Forecasting Electricity Production, Consumption, and Price: Three Novel Fractional Grey Models of a Complex System

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The opening (Title/Abstract) states that “parameters… are estimated using least squares and time-response methods” and that “the optimal two-step prediction method is applied” (lines 21–27); however, the abstract does not define what this “two-step prediction method” is, nor which specific scientific problem it solves in the state of the art. Without this, the contribution remains vague. I recommend making explicit, still in the abstract, the central research question, the precise gap (why existing multivariate fractional models are insufficient), and the nature of the “two-step” (for example, fractional-order selection/optimization + multi-step forecasting), adding 1–2 sentences that clearly indicate how the proposed method overcomes known limitations.

The Introduction offers a solid contextualization of China’s power system and of the interdependence among generation, consumption, and price (lines 32–41 and 50–63), but it lacks a clear statement of the article’s objective and the specific gap it aims to fill. The text asserts that “there is a need to build a collaborative model that can simultaneously forecast generation, consumption, and price” (lines 56–59) without demonstrating why pre-existing multivariate fractional grey models fail to meet this need. I suggest closing the Introduction with a short, direct paragraph containing: (i) a measurable objective (e.g., “to propose and validate the HFEPCSGM(1,3) model family for simultaneous forecasting with dynamic coupling”), (ii) a specific gap (“the absence of fractional models that integrate r-HAGO with coupled differential equations for x–y–w triplets under short samples”), and (iii) a contribution statement in a bullet-like continuous sentence (within the paragraph), aligned with what is later claimed.

The Literature Review is broad but overly descriptive, with overlapping domains (generation, consumption, and price) and no conceptual map leading to the gap. The text lists methods (ARIMA, LSTM, grey models) but does not compare them critically or explain why prior multivariate fractional models (e.g., fractional GM(1,N) with lag) would be insufficient—an argument essential to support novelty. I recommend adding a final subsection (“Synthesis and gap”) to the Review, contrasting model categories by: (a) data requirements, (b) ability to handle multivariate coupling, (c) robustness to short/noisy series, and (d) computational cost; conclude with a concise comparative table (5–7 rows) that highlights the gap HFEPCSGM(1,3) addresses.

The theoretical grounding (principles of grey systems and the r-HAGO operator) appears scattered across definitions and theorems, without a brief “didactic bridge” that a non-expert reader can follow. Definitions 1–5 introduce heavy notation, but the motivation for choosing the Hausdorff operator specifically (rather than other forms of fractional accumulation) is not clearly justified for a broad audience. I suggest preceding Section 3.2 with a “for non-specialists” paragraph (6–8 lines) explaining, in plain language, (i) why fractional accumulation is used with short time series and (ii) why r-HAGO is computationally attractive (vs. the traditional 1-AGO).

Methodology/modeling: while formally consistent, it lacks details for replication. The modeling flow presents steps (Step1–Step7), but several implementation parameters are missing: the actual train–test split is handled with variables s and t (lines 408–414), but the values used are not reported; for PSO, swarm size, inertia, cognitive/social coefficients, number of iterations, random seed, and search bounds for the fractional orders are not reported (lines 417–423, 423–431), which prevents reproduction. I recommend adding an “Experimental setup” subsection stating: (a) the s/t used (e.g., 2005–2016 training, 2017–2020 test), (b) preprocessing (normalization, scale handling across generation/consumption/price), (c) full PSO configuration, (d) stopping criteria, and (e) hardware/time. In addition, Theorem 1 states that the proof is omitted because it is “commonly applied,” yet the matrix construction uses specific arrangements B₁–B₃; including a brief appendix with the derivation ensures replicability and avoids index ambiguities. Data and sources: the application section reports two official sources and the 2005–2020 window (lines 462–470), but the price unit is confusing (“The Average Pin Selling Electricity Price… per 10,000 yuan”; the table repeats “Pin Selling” and mixes capital letters), and it is unclear whether the price is nominal, real (deflated), or a weighted average tariff; this undermines the economic interpretation. I recommend correcting the unit caption, standardizing the nomenclature (“Average retail electricity price, CNY/kWh, deflated to 2020”), documenting the deflator if applicable, and adding a footnote with the conversion methodology. I also suggest discussing structural stability (relevant regulatory changes in the period) and conducting at least one regime-break test or a robustness analysis by subperiods.

Evaluation and metrics: relying solely on MAPE for fitting/validation (lines 405–416 and 423) is limiting, especially because MAPE asymmetrically penalizes errors near zero and does not allow scalar comparison across series with different magnitudes (generation vs. price). I suggest also reporting MAE, RMSE, and sMAPE/MASE, along with bootstrap confidence intervals for MAPE and a Diebold–Mariano test to compare HFEPCSGM(1,3) with baselines. Without baselines (ARIMA/ARIMAX, VAR/VARX, Prophet, simple LSTM), the method’s superiority remains unsubstantiated; including 2–3 reference models with default hyperparameters would strengthen the claim.

Results and discussion: the manuscript states that the models are “applied to forecast the next five years” (lines 458–466), but (in the available excerpt) there are no uncertainty bands and no sensitivity of forecasts to the fractional orders or to coefficients p, q, r. I suggest adding forecast plots with bands (e.g., 80%/95% CIs via resampling) and a local sensitivity analysis (vary-one-parameter) for orders α, β, γ and for cross-interactions, making explicit how coupling affects predictive error. Additionally, translate insights into practical implications (e.g., if price rises by X%, the model’s predicted response of consumption/generation is Y%), connecting the applied section to the public-policy motivation.

Author Response

   We thank the reviewer for the valuable comments on our paper. We have carefully revised the manuscript according to all the comments. The responses to specific reviewer comments are included below. The revised parts of the paper have been marked in blue.

Reviewer comments:

 

Reviewer 1

Comment 1: The opening (Title/Abstract) states that “parameters… are estimated using least squares and time-response methods” and that “the optimal two-step prediction method is applied” (lines 21–27); however, the abstract does not define what this “two-step prediction method” is, nor which specific scientific problem it solves in the state of the art. Without this, the contribution remains vague. I recommend making explicit, still in the abstract, the central research question, the precise gap (why existing multivariate fractional models are insufficient), and the nature of the “two-step” (for example, fractional-order selection/optimization + multi-step forecasting), adding 1–2 sentences that clearly indicate how the proposed method overcomes known limitations.

Response: Thank you for your valuable comments. The abstract has been revised as requested, clearly highlighting the core issues of this research, the shortcomings of the fractional-order model, and the limitations that need to be overcome. Thank you again for your valuable comments on the abstract.

 

Comment 2: The Introduction offers a solid contextualization of China’s power system and of the interdependence among generation, consumption, and price (lines 32–41 and 50–63), but it lacks a clear statement of the article’s objective and the specific gap it aims to fill. The text asserts that “there is a need to build a collaborative model that can simultaneously forecast generation, consumption, and price” (lines 56–59) without demonstrating why pre-existing multivariate fractional grey models fail to meet this need. I suggest closing the Introduction with a short, direct paragraph containing: (i) a measurable objective (e.g., “to propose and validate the HFEPCSGM(1,3) model family for simultaneous forecasting with dynamic coupling”), (ii) a specific gap (“the absence of fractional models that integrate r-HAGO with coupled differential equations for x–y–w triplets under short samples”), and (iii) a contribution statement in a bullet-like continuous sentence (within the paragraph), aligned with what is later claimed.

Response: Thank you for your valuable comments. In the revised manuscript, the part 2.3 of the original paper, including the motivation, contributions, and organization of the paper, has been moved forward to the introduction section, and this part has been modified according to your suggestions, as detailed in the revised manuscript.

 

Comment 3: The Literature Review is broad but overly descriptive, with overlapping domains (generation, consumption, and price) and no conceptual map leading to the gap. The text lists methods (ARIMA, LSTM, grey models) but does not compare them critically or explain why prior multivariate fractional models (e.g., fractional GM(1,N) with lag) would be insufficient—an argument essential to support novelty. I recommend adding a final subsection (“Synthesis and gap”) to the Review, contrasting model categories by: (a) data requirements, (b) ability to handle multivariate coupling, (c) robustness to short/noisy series, and (d) computational cost; conclude with a concise comparative table (5–7 rows) that highlights the gap HFEPCSGM(1,3) addresses.

Response: Thank you for your valuable comments. The literature review section in the revised manuscript has been expanded with a new paragraph. This addition primarily addresses the limitations of statistical models and intelligent prediction models, while also explaining the rationale for proposing the model presented in this paper. The revisions have been made based on your valuable feedback, with detailed references provided in the revised manuscript.

 

Comment4: The theoretical grounding (principles of grey systems and the r-HAGO operator) appears scattered across definitions and theorems, without a brief “didactic bridge” that a non-expert reader can follow. Definitions 1–5 introduce heavy notation, but the motivation for choosing the Hausdorff operator specifically (rather than other forms of fractional accumulation) is not clearly justified for a broad audience. I suggest preceding Section 3.2 with a “for non-specialists” paragraph (6–8 lines) explaining, in plain language, (i) why fractional accumulation is used with short time series and (ii) why r-HAGO is computationally attractive (vs. the traditional 1-AGO).

Response: Thank you for your valuable comments. Section 3.2 of the revised draft now includes an explanation in plain language. For details, please refer to the revised draft.

 

Comment 5: Methodology/modeling: while formally consistent, it lacks details for replication. The modeling flow presents steps (Step1–Step7), but several implementation parameters are missing: the actual train–test split is handled with variables s and t (lines 408–414), but the values used are not reported; for PSO, swarm size, inertia, cognitive/social coefficients, number of iterations, random seed, and search bounds for the fractional orders are not reported (lines 417–423, 423–431), which prevents reproduction. I recommend adding an “Experimental setup” subsection stating: (a) the s/t used (e.g., 2005–2016 training, 2017–2020 test), (b) preprocessing (normalization, scale handling across generation/consumption/price), (c) full PSO configuration, (d) stopping criteria, and (e) hardware/time. In addition, Theorem 1 states that the proof is omitted because it is “commonly applied,” yet the matrix construction uses specific arrangements B₁–B₃; including a brief appendix with the derivation ensures replicability and avoids index ambiguities. Data and sources: the application section reports two official sources and the 2005–2020 window (lines 462–470), but the price unit is confusing (“The Average Pin Selling Electricity Price… per 10,000 yuan”; the table repeats “Pin Selling” and mixes capital letters), and it is unclear whether the price is nominal, real (deflated), or a weighted average tariff; this undermines the economic interpretation. I recommend correcting the unit caption, standardizing the nomenclature (“Average retail electricity price, CNY/kWh, deflated to 2020”), documenting the deflator if applicable, and adding a footnote with the conversion methodology. I also suggest discussing structural stability (relevant regulatory changes in the period) and conducting at least one regime-break test or a robustness analysis by subperiods.

Response: Thank you for your valuable comments. The revised draft incorporates your suggestions by adding relevant content to the modeling steps. Data sources have been revised accordingly, and all relevant content in the tables has been updated. Please refer to the revised draft for details. It is specifically noted that matrix construction employs a particular arrangement B₁-B₃. Here, since the three matrix elements respectively represent data on electricity production, consumption, and price, distinct matrices B₁-B₃ have been defined. As these three matrices are relatively straightforward and no proof is required, the revised draft does not include a corresponding appendix. Regarding the stability analysis of the model structure for the established small-sample grey model, no corresponding theories or algorithms currently exist. This warrants further research in the future.

 

Comment 6: Evaluation and metrics: relying solely on MAPE for fitting/validation (lines 405–416 and 423) is limiting, especially because MAPE asymmetrically penalizes errors near zero and does not allow scalar comparison across series with different magnitudes (generation vs. price). I suggest also reporting MAE, RMSE, and sMAPE/MASE, along with bootstrap confidence intervals for MAPE and a Diebold–Mariano test to compare HFEPCSGM(1,3) with baselines. Without baselines (ARIMA/ARIMAX, VAR/VARX, Prophet, simple LSTM), the method’s superiority remains unsubstantiated; including 2–3 reference models with default hyperparameters would strengthen the claim.

Response: Thank you for your valuable comments. The revised manuscript incorporates MAE and RMSE metrics. Given the numerous tables—including those presenting MAPE for both simulated and forecasted indicators—Tables 5 through 11 in the revised version now include MAE and RMSE metrics.

Additionally, since grey models primarily address small-sample prediction methods, this study compares only four grey models to demonstrate their effectiveness. The data for the effectiveness analysis in this paper spans 16 years, from 2005 to 2020. Given the limited data volume, the sample size is insufficient for the ARIMA/ARIMAX, VAR/VARX, Prophet, and simple LSTM models. To maintain generality, the revised version does not include these methods in the comparative experiments.

 

Comment 7: Results and discussion: the manuscript states that the models are “applied to forecast the next five years” (lines 458–466), but (in the available excerpt) there are no uncertainty bands and no sensitivity of forecasts to the fractional orders or to the coefficients p, q, r. I suggest adding forecast plots with bands (e.g., 80%/95% CIs via resampling) and a local sensitivity analysis (vary-one-parameter) for orders α, β, γ, and for cross-interactions, making explicit how coupling affects predictive error. Additionally, translate insights into practical implications (e.g., if price rises by X%, the model’s predicted response of consumption/generation is Y%), connecting the applied section to the public-policy motivation.

Response: Thank you for your valuable feedback on the Conclusions and Discussion section. The revised manuscript now includes factors influencing prediction errors in the discussion section. Additionally, we have incorporated content linking indirect conversion to practical significance and public policy motivations. Please refer to the revised manuscript for details. Furthermore, the five-year forecast employs the three models from Section 4.2's two-step forecasting approach to predict electricity generation, consumption, and prices. The values of α, β, and γ were obtained via the PSO optimization algorithm, as shown in Table 12. These results were derived through MAPE simulation versus forecast and optimal local optimization. Consequently, other values of α, β, and γ cannot achieve the effectiveness demonstrated in Section 4.2 and should not be used to predict future five-year values based on certain fractional-order values. The purpose of Section 4.2 is to obtain optimal values for α, β, and γ to demonstrate the optimal validity analysis of the three models: HFEPCSGM(1,3)pro, HFEPCSGM(1,3)c, and HFEPCSGM(1,3)pri. Adjusting other values to determine the sum of MAPE is not particularly meaningful. Therefore, the revised version of this paper does not include banded prediction charts.

Reviewer 2 Report

Comments and Suggestions for Authors

suffers from multiple weaknesses in terms of scientific reasoning, methodology, and structure. The motivation and novelty of the study are not convincingly presented.

The authors merely claim to propose new models without clearly identifying the research gap or explaining how their approach improves on existing grey prediction methods.

The literature review is descriptive rather than analytical, lacking a critical discussion of previous works and any meaningful comparison between the proposed models and established approaches. The theoretical section is overloaded with formulas that are neither well explained nor supported by practical interpretation or application to real electricity system management. The data description is missing essential details such as the source, time span, and validation process.

The experimental section is incomplete and fails to provide comparative error metrics or statistical analysis to demonstrate the models’ predictive reliability. The models appear overly theoretical, with insufficient empirical verification.

Furthermore, the structure of the text is unclear, repetitive, and requires substantial English language editing.

The title promises “three novel models,” yet the claimed novelty is neither conceptually nor quantitatively justified.

Overall, the paper lacks a clear scientific contribution, the results are weakly validated, and the work does not offer significant advancement over existing methods.

Comments on the Quality of English Language

e.g., “electricity production and consumption behaviors has contributed” → should be “have contributed”)

Author Response

We thank the reviewers for the valuable comments on our paper. We have carefully revised the manuscript according to all the comments. The responses to specific reviewer comments are included below. The revised parts of the paper have been marked in blue.

Reviewer comments:

Reviewer #2:

Comment 1: suffers from multiple weaknesses in terms of scientific reasoning, methodology, and structure. The motivation and novelty of the study are not convincingly presented.

Response: Thank you very much for your critical suggestions on this paper. While it is relatively weak in scientific reasoning, methodology, and structure, its primary innovation lies in establishing three grey models through the key dynamics of the power generation, consumption, and pricing system. Simultaneously, the power-related issues examined represent a current research focus. The modeling employs fractional-order accumulation techniques to establish corresponding fractional-order models. Scientific reasoning and methodology adhere to classical approaches for parameter estimation in grey models and discrete solution methods. The primary motivation and novelty of this research lies in developing grey models grounded in the context of power systems, demonstrating innovation in grey modeling. The revised manuscript reinterprets the abstract, key innovations, and conclusions sections to present a fresh perspective on the revised work. We sincerely appreciate your feedback.

Comment 2: The authors merely claim to propose new models without clearly identifying the research gap or explaining how their approach improves on existing grey prediction methods.  

Response: Thank you for your valuable comments. This paper establishes a grey model based on the context of power systems. Currently, few grey models have been proposed specifically for power systems, and models addressing power generation, consumption, and pricing are virtually nonexistent. Therefore, developing a grey model from the power system perspective represents a significant extension of existing grey methodologies. Methodologically, the model is primarily solved using classical least-squares parameter estimation and discrete approaches, while incorporating cumulative fractional order to enhance prediction accuracy.

Comment 3: The literature review is descriptive rather than analytical, lacking a critical discussion of previous works and any meaningful comparison between the proposed models and established approaches. The theoretical section is overloaded with formulas that are neither well explained nor supported by practical interpretation or application to real electricity system management. The data description is missing essential details such as the source, time span, and validation process.

Response: Thank you for your valuable comments. The revised manuscript has incorporated a discussion of prior work in the final paragraph of the literature review section, while also clarifying the significance of the proposed model in this paper. Section 4.1 on data sources has been added to the data description section, and adjustments have been made to the theoretical formula section. For details, please refer to the revised manuscript.

Comment 4: The experimental section is incomplete and fails to provide comparative error metrics or statistical analysis to demonstrate the models’ predictive reliability. The models appear overly theoretical, with insufficient empirical verification.

Response: Thank you for your valuable feedback on the experimental section. The purpose of this study is to demonstrate the validity of three models: the HFEPCSGM(1,3) pro model, HFEPCSGM(1,3)c model, and HFEPCSGM(1,3)pri model. Given the small sample size, comparison models are primarily grey models. For instance, the total data in this experiment comprises 16 instances. Other statistical and machine learning methods may exhibit significant computational bias with only 16 data points; hence, grey models are typically used for small-data modeling comparisons. To compare error metrics, the revised manuscript incorporates MAE and RMSE indicators to assess predictive validity. For details, please refer to the revised manuscript. 

Comment 5: Furthermore, the structure of the text is unclear, repetitive, and requires substantial English language editing.

Response: Thank you for your valuable comments. The revised draft focuses primarily on restructuring the text and eliminating redundant content. Additionally, professional language services have been engaged to address any English language issues. For details, please refer to the revised draft.

Comment 6: The title promises “three novel models,” yet the claimed novelty is neither conceptually nor quantitatively justified.

Response: Thank you for your valuable comments. The three models in the title—HFEPCSGM(1,3) pro model, HFEPCSGM(1,3)c model, and HFEPCSGM(1,3)pri model. This paper proposes these three models, utilizes grey model parameter estimation and solution methods to derive the modeling steps, and finally applies the models to forecasting electricity production, consumption, and prices. Therefore, the novelty primarily lies in establishing the models, while the proof methods for the model theorems are consistent with those of classical grey models.

 Comment 7: Overall, the paper lacks a clear scientific contribution, the results are weakly validated, and the work does not offer significant advancement over existing methods.

Response: Thank you for your feedback on the paper. While it is true that this work has some weaknesses in terms of scientific contribution and structure, the establishment of three grey models using the power system as a background represents a current extension method for grey models and holds certain significance.

 Comment 8: Comments on the Quality of English Language e.g., “electricity production and consumption behaviors has contributed” → should be “have contributed”)

Response: Thank you for your valuable comments. The revised draft has been edited as thoroughly as possible in terms of language. For details, please refer to the revised draft.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors addressed the points for revision in a thorough and concise manner.

Author Response

Thank you for your valuable comments. Thanks again for recognizing and supporting my work!

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have substantially revised the manuscript in line with the reviewers’ comments.

However, some issues remain only partially addressed.

The claimed novelty of the three proposed grey models (HFEPCSGM(1,3)pro, c, and pri) is described mainly in terms of formulation, without sufficiently demonstrating conceptual or quantitative advancement over existing fractional grey models.

Similarly, the discussion of the scientific contribution and validation of results is somewhat superficial and could better emphasize the broader methodological significance and potential applications.

Nevertheless, the justification of the models’ originality and the depth of the discussion on scientific contribution could still be strengthened before final acceptance.

 

 

Comments on the Quality of English Language

The English language has been significantly improved compared with the previous version. Grammar, syntax, and overall readability have been corrected, and the manuscript is now generally clear and coherent. However, several expressions still sound slightly unnatural (e.g., “represents a current extension method” or “theoretical formula section”), and some sentences remain overly complex. Minor polishing by a professional language editor is recommended before final publication

Author Response

Comment 1: The authors have substantially revised the manuscript in line with the reviewers’ comments. However, some issues remain only partially addressed.

The claimed novelty of the three proposed grey models (HFEPCSGM(1,3)pro, c, and pri) is described mainly in terms of formulation, without sufficiently demonstrating conceptual or quantitative advancement over existing fractional grey models.

Similarly, the discussion of the scientific contribution and validation of results is somewhat superficial and could better emphasize the broader methodological significance and potential applications.

Nevertheless, the justification of the models’ originality and the depth of the discussion on scientific contribution could still be strengthened before final acceptance.

 Response: Thank you for your recognition and critical comments on the paper. The main innovation of this study lies in establishing three gray prediction models (HFEPCSGM(1,3)pro, c, and pri) based on the modeling mechanism of gray prediction models and the chaotic models of electricity production, consumption, and prices. These three models are respectively applicable to the prediction of electricity production, electricity consumption, and electricity prices. Meanwhile, the same time-series data are used for both simulation and prediction to verify the effectiveness of the models, and the production, consumption, and prices for the same time period can be effectively predicted.  

   Simultaneously—this is also an advantage of establishing the three models through a systematic approach. Indeed, this paper adopts the existing concept of fractional accumulation for the fractional-order definition. Since the core focus of this study is to develop the three models from the perspective of the power system, and the adopted fractional accumulation only serves as an optimization, the discussion on the fractional-order concept is indeed superficial. We sincerely appreciate your critical feedback. In future research, we will conduct further in-depth studies on the concepts and theories of fractional-order gray prediction models, such as exploring gray prediction models using the definitions and properties of fractional derivatives and the stability of fractional-order systems. Admittedly, it is quite challenging for us to conduct a more in-depth discussion on fractional-order theories within a short time frame. Therefore, no modifications have been made to the model in the revised manuscript, and an in-depth discussion cannot be provided. However, your suggestions will serve as a direction for our future related research. Thank you again for your valuable comments.

 

Comments on the Quality of English Language

Comment 2: The English language has been significantly improved compared with the previous version. Grammar, syntax, and overall readability have been corrected, and the manuscript is now generally clear and coherent. However, several expressions still sound slightly unnatural (e.g., “represents a current extension method” or “theoretical formula section”), and some sentences remain overly complex. Minor polishing by a professional language editor is recommended before final publication

 Response: The paper has undergone professional language editing, and the revised manuscript is provided in detail.

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Accept in present form