Beyond R2: The Role of Polynomial Degree in Modeling External Temperature and Its Impact on Heat-Pump Energy Demand
Chenjiyu Liang
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this study, the missing data point for outdoor temperatures was recovered by fitting, thus contributing to the performance analysis of heat pumps. The innovation should be further addressed and the manuscript should be carefully checked. The specific comments are as follows:
- The innovation of the study should be addressed: Please give a comprehensive review of previous related study on this topic, whether there are other researches regarding the fitting of outdoor temperatures to find the missed data point? And what are others’ methods?
- The outdoor temperatures are fitted, how about the outdoor humidity? Should it also be considered (since this parameter also influences the system performance a lot)?
- It is recommended to first present the overall mythology and present the temperature data you used as a case study to apply the methodology.
- Please explain the acronyms (such as MAE, RMSE) when they first come.
- Every variable should be explained under each equation or in Nomenclature. For example: What is 𝑡𝑎(𝜏) in Eq. (1)?
- Please check the typos through the whole manuscript, for example, in page 3: “95% % of the long-term value”.
- What is the difference between Tobs and Tref?
Author Response
Please see the attached files.
We thank the Reviewer for the constructive comments and valuable suggestions.
A detailed point-by-point response is provided in the attached file Response_to_Reviewer_1.docx.
The revised manuscript (ENERGIES_Manuscript_REV1_tracked_2025-10-10.docx) has Track Changes enabled and contains in-text comments [R1-x] that correspond to each reviewer remark.
We believe these revisions and clarifications have significantly improved the clarity and quality of the paper.
Author Response File: 
Author Response.docx
Reviewer 2 Report
Comments and Suggestions for Authors-The study does not compare against more advanced gap-filling methods (e.g., machine learning, spline interpolation, or time-series models), which undermines its innovative value.
-The use of a single meteorological dataset (Opole, Poland) and one ASHP model reduces the robustness and generalizability of the findings.
-Fitting polynomials to sorted temperature values (duration curve) ignores temporal dependencies, which are critical in climate data.
-The authors correctly note that high-degree polynomials may lead to overfitting, but the analysis does not deeply explore this trade-off.
-The study remains descriptive and does not provide deeper insights into why certain polynomial degrees perform better under specific conditions.
-Why was a global polynomial fit chosen over time-series-aware methods (e.g., ARIMA, splines, or ML-based imputation) for hourly data with temporal structure?
-How generalizable are the findings to regions with different climatic patterns (e.g., maritime, arid, or highly variable climates)?
-Was any sensitivity analysis performed on building parameters (e.g., insulation level, thermal mass) or heat pump types?
-How does the performance of polynomial gap-filling compare to simpler methods (e.g., linear interpolation) or more complex ones (e.g., kriging, neural networks)?
-No error propagation analysis from temperature uncertainty to energy indicators is provided.
-The writing style is often repetitive and could be condensed for clarity.
-Some acronyms are not defined at first use (e.g., LWT, WFE).
Author Response
Please see the attached files.
We thank the Reviewer for the constructive comments and valuable suggestions.
A detailed point-by-point response is provided in the attached file Response_to_Reviewer_2.docx.
The revised manuscript (ENERGIES_Manuscript_REV1_tracked_2025-10-10.docx) has Track Changes enabled and contains in-text comments [R2-x] that correspond to each reviewer remark.
We believe these revisions and clarifications have significantly improved the clarity and quality of the paper.
Author Response File: Author Response.docx
Reviewer 3 Report
Comments and Suggestions for AuthorsThis manuscript addresses an interesting and relevant topic: the reconstruction of missing temperature data using polynomial methods and the evaluation of their impact on energy calculations for heat pumps. The subject is timely and can significantly improve data reconstruction methods for building energy modeling and automation systems. However, in order to enhance the quality and readability of the paper, the following aspects should be addressed:
1) Language consistency: In Chapter 2, page 3, the terms “(1 – luka, 0 – obserwacja)” should be translated into English to ensure clarity for the international readership.
2) Description of evaluation indicators: The metrics used (RMSE, MAE, and R²) are not clearly explained. Each indicator should be concisely described in Chapter 2, ideally with a short paragraph for each, outlining what it measures and why it is relevant. Furthermore, after their description, the main objective of using these indicators should be explicitly stated, so that readers can fully understand the meaning of the graphs and results presented later in the paper.
3) Visualization of reconstructed data: Since the study focuses on polynomial-based reconstruction, the paper would strongly benefit from graphical illustrations of reconstructed data in specific gap periods. At least two figures (but more will be better) are recommended:
- One figure (temperature vs. time) showing reconstructions over a selected gap period where both observed and reconstructed data are available, to demonstrate the accuracy of the method.
- One figure (temperature vs. time) showing reconstructions for an actual missing data period, with overlapping sections at the start and end to illustrate how the polynomial fills the gap.
4) Graphical presentation of energy indicators: Similar to the comment above, it would be valuable to provide figures illustrating how different polynomial degrees affect key energy indicators (e.g., Qseason, COP, or Eel). While Tables 3 and 4 present this information, graphical representations over the full analysis period would make the results easier to interpret and more impactful.
5) Conclusions and outlook: The conclusions section should not only summarize the findings but also discuss potential applications of the proposed method. Additionally, directions for future research should be outlined, indicating how this type of data reconstruction could be further improved and integrated into mainstream energy modeling and heat pump performance analysis.
In summary, I encourage the authors to address the points raised above. Implementing these suggestions will improve the manuscript’s clarity, transparency, and practical relevance, thereby strengthening its suitability for publication.
Author Response
Please see the attached files.
We thank the Reviewer for the constructive comments and valuable suggestions.
A detailed point-by-point response is provided in the attached file Response_to_Reviewer_3.docx.
The revised manuscript (ENERGIES_Manuscript_REV1_tracked_2025-10-10.pdf) has Track Changes enabled and contains in-text comments [R3-x] that correspond to each reviewer remark.
We believe these revisions and clarifications have significantly improved the clarity and quality of the paper.
Author Response File: Author Response.docx
Reviewer 4 Report
Comments and Suggestions for AuthorsThe paper studies how the polynomial degree used to reconstruct hourly outdoor temperature from its duration curve affects ASHP energy indicators. The study is relevant and the workflow is clearly organized, but the following enhancements before publication.
COP and Nel are exponentially interpolated from catalog points, but the model description does not clarify defrost cycles, crankcase heater, auxiliary resistive heat, or capacity derating at frost. These factors are important since they dominate tail behavior and Nel,max.
The peak-power errors reported (~8–22% at 35 °C) are operationally significant for electrical sizing. Please specify whether defrost and bivalence were modeled dynamically or not. If not, add a sensitivity case with realistic defrost/aux heating to test robustness of the selection.
only one season is analyzed. To support generality, test a cold and a warm year Or acknowledge this limitation.
The acceptance and stability thresholds are declared a priori operational. Provide justification (e.g., utility planning tolerances, metering uncertainty) and complement with uncertainty evaluation.
State the computational time per n to back the “complexity” argument.
Consider adding a detailed figure of the pipeline that is already described in text with inputs/outputs and the four decision criteria.
Compare your duration-curve polynomial against other baselines.
Clarify in captions whether Δ values are absolute and RE values are relative.
State the computation time per degree.
The abstract is reported in bold. Please correct.
Figure 1 b is poorly readable. Please enhance.
Figures of results section need further discussion in the text. Please describe in more detail the figures in order to convey salient information to the reader.
Discussion section lacks comparison with other literature works and also misses limitations and future developments of the current paper. Could this approach be used to other similar problems that affect buildings and related systems such as the one addressed in: A modular hybrid SOC-estimation framework with a supervisor for battery management systems supporting renewable energy integration in smart buildings?
Conclusions are too shallow. please improve this section To provide a more clear outline of the conclusions of the study
Author Response
Please see the attached files.
We thank the Reviewer for the constructive comments and valuable suggestions.
A detailed point-by-point response is provided in the attached file Response_to_Reviewer_4.docx.
The revised manuscript (ENERGIES_Manuscript_REV1_tracked_2025-10-10.pdf) has Track Changes enabled and contains in-text comments [R4-x] that correspond to each reviewer remark.
We believe these revisions and clarifications have significantly improved the clarity and quality of the paper.
Author Response File: Author Response.docx
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsIn this study, the missing data point for outdoor temperatures was recovered through data fitting, thereby contributing to the performance analysis of heat pumps. The author has addressed the reviewers’ comments well, and the manuscript can be accepted for publication; however, some typos are still present. For example, “2.1 Method overvie” should be corrected to “2.1. Method overview”. Please conduct a thorough proofread of the manuscript.
Author Response
We sincerely thank the Reviewer for the positive evaluation and final remarks. Following your kind suggestion, we have again proofread the manuscript carefully and corrected all remaining typos and minor formatting inconsistencies (e.g., “Method overview” heading and punctuation).
We are very grateful for your attentive reading throughout both rounds of review — your detailed feedback significantly helped improve the clarity and precision of the paper.
The revised manuscript (ENERGIES_Manuscript_REV2_tracked_2025-10-16.pdf) has been uploaded with all final proofreading corrections applied.
Reviewer 2 Report
Comments and Suggestions for AuthorsIn future multi-climate validation, exploring the variation pattern of the "lowest sufficient degree " across different climate zones (e.g., maritime, continental) would enrich the generalizability of the conclusions , .
Author Response
We deeply appreciate the Reviewer’s thoughtful second-round feedback and encouraging recommendation.
Your comment about the need for broader, multi-climate validation was especially valuable — we have already initiated follow-up analyses applying the same methodology to datasets from Lithuania, Croatia, and Spain. This will directly extend the scope in future publications and help quantify the variation of the “lowest sufficient degree” across climate zones.
Thank you for recognizing the value and clarity of the present study and for guiding its evolution toward a wider framework.
The revised manuscript (ENERGIES_Manuscript_REV2_tracked_2025-10-16.pdf) has been uploaded with all final proofreading corrections applied.
Reviewer 3 Report
Comments and Suggestions for AuthorsI have seen that you implemented all my recommandations and the response. Thank you, your paper quality is a lot better.
Author Response
We warmly thank the Reviewer for confirming that all previous comments were satisfactorily addressed and for acknowledging the improved quality of the paper.
Your precise and constructive suggestions in the first round (especially regarding figures, indicator explanations, and clarity) have been instrumental in shaping the final manuscript, and we are very grateful for your engagement.
The revised manuscript (ENERGIES_Manuscript_REV2_tracked_2025-10-16.pdf) has been uploaded with all final proofreading corrections applied.
Reviewer 4 Report
Comments and Suggestions for AuthorsThank you for addressing my comments
Author Response
We sincerely thank the Reviewer for the constructive comments and technical insights provided in the first review round.
Your guidance regarding the treatment of defrost processes, energy-indicator presentation, and figure clarity helped us refine the model description and improve the overall readability of the manuscript.
We are glad that the revised version now meets your expectations, and we deeply appreciate your contribution to strengthening the scientific and editorial quality of this work.
The revised manuscript (ENERGIES_Manuscript_REV2_tracked_2025-10-16.pdf) has been uploaded with all final proofreading corrections applied.
