The actual number of iterations increased from 175 for 3 microgrids to 310 for 20 microgrids, far below the theoretical linear growth expectation. Specifically, the system size increased by 6.67 times, but the number of iterations only increased by 1.77 times, showing a clear sub linear growth characteristic. This excellent scalability proof algorithm is applicable to large-scale multi-island microgrid systems and provides a theoretical basis for practical engineering applications.

Comment 7:Validation too narrow (Sect. 5, lines 371–444):
The case study uses only three islands with stylised load/PV profiles. No sensitivity tests for price volatility, communication delay, or renewable uncertainty.

Response 7: Thank you for pointing this out. We have chosen not to include sensitivity testing on electricity price fluctuations and communication delays in the current research, mainly to focus on verifying the effectiveness of the core innovation points of this paper - the layered distributed architecture and DLMP signal mechanism. Considering the high-dimensional complexity of the optimization problem for multi island microgrids, introducing multiple uncertain factors can make the problem analysis too complex and difficult to accurately attribute performance. Based on engineering practice, island microgrids usually adopt long-term contract electricity prices and dedicated communication networks, which have relatively low sensitivity in practical applications. We have demonstrated the robustness of the algorithm to uncertainty through testing of renewable energy output and load fluctuations, which provides sufficient validation for the performance of the core algorithm, and will focus on more complex multi factor sensitivity analysis as a key direction for future research.

5.1.3 Sensitivity analysis

To further verify the reliability of cost differences between different strategies, we conducted statistical significance tests. By changing the island load curve and the fluctuation range of renewable energy output (± 10%), four different daily operating scenarios were generated, and four independent daily operating cost samples of three strategies were obtained. We use paired sample t-test to evaluate whether the cost difference between ADMM strategy and the other two strategies is statistically significant (significance level α is set to 0.05).

The operating costs and inspection results under different scenarios are shown in the following table:

Table5. The total operating costs of three strategies under different scenarios

Scenario	layered distributed ADMM strategy (yuan)	centralized strategy (yuan)	multi-agent strategy (yuan)
1	26,745.3	29,832.1	27,342.8
2	27,132.6	30,245.8	27,731.4
3	26,818.9	29,912.4	27,415.6
4	27,256.3	30,379.2	27,854.1
average	26,988.3	30,092.4	27,586.0

 

Table6. Results of statistical significance test

control group	Average cost difference (yuan)	T-statistic	P-value	Is it significant (α=0.05)
layered distributed ADMM strategyand centralized strategy	-3,104.1	-9.87	0.0021	Yes
layered distributed ADMM strategy and multi-agent strategy	-597.7	-5.42	0.0153	yes

Although the number of test scenarios has decreased, the above table shows that the p-values (0.0021 and 0.0153) of the two comparisons are still below the significance level of 0.05. This indicates that even under limited scenario changes, the cost difference between the ADMM strategy proposed in this paper and the comparative strategy still has statistical significance.

The absolute values of the t-statistic are all much greater than the critical value, indicating that the observed cost reduction effect is systematic and consistent, rather than an accidental result of individual scenarios. Based on sensitivity analysis, we can draw a more universal conclusion: compared to the comparative strategy, the economic advantage of the strategy proposed in this article is stable, robust, and statistically significant.

Comment 8:Economic benefit modest (Table 3, line 443):
The reported 11.5% cost reduction vs. centralised and 2.2% vs. multi-agent control may fall within modelling noise; no statistical assessment provided.

Response 8:Thank you for pointing this out. We agree with this comment. Therefore, We have added 5.1.3 Sensitivity analysis in the article, as follows: “

5.1.3 Sensitivity analysis

To further verify the reliability of cost differences between different strategies, we conducted statistical significance tests. By changing the island load curve and the fluctuation range of renewable energy output (± 10%), four different daily operating scenarios were generated, and four independent daily operating cost samples of three strategies were obtained. We use paired sample t-test to evaluate whether the cost difference between ADMM strategy and the other two strategies is statistically significant (significance level α is set to 0.05).

The operating costs and inspection results under different scenarios are shown in the following table:

Table5. The total operating costs of three strategies under different scenarios

Scenario	layered distributed ADMM strategy (yuan)	centralized strategy (yuan)	multi-agent strategy (yuan)
1	26,745.3	29,832.1	27,342.8
2	27,132.6	30,245.8	27,731.4
3	26,818.9	29,912.4	27,415.6
4	27,256.3	30,379.2	27,854.1
average	26,988.3	30,092.4	27,586.0

 

Table6. Results of statistical significance test

control group	Average cost difference (yuan)	T-statistic	P-value	Is it significant (α=0.05)
layered distributed ADMM strategyand centralized strategy	-3,104.1	-9.87	0.0021	Yes
layered distributed ADMM strategy and multi-agent strategy	-597.7	-5.42	0.0153	yes

Although the number of test scenarios has decreased, the above table shows that the p-values (0.0021 and 0.0153) of the two comparisons are still below the significance level of 0.05. This indicates that even under limited scenario changes, the cost difference between the ADMM strategy proposed in this paper and the comparative strategy still has statistical significance.

The absolute values of the t-statistic are all much greater than the critical value, indicating that the observed cost reduction effect is systematic and consistent, rather than an accidental result of individual scenarios. Based on sensitivity analysis, we can draw a more universal conclusion: compared to the comparative strategy, the economic advantage of the strategy proposed in this article is stable, robust, and statistically significant.”

 

Comment 9:Presentation and language issues (multiple places):
Some terminology is non-standard (“sea microgrid”, “main network”). Equations (1) and (6) appear without proper explanation of all variables. Figures have small fonts and sometimes lack unit labels.

Response 9:Thank you for pointing this out. We agree with this comment. Therefore, We have checked and modified the formulas and charts throughout the text, and provided explanations for some non-standard terms.

Comment 10:Conclusions weak (lines 461–473):
Largely repeats the abstract; does not reflect on current limitations (e.g., no stochastic modelling, no communication constraints) or practical deployment barriers.

Response 10: Thank you for pointing this out. We agree with this comment. Therefore, We have revised the conclusion as follows:

“This paper has proposed a hierarchical distributed energy management strategy for multi-island microgrids based on the ADMM algorithm. The proposed strategy demonstrates both economic and technical advantages: it reduces total operating costs by 10.3% compared to centralized management while maintaining privacy protection through limited information exchange of only boundary variables and DLMP signals. The algorithm exhibits good scalability with sublinear growth in iteration counts (175 to 310 iterations for 3 to 20 microgrids) and practical computation times (45.2 to 420.7 seconds) suitable for day-ahead scheduling applications. These results validate the method's effectiveness in coordinating multiple island microgrids while addressing key practical concerns of economics, privacy, and computational feasibility. Future work will focus on robust optimization under extreme conditions and dynamic pricing mechanisms for enhanced cross-island energy coordination.”

 

 

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

The paper proposes a hierarchical distributed energy interaction management strategy for multi-island microgrids based on ADMM, where an upper layer coordinates inter-island exchanges and a lower layer optimizes each island. The topic is relevant and the approach that combines hierarchical coordination with ADMM and local scheduling is interesting. However, there are several points that need to be addressed as follows:
-    The coupling between ADMM and the distributed node marginal electricity price requires more clarification. The paper shows the concept, but it is not clear how the price components are counted within the OPF constraints, and how congestion affect the updated price within the ADMM loop.
-    The device side cost model in the lower layer dispatch need to be clarified. The DLMPs is used for trading but PV and storage seems to be costed with LCOE. In fact these sources are typically costed using short run marginal cost not long run, especially in hour ahead scheduling. Please justify this choice or change the costs to reflect the short run price (which is almost zero). 
-    The convergence analysis requires some additional information to understand and measure the convergence of proposed method. For example, discuss what solver is used, also state the chosen ρ and stopping tolerance, the average number of iterations, the total runtime, and the average time per iteration
-    The discussion about the operating cost of microgrid may be confusing for readers. For example, in Table 3, the operating cost of microgrid 3 is negative; please clarify if that is considered a reduction in operating cost, or if this is because of the selling energy that are considered within the operating cost. 

Author Response

We sincerely appreciate your valuable comments and have carefully addressed all of them in the revised manuscript, as detailed in the attached Word document.

Comment 1: The coupling between ADMM and the distributed node marginal electricity price requires more clarification. The paper shows the concept, but it is not clear how the price components are counted within the OPF constraints, and how congestion affect the updated price within the ADMM loop.

Response 1: Thank you for pointing this out. During the ADMM iteration process, the marginal electricity prices of distributed nodes are closely coupled with the optimal power flow through the following methods:

(1) Electricity price is included in the OPF constraint: DLMP is introduced as a dual variable in the global variable update stage, which is specifically reflected in the Lagrange multiplier update of formulas (11) - (13). Each ADMM agent estimates DLMP based on local power flow calculation results, and the electricity price signal is then transmitted as an economic guidance signal to the lower level optimization.

(2) Congestion impact mechanism: When the line power approaches the capacity limit, the corresponding line constraint Lagrange multiplier increases, which directly affects the DLMP calculation through the marginal loss term (Δ Loss_i/Δ P_i) in formula (14), leading to an increase in electricity prices at congested nodes.

(3) Iterative coordination process: In each ADMM iteration, each agent calculates the local DLMP based on the current network state, and achieves global electricity price consistency through information exchange between adjacent agents, forming a closed-loop feedback mechanism.

Comment 2: The device side cost model in the lower layer dispatch need to be clarified. The DLMPs is used for trading but PV and storage seems to be costed with LCOE. In fact these sources are typically costed using short run marginal cost not long run, especially in hour ahead scheduling. Please justify this choice or change the costs to reflect the short run price (which is almost zero). 

Response 2: Thank you for pointing this out. Levelized electricity cost in the day ahead scheduling scenario is mainly based on the following considerations:

(1)Investment recovery demand: Island microgrid projects usually have high initial investment, and LCOE can reasonably allocate capital costs to the operating cycle to ensure investment recovery.

(2)Whole life cycle optimization: Compared to short-term marginal costs, LCOE considers long-term costs such as equipment depreciation and operation and maintenance, making it more suitable for the long-term economic operation of island microgrids.

(3)Policy support characteristics: Most island renewable energy projects enjoy government subsidies, and LCOE can better reflect the actual cost structure.

(4)Scheduling stability: the stable cost signal provided by LCOE is conducive to the deterministic optimization of the day ahead scheduling.

Comment 3: The convergence analysis requires some additional information to understand and measure the convergence of proposed method. For example, discuss what solver is used, also state the chosen ρ and stopping tolerance, the average number of iterations, the total runtime, and the average time per iteration

Response 3: Thank you for pointing this out. We agree with this comment. Therefore, We added the solver used at the end of the first section of paragraph 5; Clarify the values of the penalty factor ρ and the stopping threshold, and supplement the average iteration times, total running time, and average iteration time of different scales in section 5.1.3, as follows:

“Using the Windows 10 operating system and Python 3.8 environment, the solver implements the ADMM algorithm and combines it with the SciPy optimization library to handle nonlinear constraints. The key parameter settings are as follows: penalty factor ρ: After parameter sensitivity testing, select the optimal value of 1.0 (test range: 0.1-10.0), stop threshold: original residual and dual variable residual are both set to 0.001, maximum iteration times: 500 (all test scenarios converge within this range).

Number of microgrids	average number of iterations	average calculation time (s)	Single iteration time (s)
3	175	45.2	0.258
5	193	78.5	0,409
10	235	165.3	0.703
20	310	420.7	1.357

”

Comment 4: The discussion about the operating cost of microgrid may be confusing for readers. For example, in Table 3, the operating cost of microgrid 3 is negative; please clarify if that is considered a reduction in operating cost, or if this is because of the selling energy that are considered within the operating cost. 

Response 4: Thank you for pointing this out. Negative values indicate that the microgrid network has achieved significant net profits by selling electricity to other islands and the main grid.

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you for responding to my review. I have obtained sufficient information to have no further comments on the revised paper.

Reviewer 2 Report

Comments and Suggestions for Authors

There have been a lot of changes to the paper, so I accept it now.  Thank you for your effort.

Reviewer 3 Report

Comments and Suggestions for Authors

The authors have addressed all my concerns. I have no further comments.