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Peer-Review Record

A Reactive Power Partitioning Method Considering Source–Load Correlation and Regional Coupling Degrees

Energies 2025, 18(8), 1960; https://doi.org/10.3390/en18081960

by Jiazheng Ding

, Xiaoyang Xu and Fengqiang Deng^*

Reviewer 1:

Viktor Denysov

Reviewer 2: Anonymous

Reviewer 3: Anonymous

Reviewer 4: Anonymous

Energies 2025, 18(8), 1960; https://doi.org/10.3390/en18081960

Submission received: 9 March 2025 / Revised: 1 April 2025 / Accepted: 7 April 2025 / Published: 11 April 2025

(This article belongs to the Section F: Electrical Engineering)

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The study proposes a joint probability distribution model based on the Copula function for the source-load correlation index. It is based on a comprehensive system for assessing the source-load relationship, taking into account the significant volume of renewable energy production. It was accounted that, unlike many studies, the source-load correlation index is not a constant value, but is a function of time. A generalized method for constructing matrices with extended sensitivity is proposed, providing for dynamic adjustment of weights using regional correction factors of the degree of connection. This study aims to overcome the theoretical limitations of using static partitioning and to provide theoretical support for dynamic zone control in modern power systems with a high volume of renewable energy generation.

The relevance of the study is beyond doubt. The claim of originality is substantiated by a sufficient number of works selected by the authors for comparison and indication of differences and listed in the review.

I agree with the authors who believe that the study, unlike those selected for comparison, not only proposes a theoretical basis for dynamic partitioning but also sets out practical ideas for optimizing reactive power dispatching in networks with a high volume of renewable energy generation.

- According to the authors of the study, compared to existing methods, the proposed approach allows achieving a 23.7% reduction of reactive power exchange between sections and improve local voltage regulation by 8.9%, which is confirmed on the 39-node IEEE system.

- The study is structured quite logically. The sequence of issues under consideration is clear. The options and method of presenting the answers to the questions indicated in the purpose of the study do not cause ambiguity.

- Based on the results of the simulation a well-founded conclusion was made that, in comparison with traditional partitioning schemes, the proposed method reduces the interregional exchange of reactive power by 23.7% and increases the suppression of local voltage fluctuations by 18.9%, and thus confirms its effectiveness in achieving the local balance of reactive power and accurate voltage regulation.

- The only remark on the list of references is that there are no references to Paper [24], and that this 1995 paper is hardly relevant to this study at all, as I indicated in the signed review.

- It may be erroneous "wind power generation units were connected to Node 38 as PV nodes." in lines 344-345.

- Figure 2. IEEE39 node system, as to me, is purely illustrative and does not carry any specific information.

- Almost the same feeling arises for Figure 4. Hierarchical clustering results.

- Regarding Figure 6. On the visualization of the partitioning results, I would like to see the decryption (legend) of the colors used.

- Tables 1 and 4 are located quite far from each other, which makes it difficult to compare the results obtained by the authors with the results obtained in work [17]. Perhaps it would be more successful to combine the compared results into one table.

Author Response

Dear Editor and Reviewers,

We are grateful for the constructive comments and valuable suggestions, which have significantly helped us improve the quality of this work, and we sincerely apologize for the oversights in the original manuscript and have now thoroughly revised the text to ensure clarity and accuracy.

In this revised manuscript, we have carefully addressed all the reviewers' concerns point by point. Below, we provide a detailed response to each comment. Please find our revisions and explanations on the following pages.

Comments 1: The only remark on the list of references is that there are no references to Paper [24], and that this 1995 paper is hardly relevant to this study at all, as I indicated in the signed review.

Response 1: Agree. We have, accordingly, modified the order and quantity of references to emphasize this point. We have deleted Reference [8] and [24], and for other references that need to be used or compared, we have made clearer marks in the paper. You can find the change in page 15, line 467-515.

Comments 2: It may be erroneous "wind power generation units were connected to Node 38 as PV nodes." in lines 344-345.

Response 2:

[wind power generation units were connected to Node 38.]

Thanks for pointing out the problem. We made a mistake in stating this sentence. We should remove “as PV nodes”, which is to connect wind turbines to 38 nodes to better simulate and integrate renewable energy units.

Comments 3: Figure 2. IEEE39 node system, as to me, is purely illustrative and does not carry any specific information. Almost the same feeling arises for Figure 4. Hierarchical clustering results.

Response 3:

[

Figure 4. Hierarchical clustering results]

Thanks for point it out. Figure 2 is used to show the ieee39 structure diagram of the calculation example in this paper, and it is convenient to explain that the wind turbine and photovoltaic generator set are connected to some special nodes as PV nodes. As for figure 4, the y axis is the normalized electrical distance without dimension and the x axis is the node index is numbered. To make it clearer, we've changed the name of the x axis. You can find the change in line 368.

Comments 4: Regarding Figure 6. On the visualization of the partitioning results, I would like to see the decryption (legend) of the colors used.

Response 4:

[

Figure 6.Visualization of partitioning results]

Agree. We have, accordingly, revised the figure to emphasize this point.We recreated Figure 6 in a variety of colors to make it look distinct. You can find the change in page 13, line 387.

Comments 5: Tables 1 and 4 are located quite far from each other, which makes it difficult to compare the results obtained by the authors with the results obtained in work [17]. Perhaps it would be more successful to combine the compared results into one table.

Response 5:

[ The source-load correlation index and the regional coupling degree index together constitute the evaluation index of reactive power zoning for this project, and calculate result. A comparative analysis is conducted between the partitioning results obtained through our method and those derived from the approach in [16] are shown in Table 1. The results of the 6 zones are shown in Table 2.

Table 1. Results of evaluation index and compare to the results of evaluation index in reference [16]

*Zone*	results of evaluation index in paper				results of evaluation index in reference [16]
*Zone*			P	J			P
1	0.6340	0.2798	19.8992	0.5107	0.3418	0.3290	20.7750
2	0.3571	0.3352			0.3699	0.3196
3	0.1621	0.3742			0.3571	0.3590
4	0.3409	0.3384			0.2600	0.4242
5	0.1949	0.3676			0.6279	0.2708
6	0.3439	0.3378			0.1954	0.2763

From the results in Table 1, it is evident that under the partitioning method proposed in this paper, the coupling degree between nodes and partitions is smaller compared to that in [16], with mean values of 0.3390 and 0.3587, respectively. Additionally, the inter-regional coupling indicators are also lower than those in [16], resulting in an ovsserall smaller total coupling degree for the proposed method. ]

Agree. We have, accordingly, combined table 1 and table 4 to emphasize this point. By combining Table 1 and Table 4, this gives a clearer comparison between this paper's approach and that of the reference. You can find the change in page 13, line 388-399.

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

Paper’s summary

Power networks with a high penetration of renewable energy sources (RES) present high volatility, making conventional partitioning strategies ineffective. In this context, this paper proposes a reactive power partitioning method that integrates dynamic source-load correlation characteristics and regional coupling degree evaluation. They construct a Copula function-based joint probability distribution model for source-load correlation.

Reviewer’s comments:

Comment:
I consider the topic relevant to the research field. However, improvements in the explanations about the papers' contributions are necessary. The authors claim three contributions, which are explained in lines 72-80. However, it is difficult to identify these three contributions in the methodological part. I suggested the authors to clearly indicate which part of the methodological section is related to each contribution of the work.

Comment:
The authors claimed that “the proposed approach achieves a 23.7% reduction in reactive power exchange between partitions and enhances local voltage regulation by 18.9%”. However, I cannot verify this affirmation based on the presented results.

Comment:
What are the values of reactive power exchange between partitions and local voltage enhancement provided by methodologies available in the literature? I suggest including a comparison between the results you get using your methodology and the results given by other methodologies.

Comment:
In figure 1, the steps are described from lines 291-316, however, the steps are not mentioned in the flowchart. I suggest numbering the steps in the flowchart, to facilitate readers understanding. The caption of figure 1 can be improved. This is the flowchart of what?

Comment:
It is missing capitalized letters in the first letter of the table captions.

Comment
Figure 4 is not understandable. It is missing the measurement units in axes y and x

Comment:
The conclusion section has only a qualitative analysis of the results, so I cannot assess the consistency of the arguments presented. I suggest to include quantitative values in the results.

Author Response

Dear Editor and Reviewers,

Comments 1: I consider the topic relevant to the research field. However, improvements in the explanations about the papers' contributions are necessary. The authors claim three contributions, which are explained in lines 72-80. However, it is difficult to identify these three contributions in the methodological part. I suggested the authors to clearly indicate which part of the methodological section is related to each contribution of the work.

Response 1:

[To resolve these issues, the proposed method aims to bridge the theoretical and practical gaps through the innovations: proposing a composite coupling index that incorporates temporal synchronization between renewable generation and load profiles, thereby optimizing local reactive power balance. The key innovations include:

In Section 2.1, a Gaussian Copula-based joint probability model captures the stochastic dependence between renewable energy sources and load demand, followed by scenario reduction via K-means clustering to generate key representative scenarios.
In Section 3, the electrical distance matrix integrated with correction factors can more effectively capture the variability of renewable energy sources, making it suitable for the current trend of increasing renewable energy sources penetration levels.
A hierarchical clustering framework driven by a composite coupling index, ensuring partitions with minimized intra-regional electrical distances and maximized inter-regional decoupling.]

Think you for pointing this out, we agree with this comment. We sincerely apologize for the previous lack of clear correspondence between our stated contributions and the specific content in the paper. In the updated manuscript, we have explicitly indicated the relationship between each contribution and its corresponding content, ensuring readers can easily understand how they align.

Comments 2: The authors claimed that “the proposed approach achieves a 23.7% reduction in reactive power exchange between partitions and enhances local voltage regulation by 18.9%”. However, I cannot verify this affirmation based on the presented results.

Response 2:

[Simulation results demonstrate that in practical case studies, compared with traditional partitioning schemes, the proposed method reduces regional coupling degree metric by 4.216% and enhances the regional reactive power imbalance index suppression by 11.082%.

...

Our case studies demonstrate a 4.216% reduction in regional coupling degree metric and 11.082% improvement in regional reactive power imbalance.]

Thank you for pointing this out. We agree with this comment. We have re-compared the results obtained by the method in this paper with those in the references, and found that the previous calculation was wrong, and now we have revised it. According to the result of P and , we find our case studies demonstrate a 4.216% reduction in regional coupling degree metric and 11.082% improvement in regional reactive power imbalance compared with reference [16]. You can find the change in page 1, lines 23-24 and page15, lines 458-459.

Comments 3: What are the values of reactive power exchange between partitions and local voltage enhancement provided by methodologies available in the literature? I suggest including a comparison between the results you get using your methodology and the results given by other methodologies.

Response 3:

[The source-load correlation index and the regional coupling degree index together constitute the evaluation index of reactive power zoning for this project, and calculate result. A comparative analysis is conducted between the partitioning results obtained through our method and those derived from the approach in [16] are shown in Table 2. The results of the 6 zones are shown in Table 3.

Results of evaluation index and compare to the results of evaluation index in reference [16]

*Zone*	results of evaluation index in paper				results of evaluation index in reference [16]
*Zone*			P	J			P
1	0.6340	0.2798	19.8903	0.5107	0.3418	0.3290	20.7750
2	0.3571	0.3352			0.3699	0.3196
3	0.1621	0.3742			0.3571	0.3590
4	0.3409	0.3384			0.2600	0.4242
5	0.1949	0.3676			0.6279	0.2708
6	0.3439	0.3378			0.1954	0.2763

From the results in Table 2, it is evident that under the partitioning method proposed in this paper, the coupling degree between nodes and partitions is smaller compared to that in [16], with mean values of 0.3390 and 0.3587, respectively. Additionally, the inter-regional coupling indicators are also lower than those in [16], resulting in an ovsserall smaller total coupling degree for the proposed method. ]

Thank you for raising these important points. In preparing the initial draft, we mistakenly translated η and P as "reactive power exchange between partitions" and "local voltage enhancement" respectively. These have now been corrected. During our re-evaluation of these two metrics, we identified some computational errors in the implementation, all of which have been rectified. We sincerely apologize for these fundamental oversights.

Tables 2, 4, and the newly added Table 5 all present comparative results between our proposed method and other benchmark methods.

Comments 4: In figure 1, the steps are described from lines 291-316, however, the steps are not mentioned in the flowchart. I suggest numbering the steps in the flowchart, to facilitate readers understanding. The caption of figure 1 can be improved. This is the flowchart of what?

Response 4:

[Step 1: Obtain the output data of all grid-connected renewable energy sources equipment and the electricity consumption load data on the user side within the power grid to be partitioned. These data are referred to as source-load data. The resolution of the source-load data used in this paper is 1 hour.

Step 2: Fit the source-load data. Calculate the marginal distribution of the source-load data and construct a Gaussian Copula-based correlation model for the source-load data based on their marginal distributions.

Step 3: Use the Copula model constructed in Step 2 to generate base scenarios, and then cluster these base scenarios into K scenarios using the K-means clustering algorithm.

Step 4: Calculate the index J according to Equations (4-7).

Step 5: Compute the power flow for the K generated scenarios to obtain K power flow operating states, and determine the correction factor and the full-dimensional electrical distance matrix based on Equations (8-14).

Step 6: Based on the calculation results from Steps 4 and 5, compute the modified full-dimensional electrical distance matrix that accounts for source-load correlations.

Step 7: Perform hierarchical clustering.

Step 8: Select the number of partitions.

Step 9: Calculate the regional coupling degree metrics according to Equations (16-18).

Step 10: Output the partitioning results.

The flowchart of reactive power partitioning method considering source-load correlation and regional coupling degree is shown in Figure 1.

Figure 1. The flowchart of reactive power partitioning method considering source-load correlation and regional coupling degree]

Agree. We have modified the description of part of the process in the figure and in the article to make it clear, but because some steps are implemented in combination, it is difficult to reflect them in the flow chart, so it is very difficult to number them on the figure. At the same time, we have renamed Figure 1 as “The flowchart of reactive power partitioning method considering source-load correlation and regional coupling degree”. You can find the change in pages 8-10, lines 298-322.

Comments 5: It is missing capitalized letters in the first letter of the table captions.

Response 5: Agree. We have, accordingly, checked the title of the entire article and made changes to the first letter to conform to the specification.

Comments 6: Figure 4 is not understandable. It is missing the measurement units in axes y and x.

Response 6:

[

Figure 4. Hierarchical clustering results]

In figure 4, the y axis is the normalized electrical distance without dimension and the x axis is the node index is numbered. To make it clearer, we've changed the name of the x axis. You can find the change in line 368.

Comments 7: The conclusion section has only a qualitative analysis of the results, so I cannot assess the consistency of the arguments presented. I suggest to include quantitative values in the results.

Response 7:

[Our case studies demonstrate a 4.216% reduction in regional coupling degree metric and 11.082% improvement in regional reactive power imbalance. However, the Gaussian Copula’s limitations in capturing tail dependencies warrant further investigation.]

Thanks for pointing it out. We quite agree that conclusions need to be quantified with some data. So at the end of the paper, we add a discussion section to point out the achievements of this method and the parts that need to be improved through data. You can locate our updates at line 425-429.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Dear authors,

Here are my observations.

Comments for author File: Comments.docx

Author Response

Dear Editor and Reviewers,

Comments 1: The article format is not correct: equations are aligned incorrectly; sections are not correctly established (e.g., 5 Selection of the number of partitions should not be a section). The sections should clearly highlight the introduction, methodology, experimental, results and related discussions, which is not valid in this article. It needs to be restructured; CONCLUSION should be written in lowercase; The final sections are missing: funding, acknowledgment, data statement, etc…

Response 1: Thank you for pointing this out. We agree with this comment. Therefore, we have:

Revised the format of equations;
Reset the section, the fourth section of the original text is merged into the third section;
Checked the title of the full text for capitalization and modified to conform to the specification.
Added the missing sections at the end of the article: Patents, Supplementary Materials, Author Contributions, Funding, Data Availability Statement, and Conflicts of Interest. You can find the change in page 14, line 445-458.

Comments 2: In the introduction, it should be clearer what is the difference brought by this article compared to other mentioned works. Also, the discussion about gaps should have a purpose.

Response 2:

[To resolve these issues, the proposed method aims to bridge the theoretical and practical gaps through the innovations: proposing a composite coupling index that in-corporates temporal synchronization between renewable generation and load profiles, thereby optimizing local reactive power balance. The key innovations include: …

The electrical distance matrix integrated with correction factors can more effectively capture the variability of renewable energy sources, making it suitable for the current trend of increasing renewable energy sources penetration levels.]

Thank you for this constructive suggestion. We fully agree that the introduction should better highlight our novel contributions. In the revised manuscript (Page 2, Section 1, Paragraphs 3–4, line 71-74 and line 78-80), we have made the following key improvements:

Unlike existing methods [1–6], our work makes 2 fundamental advances:
(1) A dynamically weighted electrical distance matrix with renewable penetration correction factors, overcoming the static impedance limitations in [3,16];
(3) A composite coupling index (P) that integrates both intra-region synchronization and inter-region decoupling, resolving the incomplete evaluation framework noted."

These innovations directly addres critical gaps in renewable-rich grids: Existing partitioning methods ignore time-varying source-load dependencies, leading to reactive power imbalance during renewable fluctuations.

These revisions clarify both the methodological distinctions and the practical significance of bridging these gaps. The updated introduction links directly to the case study results (Section 5) to validate the claims.

Comments 3: The assumptions of the Copula model are not clearly explained – it should be detailed plus mentioned if there are limitations in applicability.

Response 3:

[The Gaussian Copula assumes linear dependencies between variables and sym-metry in tail correlations. While effective for modeling renewable energy sources and load dependencies, it may underestimate extreme events (e.g., simultaneous low wind and high load). Future work could explore vine copulas for asymmetric dependencies.]

We sincerely appreciate this insightful comment. To address this concern, we have expanded the discussion of the Gaussian Copula's assumptions and limitations in Section 2.1 (Page 3, Lines 121-124), with explanations in below:

Assumptions:
(1) Linear dependence structure: The correlation between variables is characterized by a covariance matrix, which may not capture nonlinear interactions.
(3) Gaussian marginal transforms: All input variables must be transformable to standard normal distributions via probability integral transforms.
Limitations and Mitigations:

Underestimation of tail risks: The model may undervalue the probability of concurrent extreme events (e.g., <5th percentile wind generation coinciding with >95th percentile load), as noted in our case study (Section 5.3, Table 4).
Stationarity requirement: The correlation structure assumes temporal invariance, which may not hold during seasonal transitions.
To partially mitigate these issues, we incorporate a dynamic weighting mechanism (Section 3.2) that adjusts for observed deviations in the scenario sets."*

Future work direction: for systems requiring asymmetric dependence modeling (e.g., offshore wind farms with hurricane risks), vine copulas or non-parametric Copulas could be explored, albeit at higher computational cost. This will be investigated in our subsequent research on multi-energy systems.

Comments 4: In the article I encounter the notation Copula and copula. Standardize the writing.

Response 4: We thank the reviewer for catching this inconsistency. We agree this comment. In the revised manuscript, we have standardized all references to this term. You can find the change in line 103.

Comments 5: The K here is determined 127 by the elbow method” at line 127. This must be detailed.

Response 5:

[The K here is determined by the elbow method. The Elbow Method is an intuitive ap-proach used to determine the optimal number of clusters in clustering algorithms. Its core idea lies in identifying an "elbow point" by analyzing how clustering performance changes with different K values (number of clusters), thereby selecting the most ap-propriate cluster quantity.

Specifically, we calculate the Within-Cluster Sum of Squares (WCSS) minimized for different K values to identify the optimal scenario. When the rate of WCSS decrease significantly slows down, the corresponding number of clusters at this inflection point is considered the optimal choice.]

Think for you pointing this out. We agree with this comment. In response, we have significantly expanded the explanation of the elbow method in Section 2.2 (Pages 5-6) with the following specific revisions:

Added two new paragraphs (Page 4, paragraph 1 and 2, Lines 133-141).

The elbow method's application is now fully reproducible based on the provided details. We believe these modifications substantially strengthen the methodological rigor of our approach.

Comments 6: Is there any criterion for choosing kNN over another algorithm?

Response 6:

Think for you pointing this out. The algorithm used in this study for scenario reduction is K-means clustering, not k-nearest neighbors (kNN). The reviewer might have made a typo. Below, we provide a detailed rationale for selecting K-means over kNN and other clustering algorithms:

Fundamental Differences Between K-means and kNN

K-means is an unsupervised learning algorithm designed to partition unlabeled data into clusters by minimizing intra-cluster variance. It directly generates representative scenarios (centroids) that preserve key statistical and temporal features of the original data, aligning with our goal of dynamic source-load correlation modeling.
kNN is a supervised learning algorithm requiring labeled data to classify or predict new instances based on distance metrics. Since our scenario reduction task involves unsupervised clustering (no predefined labels), kNN relies on existing data points for inference and cannot synthesize new representative scenarios. So kNN is unsuitable.

Task-Specific Advantages of K-means

Representative Scenario Generation:
K-means minimizes the within-cluster sum of squared errors (SSE), ensuring that scenarios within the same cluster are highly similar, which matches our requirement for reduced yet representative scenarios.

Comparative Analysis with Other Clustering Algorithms

Hierarchical Clustering: We generated over 1,000 base scenarios using the Copula model, and if hierarchical clustering is used high computational cost (O(N³)) makes it impractical for large datasets.
DBSCAN: Sensitive to density parameters and does not directly support predefined cluster counts, conflicting with our need for fixed partitions.

Thank you again for your valuable feedback. We hope this clarification addresses your concern.

Comments 7: In section 3, the proposed method should be compared with other methods for calculating electrical distance (e.g. based on admissibility). And the criterion for choosing this approach should be detailed.

Response 7:

[Unlike the aforementioned methods, this paper constructs a Copula-based joint probability model to not only quantify the impact of spatio-temporal source-load correlations on partition stability but also introduces dynamic correction factors and composite coupling metrics, addressing the incompleteness of existing partition coupling degree evaluation frameworks.]

Think for you pointing this out. We agree with this comment.

The method proposed in this paper for calculating the electrical distance matrix differs from established approaches (such as admittance matrix-based or impedance-based methods) in the following key aspects:

Computational Complexity, Dynamic Condition Adaptability, and Integration of Renewable Energy Variability. The proposed sensitivity matrix method offers distinct advantages over conventional methods in terms of computational efficiency, adaptability to dynamic conditions, and incorporation of renewable energy fluctuations.
Dynamic Correction for Renewable Variability. Our approach integrates dynamic correction factors to account for fluctuations in renewable energy generation, enhancing accuracy under real-world operating conditions.
Through the matrix formulation, the sensitivity calculation is extended to all node types. This represents a significant advantage, as traditional impedance-based methods often struggle to handle such cases effectively.

You can locate it at Paper 1, Paragraph 1, line 48-52 and section 3.

Comments 8: The section on extending sensitivity to all nodes is confusing. An explanatory figure would be useful.

Response 8:

[The pseudocode for the aforementioned process is shown in Algorithm 1.]

Think for you pointing this out. We agree with this comment. We have added Algorithm 1 to more intuitively describe the process of expanding the sensitivity matrix. You can locate it at line 241.

Comments 9: The impact of choosing the number of clusters on the system performance is not discussed.

Response 9: [

Table 1 show the impact of choosing the number of clusters on the system performance.

the impact of choosing the number of clusters on the system performance.

*Cluster Number*	P
8	19.8927
16	19.8903
20	19.8895

We can easily observe that although the index P obtained by the optimal clustering determined by the elbow method is not the smallest, when the number of clusters exceeds this value, the growth trend of index P becomes slow. At the same time, it should be noted that the increase of the number of clusters will bring more scenes, and the calculation of scenes is very time-consuming. So let K=16 is a good choice.]

Think you for pointing this out. We agree with this comment. We have added a dedicated analysis in Section 5 to clarify how the cluster number (K) affects system performance.

our updates demonstrate that our K=16 choice is empirically justified, avoiding both under-/over-clustering pitfalls. Thank you for prompting this critical discussion.

Comments 10: The formula for P (Eq. 18) combines several metrics, but it is not clear how the results vary, what it reflects, limits, etc.

Response 10:

[Lower X_l which means tighter intra-region couplingand higher Y_lwhich means stronger inter-region separation will make P lower. If X_lincreases or Y_l decreases then index P will rise sharply. It reflects the comprehensive integration of coupling degrees within partitions and decoupling levels between regions. The larger the total evaluation index P of regional coupling degree, the more effective the zoning is. However, P is susceptible to extreme X_l and Y_l, which is where attention should be paid.]

We sincerely appreciate this insightful comment. To address this concern, we have detailed the function and meaning of each element in the equation in Section 3.3 (Page 8, Lines 272-277), which will help to understand the formula.

Comments 11: What happens if there are sudden changes in energy generation? How does this affect the coupling between regions?

Response 11:

[After the above partition is completed, in order to further partition the effectiveness, we consider the new energy output scenario as shown in Table 4, and calculate the imbalance index under the corresponding scenario.

It is not difficult to see from Table 4 that the new energy output gradually decreases with the new energy output, and the imbalance is constantly increasing, which is obvious. At the same time, when the new energy fluctuation range is large, the degree of imbalance brought by the proposed zoning is higher than that of reference 16, which indicates that the internal balance of the zoning obtained by the proposed method is closer.]

Thank you for raising this important question. We agree that sudden changes in renewable energy generation can significantly impact regional coupling. To address this, our method incorporates dynamic correction factors and scenario-based analysis to adapt to such fluctuations. Specifically:

Dynamic Correction Factors: The correction factor in Eq. (13) quantifies the impact of renewable energy volatility on electrical distances. This factor adjusts the partitioning weights in real-time based on observed generation changes, ensuring the model remains responsive to sudden shifts.
Validation Under Variability: By generating and clustering scenarios, our method captures a wide range of generation variability. However, the data we obtained lacked the more extreme cases, so we reduced the renewable energy output in the calculation of our proposed indicators. Table 4 in Section 5 demonstrates the method’s robustness under reduced renewable output. The results show that while imbalance increases with larger fluctuations, our partitions maintain tighter intra-regional coupling compared to static methods.

These revisions are highlighted in: Section 5 (Page 14, Table 4, Line 411-419):

Comments 12: How does a larger/smaller number of regions affect the efficiency of the network?

Response 12:

Think you for pointing this out, as illustrated in Figure 5, we present the variation of metric P with respect to the number of partitions. From the figure, it can be clearly observed that P exhibits a distinct minimum value—specifically, reaching its lowest point when the partition number equals 6. Furthermore, any deviation from this optimal partition count (whether increasing or decreasing the number of partitions) leads to a rise in P. Given that P serves as a comprehensive indicator reflecting both intra-regional and inter-regional coupling degrees, this suggests that partition numbers other than 6 may yield suboptimal outcomes. In such cases, neither internal nor external regional divisions would achieve maximal effectiveness in zoning and control.

Thank you for your insightful comment.

Comments 13: Line 280 "The optimal number of partitions is determined 280 when P reaches a local minimum, indicating the most stable coupling structure." - This must be detailed.

Response 13:

[In this case, the coupling within the region has a higher value, and the coupling between the regions has a smaller value, which is a best compromise for both within and outside the region.]

We sincerely appreciate this insightful comment. To address this concern, we have detailed the reason why we say that “P reaches a local minimum, indicating the most stable coupling structure” in Section 3.3, Lines 287-288. This is because when P reaches the local minimum, the coupling within the region has a larger value, and the coupling between regions has a smaller value, and these two values are better than other partitions, so choose to change the number of partitions. This has some relevance to our response in Comment 12.

Comments 14: In section 6 some steps are described vaguely. The computational requirements of the method and the feasibility of implementation are not discussed. Figure 1 should be explained more clearly.

Response 14:

[Step 1: Obtain the output data of all grid-connected renewable energy sources equipment and the electricity consumption load data on the user side within the power grid to be partitioned. These data are referred to as source-load data. The resolution of the source-load data used in this paper is 1 hour.

Step 3: Use the Copula model constructed in Step 2 to generate base scenarios, and then cluster these base scenarios into K scenarios using the K-means clustering algo-rithm.

Step 4: Calculate the index J according to Eq (4-7).

Step 5: Compute the power flow for the K generated scenarios to obtain K power flow operating states, and determine the correction factor and the full-dimensional electrical distance matrix based on Eq (8-14).

Step 6: Based on the calculation results from Steps 4 and 5, compute the modified full-dimensional electrical distance matrix that accounts for source-load correlations.

Step 7: Perform hierarchical clustering.

Step 8: Select the number of partitions.

Step 9: Calculate the regional coupling degree metrics according to Eq (16-18).

Step 10: Output the partitioning results.

The flowchart of reactive power partitioning method considering source-load cor-relation and regional coupling degree is shown in Figure 1.]

Thank you for your valuable feedback. We have revised Section 6 to clarify the methodological steps, computational requirements, and implementation feasibility, and expanded the explanation of Figure 1.

We sincerely hope that these updates could address the reviewer’s concerns about vagueness and feasibility while maintaining methodological rigor. Thank you for the constructive critique. You can locate our updates at line 291-311.

Comments 15: What are the characteristics of the dataset used? What are the data sources?

Response 15:

The data is characterized by wind data with a resolution of 15 minutes, including wind speed, wind direction and wind output at different altitudes. Photovoltaic power generation data with one-hour resolution only includes time and photovoltaic output.

Data source is not convenient to disclose, you can contact the corresponding author to obtain.

Comments 16: The experiment is not clear, all conditions, equipment, structure should be detailed, and possibly a photo should be introduced, at the beginning of section 7.

Response 16:

Thank you for your suggestion to enhance the experimental clarity. While our study is currently based on simulation (as field data from practical grid equipment is challenging to obtain), we have added the following details to Section 5 to improve reproducibility and transparency:

Test System: IEEE 39-bus system (Figure 2), modified with:

Renewable Integration: PV plant at Bus 35 wind farm at Bus 38.

Software/Hardware:

Platform: MATLAB R2023a.
Hardware: AMD Ryzen 7 5800H, 16 GB RAM.

Scenario Parameters:

1,000 base scenarios → reduced to K=16 via K-means (elbow method).
Copula margins: Weibull (wind), beta (PV), normal (load) – parameters in Figure

Your feedback has helped us better contextualize the study’s scope while maintaining rigor. We hope these revisions address your concerns.

Comments 17: More scenarios should be added, including extreme conditions or failures.

Response 17:

Imbalance index under different renewable energy sources output

Think you for pointing this out. We agree with this comment.

We sincerely apologize for the limitations in our dataset. The available data did not include extreme operating conditions, and obtaining such data within a short timeframe proved challenging. To address this, we have instead evaluated the impact of renewable energy output variations within each partitioned zone, using the regional reactive power imbalance index as our quantitative metric.

The results presented in Table 4 demonstrate that:

1) Our partitioning method effectively accounts for renewable generation fluctuations

2) The proposed approach achieves stronger intra-zone connectivity compared to conventional methods, as evidenced by the reactive power imbalance metrics

This suggests our zoning results maintain better electrical coherence during normal operation scenarios, though we acknowledge further validation under extreme conditions would be beneficial.

Comments 18: Are the results obtained applicable only to the tested system or are they generalizable?

Response 18:

Thank you for this critical question. While our current validation is based on the IEEE 39-bus system (Section 5), the method’s theoretical framework is designed for generalizability. Here’s why:

The Copula-based source-load modeaccommodates any marginal distributions, validated for Weibull/beta/normal cases but extensible to others.
The modified electrical distance matrix auto-adjusts to grid topology via sensitivity matrices independent of specific bus configurations.
The composite coupling index P is topology-agnostic, balancing intra/inter-regional coupling universally.

While the method’s principles are general, real-world deployment requires testing on diverse grids. Future work will validate this.

Comments 19: A discussion section is missing where the authors can detail certain aspects related to the work done: advantages, limitations, complexity, computational time, generalization, practical implementation, etc.

Response 19:

[In this paper, the computing platform we use is MATLAB under AMD Ryzen 7 5800H. It took 1.23s for scene generation and clustering, 0.02337s for hierarchical clustering, and 15.221086s for parallel calculation to correct the electrical distance matrix. While the computational workflow in Reference [16] took 20 seconds to complete. The calculation of this method mainly focuses on the steps of generating the sensitivity matrix, but the overall speed is still very impressive. Compared with the existing methods, the method has the characteristics of faster calculation time and better results.]

Think you for pointing this out. We agree with this comment. We have, accordingly, added a discussion part at the beginning of section 5 to emphasize this point. The discussion we made mainly included program runtime. You can find the change in page 13, line 319-325.

Comments 20: The conclusions should be reformulated, they are currently not suitable for the context of the article.

Response 20:

[This paper addresses the challenges posed by the expanding integration of renewable energy and enhanced power system interconnectivity, and we propose a dynamic reactive power partitioning method that integrates Copula-based source-load correlation and composite coupling metrics. A multidimensional partitioning criterion is established by combining time-series correlation indices of source-load interactions with an evaluation framework for regional electrical coupling. The implementation involves constructing a modified electrical distance matrix that accounts for power fluctuations in renewable generation units. Dynamic coupling relationships are modeled by incorporating time-series correlation analysis between source and load. A spectral clustering algorithm is applied to solve the optimized partitioning model. The resulting partition achieves strong electrical coupling between nodes and regions. This method provides effective zoning boundaries for reactive power and voltage optimization control. It enhances system operational economy and voltage stability through rational regional decoupling.

The effect of reactive power partitioning to achieve the purpose of reactive power optimization, which is conducive to the realization of the local balance of reactive power and the accurate control of node voltage. This method is a scientific and easy to implement multi-objective reactive power optimization method in power system, which is widely applicable to power networks, has a significant engineering practical value and has a wide range of application prospects.

Our case studies demonstrate a 4.216% reduction in regional coupling degree metric and 11.082% improvement in regional reactive power imbalance. However, the Gaussian Copula’s limitations in capturing tail dependencies warrant further investigation.]

Think you for pointing this out. We agree with this comment. We have, accordingly, revised the whole conclusion to emphasize this point. We split the long sentence in the conclusion part, added quantified data, and modified some inappropriate expressions, You can find the change in section 6, page 15, lines 437-459.

Author Response File: Author Response.docx

Reviewer 4 Report

Comments and Suggestions for Authors

This paper contributes to reactive power management in power systems with high renewable energy penetration by introducing a reactive power partitioning method that considers source-load correlation and regional coupling, utilizing a Copula function-based source-load correlation model and a modified electrical distance matrix to improve inter-cluster distance evaluation. Employing a bottom-up agglomerative algorithm for optimal node clustering, the proposed methods are validated through case studies, including the IEEE 39-node system, demonstrating enhanced reactive power exchange and local voltage regulation.

After a thorough review of the manuscript, I have identified several critical weaknesses in its technical content, use of abbreviations, writing quality, and typographical accuracy. Below, I provide a detailed critique highlighting specific examples from the paper.

The introduction fails to explicitly state the research gap and how this study advances the field. For example (from the introduction, Lines 31–41), the text discusses challenges in reactive power partitioning due to high-penetration renewable energy, but it does not clearly define what new solution is provided. Moreover, the phrase: "This research breaks through the theoretical limitations of static partitioning..." is vague. What specific theoretical limitations? How does this method address them better than previous approaches?
The methodology lacks sufficient details, making it difficult to replicate the study or assess its validity. For example (Section 2, Lines 92–115), the paper states: "Construct a source-load correlation model based on Copula function based on the marginal distribution of source-load data.", the exact steps for constructing the Copula function are not clearly explained such as what dataset was used?, what specific assumptions were made?, and how were the parameters estimated?
Self-citation issues reduce the credibility of the work by overemphasizing the authors’ previous studies rather than engaging with broader, independent research. A significant proportion of references (e.g., References [8], [9], [14], [17]) are authored by the same research group.
Several abbreviations are used without prior definition, leading to confusion. For example (Abstract, Lines 28–29), the terms "RES" (Renewable Energy Sources), "DoD" (Depth of Discharge), and "LVRT" (Low Voltage Ride-Through) appear without being defined.
Some abbreviations change formats throughout the paper, reducing clarity. Such as in (Lines 162–171), "ML" (Machine Learning) is sometimes written as "Ml."
Many sentences are unnecessarily long and complex, making them hard to read such as in (Lines 416–421) too long (68 words in one sentence).:

"Due to the increasing scale of new energy grid-connectedness and the improvement of power system connectivity, this paper proposes a reactive power partitioning method considering the source-load correlation and regional coupling degree, combining the source-load correlation and regional coupling degree indexes, and solving the modified electrical distance matrix considering the fluctuation of new energy power generating equipments and the time correlation between them and loads, so as to realize the reactive power partitioning with the effect of achieving a high degree of coupling between each node and each region, and to achieve the purpose of reactive power optimization."

Comments on the Quality of English Language

Several abbreviations are used without prior definition, leading to confusion. For example (Abstract, Lines 28–29), the terms "RES" (Renewable Energy Sources), "DoD" (Depth of Discharge), and "LVRT" (Low Voltage Ride-Through) appear without being defined.
Some abbreviations change formats throughout the paper, reducing clarity. Such as in (Lines 162–171), "ML" (Machine Learning) is sometimes written as "Ml."
Many sentences are unnecessarily long and complex, making them hard to read such as in (Lines 416–421) too long (68 words in one sentence).:

Author Response

Dear Editor and Reviewers,

Comments 1: The introduction fails to explicitly state the research gap and how this study advances the field. For example (from the introduction, Lines 31–41), the text discusses challenges in reactive power partitioning due to high-penetration renewable energy, but it does not clearly define what new solution is provided. Moreover, the phrase: "This research breaks through the theoretical limitations of static partitioning..." is vague. What specific theoretical limitations? How does this method address them better than previous approaches?

Response 1:

…

To resolve these issues, the proposed method aims to bridge the theoretical and practical gaps through the innovations: proposing a composite coupling index that incorporates temporal synchronization between renewable generation and load profiles, thereby optimizing local reactive power balance. The key innovations include:

A Copula-based joint probability model to capture the stochastic dependencies between renewable energy sources generation and load demand, enabling the generation of reduced scenario sets via K-means clustering.
The electrical distance matrix integrated with correction factors can more effectively capture the variability of renewable energy sources, making it suitable for the current trend of increasing renewable energy sources penetration levels.
A hierarchical clustering framework driven by a composite coupling index, ensuring partitions with minimized intra-regional electrical distances and maximized inter-regional decoupling.]

Thank you for pointing this out. We agree with this comment. We fully agree that the introduction should better highlight our novel contributions. In the revised manuscript (Page 2, Section 1, Paragraphs 3–4, line 71-74 and lines 78-80), we have made the following key improvements:

Unlike existing methods [1–6], our work makes 2 fundamental advances:

(1) A dynamically weighted electrical distance matrix with renewable penetration correction factors, overcoming the static impedance limitations in [3,16];

(3) A composite coupling index (P) that integrates both intra-region synchronization and inter-region decoupling, resolving the incomplete evaluation framework noted."

Comments 2: The methodology lacks sufficient details, making it difficult to replicate the study or assess its validity. For example (Section 2, Lines 92–115), the paper states: "Construct a source-load correlation model based on Copula function based on the marginal distribution of source-load data.", the exact steps for constructing the Copula function are not clearly explained such as what dataset was used?, what specific assumptions were made?, and how were the parameters estimated?

Response 2:

[The Gaussian Copula assumes linear dependencies between variables and symmetry in tail correlations. While effective for modeling renewable energy sources and load dependencies, it may underestimate extreme events (e.g., simultaneous low wind and high load). Future work could explore vine copulas for asymmetric dependencies…….

Based on Equations (1-2) and the marginal distributions, we determined the required parameters of the Gaussian Copula function using the undetermined coefficients method, thereby establishing a Gaussian Copula-based correlation model for source-load dependencies.]

Thank you for pointing this out. We agree with this comment. We have supplemented the fundamental assumptions underlying our use of the Gaussian Copula function. The methodology employs hourly-resolution source-load data, where the probability distributions of photovoltaic generation, wind power, and load demand - along with the subsequent Gaussian Copula function - are estimated using the method of undetermined coefficients.

We sincerely apologize for the previous lack of clarity in our presentation. Your insightful comments have significantly improved the quality and rigor of our paper.

You can locate our updates at Paper 3,Paragraph 7,Line 120-123 and Paper 10, Paragraph 4 , Line 340-343.

Comments 3: Self-citation issues reduce the credibility of the work by overemphasizing the authors’ previous studies rather than engaging with broader, independent research. A significant proportion of references (e.g., References [8], [9], [14], [17]) are authored by the same research group.

Response 3: Agree. We have, accordingly, modified the order and quantity of references to emphasize this point. We have deleted Reference [8] and [24], and for other references that need to be used or compared, we have made clearer marks in the paper. You can find the change in page 15, line 467-515.

Comments 4: Several abbreviations are used without prior definition, leading to confusion. For example (Abstract, Lines 28–29), the terms "RES" (Renewable Energy Sources), "DoD" (Depth of Discharge), and "LVRT" (Low Voltage Ride-Through) appear without being defined.

Response 4: Agree. We have, accordingly, deleted all the abbreviations, and all use full name to emphasize this point. You can find the change in section 1, Introduction part, we changed all the “RES” into “renewable energy sources”.

Comments 5: Some abbreviations change formats throughout the paper, reducing clarity. Such as in (Lines 162–171), "ML" (Machine Learning) is sometimes written as "Ml."

Response 5: Agree. We have, accordingly, checked the whole article for capitalization issues, and revised “copula” to “Copula” to emphasize this point. You can find the change in Line 104.

Comments 6: Many sentences are unnecessarily long and complex, making them hard to read such as in (Lines 416–421) too long (68 words in one sentence).:

Response 6:

Thank you for this constructive suggestion. We fully agree that long sentence should be split into short sentences to conform to English norms. You can find the change in Page 15, Section 6, lines 438-451.

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Dear authors,

Thanks for responding to my comments on the previous review process. The majority of my comments have been addressed. However, there is one question that still remains. It is written:

“Our case studies demonstrate a 4.216% reduction in regional coupling degree metric and 11.082% improvement in regional reactive power imbalance.”.

I see these values in the abstract and conclusion, however, I cannot verify this from the results section. Could you briefly explain how to reach those values? I also kindly advise including this explanation in the results section.
In addition, there is a type in line 395, where is written “ovsserall”.

Author Response

Comments 1: I see these values in the abstract and conclusion, however, I cannot verify this from the results section. Could you briefly explain how to reach those values? I also kindly advise including this explanation in the results section.

Response 1: [A comparative analysis shows that the proposed partitioning method achieves a lower regional coupling degree, reducing it by 4.216% compared to reference [16], while also decreasing regional reactive power imbalance by 11.082%. This effectively enhances intra-regional connectivity and reduces inter-regional reactive power dependency, thereby ensuring local reactive power balance within each partition.] Thank you for pointing this out, we agree with this comment. The comparative values are derived using the formula (X − Y)/Y, where X denotes the results obtained by our proposed method and Y those from the referenced approach. This calculation highlights the relative improvement of our method over the existing ones. Thank you for your valuable feedback. You can locate this at Paper 14, Paragraph 1, Line 422-427.

Comments 2:In addition, there is a type in line 395, where is written “ovsserall”.

Response 2: Thank you for your careful review and for pointing out the spelling error. We sincerely appreciate your attention to detail, which has helped improve the quality of our manuscript. The correction has been made accordingly.

Reviewer 3 Report

Comments and Suggestions for Authors

No more comments. the article can be accepted.

Author Response

Dear Editor and Reviewers,

We sincerely appreciate the time and expertise that the reviewers have dedicated to evaluating our work throughout the review process.

We are grateful to learn that the reviewers have no further concerns and recommend acceptance of the manuscript in its current form. It has been a privilege to incorporate the valuable feedback provided during previous rounds of review, which undoubtedly strengthened the quality of our research.

Thank you once again for your constructive guidance and for facilitating this scholarly exchange.

Sincerely,

On behalf of all co-authors

Reviewer 4 Report

Comments and Suggestions for Authors

After a thorough review of the authors' responses, I am pleased to see that the paper has been significantly enhanced. The revisions address all the concerns raised in the previous review.

Comments on the Quality of English Language

After a thorough review of the authors' responses, I am pleased to see that the paper has been significantly enhanced. The revisions address all the concerns raised in the previous review.

Author Response

Dear Editor and Reviewers,

We sincerely appreciate the time and expertise that the reviewers have dedicated to evaluating our work throughout the review process.

Thank you once again for your constructive guidance and for facilitating this scholarly exchange.

Sincerely,

On behalf of all co-authors

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A Reactive Power Partitioning Method Considering Source–Load Correlation and Regional Coupling Degrees

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