Incorporating Renewable Generation Uncertainty Into Multi-Objective Dispatch Optimization

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors propose a methodology to analyze the trade-offs between generation costs and CO_2 emissions in a power system with high wind penetration. The study utilizes the IEEE-57 bus system, modifying it to include wind farms. The core contribution involves using Kernel density estimation to model wind forecast errors (based on historical data) and applying the epsilon-constraint method to solve the Multi-Objective Optimal Power Flow (MO-OPF). The results visualize Pareto fronts under different wind deviation scenarios. The topic is relevant to the current energy transition. The choice to use Kernel distribution for wind error modeling rather than assuming a standard Gaussian distribution is a positive aspect of the paper, as it better captures real-world skewness in wind data. However, there are several points where clarifications, justifications, and expansions can further improve the quality.

 

1. Multi-objective optimization for economic emission dispatch (EED) has been extensively studied. The paper should clearly articulate its distinct contributions relative to the substantial body of work published over the past decade.
2. The Introduction requires a more comprehensive literature review. Currently, the discussion does not cover studies addressing RES uncertainty. For example, “Enabling high-efficiency economic dispatch of hybrid AC/DC networked microgrids: Steady-state convex bi-directional converter models” demonstrates how convex modeling can efficiently incorporate 8,760 scenarios. Incorporating and discussing such works would greatly enhance the technical depth and completeness of the introduction.
3. The research gaps should be presented more explicitly and systematically. A clear point-by-point articulation would improve the clarity and depth of the gap analysis.
4. The paper would be strengthened by comparing the proposed kernel density estimation approach with a conventional Gaussian assumption, thereby quantifying the reduction in “error” or “risk” attributable to the proposed method.
5. Many figures, particularly Figures 16 through 22, are too blurry to interpret clearly. It is recommended to replace them with higher-resolution versions.
6. The conclusion should summarize the key quantitative findings of the study to provide readers with a clear understanding of the main outcomes.

Author Response

Comments 1: Multi-objective optimization for economic emission dispatch (EED) has been extensively studied. The paper should clearly articulate its distinct contributions relative to the substantial body of work published over the past decade.

Response 1: 

We fully agree that multi-objective economic–emission dispatch has been widely investigated in the literature, and that clearly positioning the contribution of the present work with respect to previous studies is essential.

 

In response to this comment, the manuscript has been revised to explicitly clarify the distinct contributions of this work relative to existing EED and MO-OPF approaches. In particular, the novelty of the proposed methodology does not lie in the formulation of a new EED objective, but rather in the integrated treatment of renewable generation uncertainty and operational decision-making within a multi-objective OPF framework.

 

The main distinguishing aspects of this work are now explicitly emphasized in the revised Introduction and Discussion sections, and can be summarized as follows:

 

a) Unlike most EED studies that assume deterministic renewable outputs or Gaussian forecast errors, this work employs kernel density estimation (KDE) based on historical wind data to capture non-Gaussian and asymmetric forecast error distributions, providing a more realistic representation of wind uncertainty.

b) The proposed approach integrates this data-driven uncertainty modeling directly into a parametric MO-OPF framework solved using the ε-constraint method, enabling a systematic exploration of cost–emission trade-offs under different wind deviation scenarios.

c) Rather than focusing on a single optimal dispatch solution, the methodology provides Pareto fronts conditioned on renewable uncertainty, offering decision-makers valuable insight into the sensitivity of economic and environmental performance to wind forecast deviations.

d) The approach is conceived from an operational perspective, aiming to support preventive decision-making in systems with high renewable penetration, while maintaining moderate computational complexity compared to fully stochastic or scenario-based optimization methods.

 

These aspects have been clarified and highlighted in the revised manuscript, particularly in the final paragraph of the Introduction and in the Discussion section, in order to better distinguish the proposed methodology from existing EED literature published over the past decade. We think that these aspects differentiate the paper from works focused on algorithmic novelty.

Comments 2: The Introduction requires a more comprehensive literature review. Currently, the discussion does not cover studies addressing RES uncertainty. For example, “Enabling high-efficiency economic dispatch of hybrid AC/DC networked microgrids: Steady-state convex bi-directional converter models” demonstrates how convex modeling can efficiently incorporate 8,760 scenarios. Incorporating and discussing such works would greatly enhance the technical depth and completeness of the introduction.

Response 2:

The Introduction has been significantly expanded to include recent literature (last five years) on renewable uncertainty, reserve requirements, and system flexibility. In particular, we now position our work relative to stochastic programming, scenario‑based approaches, and convex formulations. We explicitly discuss that,  our contribution emphasizes realism of uncertainty modeling based on real data and interpretability of results for operators.

 

Comments 3: The research gaps should be presented more explicitly and systematically. A clear point-by-point articulation would improve the clarity and depth of the gap analysis.

Response 3:

In response to this comment, the Introduction has been revised to explicitly identify and structure the main research gaps addressed by this work in a clear, point-by-point manner. Rather than implicitly embedding the gaps within the literature discussion, the revised manuscript now includes a dedicated paragraph that synthesizes the limitations of existing studies and directly motivates the proposed approach.

Specifically, the revised gap analysis highlights the following aspects:

a) Scenario-based and stochastic optimization approaches, while powerful, frequently require a large number of scenarios, leading to increased computational complexity and limiting their applicability in operational or near-real-time decision-making contexts.

b) Many studies focus on identifying a single optimal dispatch solution, providing limited insight into how cost–emission trade-offs evolve under different levels of renewable uncertainty.

c) There is a lack of operationally oriented methodologies that integrate realistic renewable uncertainty modeling with multi-objective OPF in a way that is both interpretable and computationally tractable for system operators.

These gaps are now explicitly articulated in the revised Introduction, immediately preceding the statement of contributions. This restructuring improves the clarity of the gap analysis and provides a direct and logical link between the identified limitations in the literature and the proposed methodology.

Comments 4: The paper would be strengthened by comparing the proposed kernel density estimation approach with a conventional Gaussian assumption, thereby quantifying the reduction in “error” or “risk” attributable to the proposed method.

Response 4:

In the revised manuscript, we explicitly justify the use of KDE by showing that real wind deviations exhibit skewness and heavy tails that are not well captured by Gaussian models. While a full quantitative risk comparison is beyond the scope of the paper, we now include a qualitative discussion and literature justifications highlighting how Gaussian assumptions underestimate extreme deviations.

We acknowledge that a direct numerical comparison between Gaussian-based and KDE-based uncertainty models, in terms of dispatch risk or cost variance, would be an interesting extension of this work. This point has now been explicitly noted in the Conclusions as a direction for future research. Nevertheless, we believe that the current formulation already demonstrates the operational relevance of using KDE to capture realistic wind uncertainty in multi-objective dispatch problems.

Comments 5: Many figures, particularly Figures 16 through 22, are too blurry to interpret clearly. It is recommended to replace them with higher-resolution versions.

Response 5:

All figures have been regenerated and replaced with high‑resolution versions. In particular, Figures 16–22 have been substantially improved for readability.

Comments 6: The conclusion should summarize the key quantitative findings of the study to provide readers with a clear understanding of the main outcomes.

Response 6:

The Conclusions section has been rewritten to include explicit qualitative insights focused on the following topics:

a) Enhancing that wind power forecast errors are asymmetrically distributed and can be accurately reproduced using kernel density estimation, enabling the generation of realistic short-term deviation scenarios.

b) By embedding these stochastic representations into a multi-objective AC Optimal Power Flow framework, the results show that optimal redispatch of thermal generation can be identified in advance to accommodate wind variability while strictly respecting predefined emission limits.

c) The proposed approach provides system operators with actionable guidance on how forecast uncertainty translates into specific generation adjustments, which can be implemented through economic dispatch or ancillary service mechanisms, thereby improving operational efficiency under increasing renewable penetration.

 

Reviewer 2 Report

Comments and Suggestions for Authors

Although the manuscript discusses a noteworthy and fascinating subject, it must be rewritten to comply with the traditional scientific writing standard. First of all, a comprehensive review of the state of the art must be included in the manuscript. The newest bibliographic reference present in the manuscript is from 2015. The state of the art must be constructed with references published within the last 5 years. The citation appears only at the end of page 3. All the hypotheses used must be clearly described and their foundations justified. It is not enough to add a table/figure in the manuscript and use it as a base determining, for example, heat rates, costs, etc., without the proper description of the conditions that were used to construct the table/figure, its validity, its limitations, the references, etc. The lack of precise information on the data used, assumptions, and constraints appears systematically throughout the manuscript, from section 3, through the optimization methodology, and even in the results. This type of problem makes it virtually impossible to reproduce the results obtained. The authors indicate the presence of sulfur in fuels, but do not consider its impact on emissions. In fact, the way they express emissions, considering the conversion of carbon entirely into CO2, is very simplistic. What would be the impact in terms of CO, NOx, SOx, particulates, etc.? Considering the preliminary description made by the authors, hydroelectric plants should have been included in the study.

Comments on the Quality of English Language

Authors should prefer the use of direct technical-scientific language. They should avoid the use of the first person (singular and plural). Eliminate coordination errors. There are not many, but they do exist.

Author Response

Comment 1: A comprehensive and up‑to‑date state of the art is missing.

Response 1:

 

The reviewer is correct in noting that earlier versions of the manuscript did not sufficiently emphasize recent literature.

In the revised manuscript, the Introduction has been substantially expanded to include a comprehensive and up-to-date review of studies published within the last five years, covering:

a) multi-objective OPF and economic–environmental dispatch,

b) renewable energy source uncertainty modeling,

c) scenario-based, stochastic, and data-driven dispatch methods,

d) hybrid AC/DC networks and flexibility-oriented dispatch.

We have substantially expanded the literature review, adding recent references (2019–2024) on economic‑emission dispatch, renewable uncertainty modeling, reserve allocation, and system flexibility. The review is now distributed across the Introduction and methodological motivation sections.

 

Comment 2: Hypotheses, data sources, and assumptions are insufficiently described, hindering reproducibility

Response 2:

This concern has been carefully addressed. The revised manuscript now explicitly states that heat‑rate curves of thermal units are derived from real operational measurements at different load levels (full, 75%, and 50% load). Fuel analyses are real and are used to derive cost and CO₂ emission curves, and the bibliographic sources have been added.

In fact, the authors believe that the paper is actually stronger in this respect because it uses real data taken from real power plant tests and real fuel analyses, whereas many papers do not dwell too much on justifying this aspect, because in them this is a secondary aspect when it only serves to support the method described, or because they directly take them from other published articles.

A comment has been added in relation to the assumption of complete carbon oxidation to CO₂: while acknowledging the typically low figure of the unburned carbon percentage, the authors encourage its use if such information is available, providing the mathematical expression included in the article. The authors lacked detailed information on this matter but include the expression for a power generation operator to facilitate its inclusion in the study.

Comment 3: Sulfur and other emissions are ignored.

Response 3:

 

We agree that a full emission inventory would be valuable. However, the focus of this paper is on CO₂, consistent with most EED literature and climate‑driven operational constraints.

The authors agree with this assessment, and the study would be more comprehensive by integrating all types of emissions. However, we did not want to lose focus or become diluted, given that the aim was to study the influence of wind runcertainty on multi-objective economic-environmental optimization. Since the largest volume of emissions from a power plant is CO2, given the significant global environmental concern about the greenhouse effect, and the historically substantial effort to enroll all countries in its regulation (from the Kyoto Protocol to the present day), CO2 seems to be the most representative emissions factor. Furthermore, it is an inherent emission of all thermal power plant, with SO2 and NOx emissions being significant depending on the type of plant (for example, a combined cycle plant typically has very little or none SO2 emissions, while a plant using low-quality coal may have them).

In fact, one of the authors' theses focused on the study of the three emissions CO2, NOx and SO2, apart from the production costs (see reference 53 of the paper), and the interference between them (influence of desulfuration in CO2 emissions) was also studied, as also suggested by the reviewer, and an article on this subject is in development, but this paper has a different approach.

Comment 4: Hydropower should have been included.

Response 4:

We acknowledge the reviewer’s observation regarding the potential inclusion of hydroelectric plants.

In the present study, hydropower generation was intentionally excluded in order to isolate and analyze the interaction between thermal generation and wind uncertainty under emission constraints. Hydropower introduces intertemporal coupling, reservoir dynamics, and water management constraints, which would significantly alter the problem formulation.

The revised manuscript now clarifies this modeling choice and explicitly states that extending the framework to include hydropower and other flexible resources (e.g., storage) constitutes a natural and relevant extension of the proposed methodology.

Comment 5: Overall manuscript revision

Response 5:

In summary, the manuscript has been thoroughly revised to comply with traditional scientific writing standards by:

a) updating and restructuring the literature review,

b) clearly stating hypotheses, assumptions, and constraints,

c) improving methodological transparency and reproducibility,

d) explicitly defining the scope and limitations of the study.

We believe these revisions have substantially strengthened the manuscript and addressed the reviewer’s concerns.

 

Reviewer 3 Report

Comments and Suggestions for Authors

I think this paper has important weaknesses, such as:

• There is no literature review on current papers in the specific topic, only some reference to fundamental works that are very old (such as Pareto, V. Cours d’économie politique; F. Rouge, 1896.). An overall literature review on current papers must be added. For example similar work is carried out in

1. F. Krommydas, C. N. Dikaiakos, G. P. Papaioannou and A. C. Stratigakos, “Flexibility study of the Greek power system using a stochastic programming approach for estimating reserve requirements,” in Electric Power Systems Research, vol. 213, 2022.

• The innovation and added value of the paper is not explained compared to similar works on the field. Please add this section.
• The language must be substantially improved, for example the following phrases must be corrected (and many others)

“have been gaining weight”

“they go in opposite directions”

• Substantially improve figures, Figure 16 for example is very bad quality
• Using data from 1988 is to outdated and thus the result lack validity

• Explain what are the parameters in Table 1

• It would be interesting to examine a scenario with wind energy having a cost (e.g. feed-in tariff)

• In line 494, the number of Figure is missing

Author Response

Comment 1: Lack of modern literature review and positioning.

Response 1: 

We fully agree in that comment. This point has been addressed by expanding the literature review and explicitly comparing our work with recent stochastic and flexibility‑oriented studies. We now clarify that our approach complements, rather than competes with, stochastic programming by offering a data‑driven, operator‑oriented analysis framework.

Comment 2: Innovation and added value are unclear.

Response 2:

A clear contribution subsection has been added. The novelty lies in the use of real wind deviation data, KDE‑based modeling across offer sizes, and the generation of interpretable multi‑objective operating maps useful for useful management of the thermal power plants, and for secondary/tertiary regulation studies.

Comment 3: Language and figure quality issues.

Response 3:

We acknowledge the reviewer’s comment regarding language quality.

The manuscript has been thoroughly revised for English clarity and scientific style, and informal or imprecise expressions (e.g., “have been gaining weight”, “they go in opposite directions”) have been corrected throughout the text. Additional language polishing has been performed to comply with standard scientific writing conventions.

We agree that figure quality is essential. All figures, including Figure 16, have been regenerated at higher resolution and replaced in the revised manuscript to ensure clear readability of axes, labels, legends, and data points

Comment 4: Use of outdated data (1988).

Response 4:

The heat-rate curves of some power plants were taken from the MINER publication of 1988, and also the fuels analysis may be quite old (but basically remain the same). However, they are real data, taken from operative tests and measurements of real power plants, which is a point not usually justified in OPF papers.

Also the wind data are all real, the forecasted values and the actual data of production the day ahead, and so the deviation data.

We clarify that 1988 MINER data were used to parameterize heat-rate behavior as a methodological baseline; for combined cycle, we rely on more recent plant data. The focus is on the dispatch methodology and uncertainty propagation rather than absolute year-specific costs. The KDE-based deviations originate from operational datasets, and scaling is explained.

Although older, the efficiency curves of subcritical Rankine cycle thermal power plants have not changed drastically over time, and many of them have continued to operate as is, except for those that have undergone significant repowering. In fact, the addition of exhaust gas purification systems may even have negatively impacted their efficiency.

On the other hand, this study does not aim to provide absolute or exact figures for operating points, but rather to illustrate how to manage a portfolio of thermal power plants in the face of wind uncertainty within the same system.

Comment 5: Explain Table 1 parameters and missing figure reference.

Response 5: 

Table 1 has been fully explained, and all figure numbering errors have been corrected.

Comment 6: Costs and scenario with wind cost (feed-in tariff):

Response 6:

Our base-case assumes zero marginal cost for wind. We note that the framework readily accommodates non-zero wind bids (e.g., feed-in tariffs) as an added cost term.

We believe it could be easily incorporated, including a sensitivity where wind carries a modest marginal cost or other wind-related strategies used in different systems, thus obtaining new Pareto fronts. However, this study assumed a zero cost, which is clearly stated as the initial hypothesis, and incorporating a wind cost would require exhaustively repeating thousands of simulated points. Nevertheless, we appreciate the comment, and we will take the idea into consideration as a factor in our future approaches to simulations and studies.

Comment 7: Minor editorial issue (missing figure number)

Response 7: We thank the reviewer for pointing out this oversight.

The missing figure reference has been corrected in the revised manuscript.

Final remark

In summary, the manuscript has been extensively revised to address all points raised by the reviewer, including updating the literature review, clarifying innovation and assumptions, improving language and figure quality, and correcting editorial issues. We believe these changes have significantly strengthened the manuscript.

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

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

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript has improved compared with the first version, and most of my suggestions were incorporated. These versions contain a stronger bibliographic foundation and justification of the methodology chosen.
Although the authors did not accept my suggestion of including the hydroelectric plants in their analyses, I can buy their justification, especially considering that the manuscript explicitly states that their analysis is limited to "thermal generation plants".

Reviewer 3 Report

Comments and Suggestions for Authors

No further comments