Frequency-Constrained Economic Dispatch of Microgrids Considering Frequency Response Performance

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article proposes a distributionally robust frequency-constrained economic dispatch (DRFC-ED) model for microgrids, which incorporates frequency constraints to enhance system stability under power imbalances, particularly in scenarios with high penetration of renewable energy sources (RESs).

The similarity percentage with previous works is very low, indicating that the paper is original.

The abstract provides a concise description of the problem and the methods used to address it. However, it would be beneficial to explicitly state the main results achieved.

The introduction could be improved by including a comparative table that highlights the contributions of previous studies versus the novel aspects introduced by this paper. Additionally, the term FRP in line 44 should be defined upon its first appearance.

2.1 Center-of-inertia Frequency Response Model

Replace phrases such as “Here, the governor model of diesel generators is approximated as a first-order inertial dynamic” with clearer expressions like: "The governor dynamics of diesel generators are modeled as a first-order inertial system."

In equation (5), the use of notations such as Δf(s), ωₙ, and ζ should be clarified with a table or a note indicating their physical meaning.

Include a brief explanation of the physical significance of H_sys, K_sys, and D_sys immediately after equation (6), to aid comprehension.

2.2 Modeling of Frequency Constraints

2.2.1 RoCoF Constraint:

Improve the wording of the following sentence: Instead of "RoCoF is maximized at the initial time of the disturbance and must not exceed the permissible threshold," consider using: "The RoCoF reaches its maximum value immediately after the disturbance and must remain below the permissible threshold."

2.2.2 MFD Constraint:

Simplify unnecessarily complex phrasing:

Replace "Due to the different expressions in the overdamped and underdamped cases..." with:
"Since MFD differs between overdamped and underdamped cases..."

Although the terms ψ₁ and ψ₂ are mentioned after their usage, they have not been explicitly defined. It is recommended to define these terms before or immediately at the point of their first appearance to improve clarity and ensure proper understanding.

2.2.3 QSSFD Constraint:

Clarify the dependencies of each term in equation (9), particularly whether f_qss is a fixed parameter or an optimization variable.

2.3 Modeling of ITAE of Frequency

The use of Simpson’s Rule is mentioned, but no justification is provided for choosing this numerical integration method. It is recommended to briefly explain why Simpson’s Rule is appropriate

2.4 Convexification Method for MFD and ITAE of Frequency

Equation (11) is not clearly.

The phrase “This ‘max(x, y)’ form can be easily linearized…” could be improved by using more formal language.

3.3 Objective Function

Equation (16) is long and difficult to read. It is recommended to divide the cost function into subcomponents (e.g., C₁ to C₆), and provide an explanation of each term before presenting the complete expression. This will enhance clarity and allow the reader to better understand the role of each component.

Additionally, consider including a summary table that outlines:

Cost component (C₁–C₆)

Description / Interpretation

Variables involved

The constraint formulation is clearly structured and contributes to the overall readability of the model.

4. Solution Methodology (Introductoria)

4.1 Modeling of Wasserstein Distance-Based Ambiguity Set

Include an explanation of the parameter θ, clarifying its role and interpretation

4.2 Linearization of Objective Function and DR Chance Constraints

Define all functions used, including f(x, ξ), a(x), b(x), etc., to ensure clarity for the reader.

Additionally, auxiliary variables such as χ, λ, and β are introduced without definition. These should be clearly described, including their purpose and domains, to avoid ambiguity in the formulation.

5. Case Study (Intro)

The sentence “…is validated in a microgrid with 4 diesel generators…” could be improved for clarity and formality. A better alternative would be:

“…is evaluated using a test microgrid system composed of four diesel generators...”

5.2 Comparison of Optimization Results

The sentence  “...demonstrates that the multi-objective optimization method can achieve a synergistic optimization...” could be improved for clarity and formality. A better alternative would be:
“...demonstrates that the proposed multi-objective formulation enables a balance between economic efficiency and frequency performance.”

Clarify what the terms d⁺, d⁻, and Δd² represent in Table 2.

6. Conclusions

The conclusions are appropriate; however, they could be written in a more technical manner and avoid repetition of terms. For instance, the sentence “Furthermore, it demonstrates that damping has a more significant impact on improving ITAE of frequency compared to inertia. Appropriately increasing damping can alleviate concerns...” could be rephrased using alternative wording. Similarly, “This study contributes to the economic and secure operation of microgrids with high proportions of RESs...” could benefit from a more formal and precise formulation.

References

The references used are appropriate.

Comments on the Quality of English Language

The English is not incorrect, but it could be improved by using more technical language.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper presents a novel approach to improving microgrid frequency response through a distributionally robust frequency-constrained economic dispatch (DRFC-ED) model. It highlights the value of integrating virtual inertia and damping for inverter-based resources.

1. The proposed model is mathematically robust; there is insufficient discussion regarding the practical obstacles in applying the DRFC-ED model within actual microgrids. For instance, the paper fails to investigate the technical or operational challenges associated with incorporating virtual inertia and damping from energy storage systems (ESSs) and diesel generators. A more thorough examination of possible limitations would enhance the model's relevance.

2. In this paper, deep neural networks (DNNs) are discussed for convexification, but the computational complexity of the optimization process is not discussed in detail. Due to the complexity of the optimization, a more detailed discussion of computational costs and scalability would be beneficial, especially for large-scale microgrids.

3. RES uncertainty is addressed using a Wasserstein distance-based ambiguity set. It does not fully explain how this uncertainty model compares to other approaches, especially when it comes to computational efficiency and robustness. There needs to be further discussion about how this method compares to other methods (e.g., Gaussian-based methods).

4. Static optimization and economic factors are primarily discussed in the paper. It lacks an in-depth analysis of microgrid dynamics during sudden disturbances, especially in terms of transient stability. A more comprehensive evaluation of the proposed model would include simulations or results under dynamic disturbances.

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Overall, the paper addresses an important topic and the methodology seems sound, but there are areas where clarity, justification of parameters, and discussion of limitations could improve.

 

1. This paper presents a meaningful approach by transforming non-convex optimization into convex optimization in microgrids. In this context, several relevant studies have been conducted, including "Enabling high-efficiency economic dispatch of hybrid AC/DC networked microgrids: Steady-state convex bi-directional converter models," which introduces a convex relaxation technique to accommodate a large number of scenarios representing RES uncertainties. Citing and discussing this work in the introduction section would help strengthen the paper’s foundation and enhance its overall quality.
2. It would be beneficial to provide additional details on the DNN architecture, such as the number of layers, neuron selection, and training data size. Furthermore, including validation metrics like RMSE and R² would help justify the choice of linear layers and ensure that the convex approximations are sufficiently accurate.
3. Conducting a sensitivity analysis to examine how variations in θ impact economic costs, ITAE, and constraint violations would be valuable. This analysis could help operators strike a balance between conservatism and efficiency when selecting θ in practical applications.
4. A more detailed explanation of the rationale for excluding the frequency support capabilities of RESs would be helpful. Additionally, discussing whether this assumption remains valid in microgrids with advanced grid-forming inverters and how it might affect the model’s applicability would strengthen the paper.
5. In the conclusion, it would be helpful to explicitly outline the model’s limitations and suggest potential future research directions, such as incorporating RES frequency support or exploring hybrid ambiguity sets for uncertainty modeling.

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The work presented complements and contributes to the state of the art in the field of frequency stability and falls within the scope of the journal.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors adequately address all comments.

Thanks

Reviewer 3 Report

Comments and Suggestions for Authors

Thanks for the careful revisions. The authors have fully addressed all my concerns. Great work!