Optimal Sizing of Residential PV and Battery Systems Under Grid Export Constraints: An Estonian Case Study

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Comments:

This paper developed and evaluated three distinct sizing models for photovoltaic (PV) and battery (BAT) systems for Estonian households operating under grid constraints that prevent the sale of surplus energy. However, in order to further enhance the quality of the article and the depth of the research, the following points are suggested:

1. The basis for the values of the key parameters α (PV capacity factor) and β (battery capacity factor) in Model 1 is very weak. It is mentioned in the paper that α is set to “approximate the results of more complex models”, while β is directly quoted from the reference [15]. This approach lacks detailed theoretical support. Besides, the following related research can be compared: a: Two-Stage Coordinated Robust Planning of Multi-Energy Ship Microgrids Considering Thermal Inertia and Ship Navigation b: Extension of pole differential current based relaying for bipolar LCC HVDC lines
2. The paper reaches the compelling conclusion that “simplicity is better than complexity”, but the reasons behind this are not sufficiently explored. Why is a complex model based on full optimization (Model 3) inferior to a simple rule of thumb (Model 1)?
3. The optimization model of the paper should give more detailed objective function, constraints and network model.
4. The paper should refer to other literature and use an optimization method such as a solver to build a model to solve the problem instead of using an exhaustive approach.
5. The authors assign weights to the four MCDA indicators, explaining that their purpose is to “balance the scales” and “reflect importance”. However, this weighting process is inherently subjective. A sensitivity analysis of the MCDA weights, or some other methodology, is needed to determine the optimal weights.
6. As an “Estonian case study”, the generalizability of its findings needs to be carefully defined. The Estonian electricity price level, grid policy (especially the ban on electricity sales), light resources and household load characteristics together determine the results of this study.

Comments on the Quality of English Language

good

Author Response

Comment 1: The basis for the values of the key parameters α (PV capacity factor) and β (battery capacity factor) in Model 1 is very weak. It is mentioned in the paper that α is set to “approximate the results of more complex models”, while β is directly quoted from the reference [15]. This approach lacks detailed theoretical support. Besides, the following related research can be compared: a: Two-Stage Coordinated Robust Planning of Multi-Energy Ship Microgrids Considering Thermal Inertia and Ship Navigation b: Extension of pole differential current based relaying for bipolar LCC HVDC lines

Response: We thank the reviewer for this valuable feedback. We acknowledge that these simple heuristics are not derived from first principles but are intended to serve as pragmatic, data light starting points for preliminary assessments. We have revised the manuscript to provide a clearer justification for their calibration and validation.

For the PV sizing factor α, we have clarified the text in Section 3.1:

• "While based on a simple heuristic, the sizing factor was calibrated to ensure the model produces outputs in a realistic range compared to the more complex models, making it a pragmatic and accessible starting point for preliminary assessments."

For the battery sizing factor β, we have added:

• "This value, drawn from [15], was validated against our more complex models to ensure it represents a reasonable baseline for storing a fraction of the daily load (approximately six hours of average consumption)."

We also appreciate the literature suggestions. While the specific topics of the suggested papers are different from our residential focus, we agree on the importance of a strong theoretical foundation and have expanded our literature review to better contextualize our approach within existing research on sizing methodologies.

 

Comment 2: The paper reaches the compelling conclusion that "simplicity is better than complexity", but the reasons behind this are not sufficiently explored. Why is a complex model based on full optimization (Model 3) inferior to a simple rule of thumb (Model 1)?

Response: We thank the reviewer for highlighting this important point. We have expanded the discussion in the "Comparative Analysis of Baseline Results" section to provide a clearer explanation for this key finding. The core reason lies in the trade off between maximizing absolute profit and maintaining capital efficiency, which our MCDA framework is designed to capture.

We have added the following paragraph to the manuscript:

• "The conclusion that the simplest model outperforms the most complex one stems from the critical trade off between maximizing absolute returns and maintaining capital efficiency. Model 3, through its exhaustive optimization, successfully identifies configurations with the highest absolute NPV. However, achieving these returns requires substantial initial investments, which reduces overall investment efficiency (i.e., lower IRR, PIR, and longer ROI). In contrast, Model 1 recommends smaller, more affordable systems that deliver a better balance of returns relative to the capital invested. Our MCDA framework, which places significant weight on efficiency and rapid payback, consequently favors the more balanced and less capital intensive approach of Model 1."

Comment 3: The optimization model of the paper should give more detailed objective function, constraints and network model.

Response: We agree with the reviewer that this information is crucial for reproducibility. We have expanded Section 3.3 (Model 3) to include a detailed description of the objective function and the key operational constraints of the simulation framework.

The following details have been added to the manuscript:

• "The framework's objective is to minimize the net cost of electricity over the project lifetime, formulated as:
• 
   where G_tⁱⁿ is the energy drawn from the grid, p_t^buy is the hourly purchase price, and I is the total investment. This optimization is subject to several key operational constraints:
  - No grid export:  Energy sold to the grid, G_t^out, must be zero for all hours t.
  - Battery state of charge:  SOC_min ≤ SOC_t  ≤ SOC_max
  - Battery power limits: The charging and discharging power, P_ch,t and P_dis,t, cannot exceed the battery's maximum power ratings.
  - Hourly energy balance: The household's load L_t plus any energy used to charge the battery must be met by a combination of grid imports, PV generation (P_PV,t), and battery discharge: L_t + P_ch,t = G_tⁱⁿ + P_PV,t + P_dis,t."

 

Comment 4: The paper should refer to other literature and use an optimization method such as a solver to build a model to solve the problem instead of using an exhaustive approach.

Response: We thank the reviewer for this suggestion. We agree that for larger, more complex problems, using a dedicated optimization solver would be more efficient. However, for the scope of this study, the iterative search was computationally feasible and had the distinct advantage of guaranteeing a global optimum within the defined discrete search space. We have added a sentence to the manuscript to justify our choice and acknowledge its limitations.

The text now includes:

• "While advanced solvers could also be used, our use of a discrete, iterative search over a defined and manageable range of system sizes was computationally feasible for this study and guarantees finding the global optimum within the specified discrete search space."

We have also added this as a point for future work in the "Limitations and Future Work" section.

Comment 5: The authors assign weights to the four MCDA indicators, explaining that their purpose is to "balance the scales" and "reflect importance". However, this weighting process is inherently subjective. A sensitivity analysis of the MCDA weights, or some other methodology, is needed to determine the optimal weights.

Response: This is an excellent point. We fully agree that the choice of weights is subjective and can influence the final rankings. We have added a paragraph on "Weighting subjectivity" in Section 4.1 to explicitly acknowledge this.

The new paragraph states:

• "It is important to acknowledge that the choice of weights in the MCDA is inherently subjective and reflects a specific set of priorities. The weights used here represent a typical homeowner's preference for a quick payback and strong investment efficiency. A different set of priorities (e.g., maximizing long term absolute profit) would result in a different weighting scheme and could alter the final rankings. A formal sensitivity analysis of these weights or the use of methods like the Analytic Hierarchy Process (AHP) to derive them more objectively would be a valuable extension for future work but falls outside the scope of this study."

Comment 6: As an "Estonian case study", the generalizability of its findings needs to be carefully defined. The Estonian electricity price level, grid policy (especially the ban on electricity sales), light resources and household load characteristics together determine the results of this study.

Response: We thank the reviewer for this important comment. We have added a dedicated paragraph in the new "Limitations and Future Work" section to clearly define the context and boundaries for the generalizability of our findings.

The text now reads:

• "First, our findings are specific to the Estonian context, characterized by high retail electricity prices, no feed-in tariffs, and a northern European climate. The generalizability of our results to other markets will depend on the similarity of these conditions."

Reviewer 2 Report

Comments and Suggestions for Authors

1. The paper assumes a battery cost of €300/kWh and a PV cost of €600/kW. Please provide citations for these figures, ideally from Estonian for 2023. The cost of installation can be a significant portion of the total investment, should also be mentioned as being included in these figures.
2. While the Estonian case study is novel, the broader methods of PV and battery sizing are well established. The literature review could be expanded to a more current work within the existing body of research on sizing methodologies and MCDA applications in this field.
3. The optimization relies on brute-force search, which is computationally intensive and may not be scaled well.
4. Some methodological assumptions (e.g., parameter values in Models 1 and 2, MCDA weights) lack sufficient justification or sensitivity testing.

Author Response

Comment 1: Cost Citations Needed: The paper assumes a battery cost of €300/kWh and PV cost of €600/kW. Please: Provide citations for these figures (ideally Estonian sources for 2023). Clarify whether installation costs are included in these values, as they represent a significant portion of total investment.

Response: We thank the reviewer for pointing out this omission. We have updated Section 3.3 (Model 3) to include recent, local citations for the cost assumptions and to clarify that these are all inclusive turnkey costs.

The updated text is:

• "Investment cost assumptions: battery cost approximately 300 EUR/kWh and PV cost approximately 600 EUR/kW. These estimates are for turnkey systems, inclusive of all hardware and installation costs. The figures are based on recent 2023–2024 Estonian market data, including the Auvere utility scale battery project and reports from Eesti Energia and the Estonian Renewable Energy Chamber [21–23]."

Comment 2: Expand Literature Review: While the Estonian case study is novel, broaden the literature review to: Cover recent work on PV/battery sizing methodologies. Discuss current MCDA (Multi-Criteria Decision Analysis) applications in this field.

Response: We thank the reviewer for this suggestion. We have expanded the literature review in the Introduction to better situate our work. We have included additional references that cover modern sizing methodologies and recent applications of MCDA frameworks in the field of renewable energy systems.

The updated introduction now includes:

•  "Emerging research expands on these methods by incorporating multi criteria decision analysis (MCDA) frameworks. \citet{jiang2024} provide a comprehensive review of optimization models for battery sizing, highlighting hybrid mathematical programming and heuristic approaches. \citet{nematirad2023} proposed a statistical methodology for sizing systems to reduce peak demand using Monte Carlo simulations, emphasizing robustness. Importantly, \citet{sandelic2020} evaluated simplified sizing rules against comprehensive modeling, finding that heuristic based approaches can provide robust results with much lower data complexity, a finding that supports the direction of our study."

Comment 3: Optimization Method Limitation: The brute-force search approach is computationally intensive and scales poorly. Address this limitation or suggest alternatives for larger-scale applications.

Response: We thank the reviewer for this comment. We agree that the brute-force (iterative search) method has scalability limitations. We have addressed this by adding a justification for its use within the scope of our study and by acknowledging this limitation in the new "Limitations and Future Work" section.

In Section 3.3, we now state:

• "While advanced solvers could also be used, our use of a discrete, iterative search over a defined and manageable range of system sizes was computationally feasible for this study and guarantees finding the global optimum within the specified discrete search space."

Comment 4: Justify Assumptions: Key assumptions (e.g., parameter values in Models 1–2, MCDA weights) lack sufficient justification. Add: Rationale for chosen values. Sensitivity analysis to test robustness.

Response: We thank the reviewer for this feedback. We have revised the manuscript to provide stronger justification for our key assumptions.

Model 1 Parameters (Section 3.1): We clarified the calibration and validation of the heuristic parameters.

• For the PV sizing factor, we added:  "While based on a simple heuristic, the sizing factor was calibrated to ensure the model produces outputs in a realistic range compared to the more complex models, making it a pragmatic and accessible starting point for preliminary assessments."
• For the battery sizing factor, we added:  "This value, drawn from [15], was validated against our more complex models to ensure it represents a reasonable baseline for storing a fraction of the daily load (approximately six hours of average consumption)."

MCDA Weights (Section 4.1): We added a new paragraph on "Weighting subjectivity" to transparently discuss the rationale and its limitations:

• "It is important to acknowledge that the choice of weights in the MCDA is inherently subjective and reflects a specific set of priorities. The weights used here represent a typical homeowner's preference for a quick payback and strong investment efficiency. A different set of priorities (e.g., maximizing long term absolute profit) would result in a different weighting scheme and could alter the final rankings. A formal sensitivity analysis of these weights or the use of methods like the Analytic Hierarchy Process (AHP) to derive them more objectively would be a valuable extension for future work but falls outside the scope of this study."

 

Reviewer 3 Report

Comments and Suggestions for Authors

In the context of the energy transition and the increasing number of prosumers, the topic of optimal selection of PV and battery systems under restrictions on energy export to the grid is in high demand. In their article, the authors consider three approaches of varying complexity (rough heuristics, profiling, simulation modeling) and conduct a comparative analysis using multi-criteria assessment (MCDA). They provide results applicable not only to the Estonian context, but also to countries with similar infrastructure and restrictions on electricity export. The article is generally positive, but for me some issues remained unclear.

Remarks
1. There is no assessment of the uncertainty of the input data parameters: For example, consumption and solar radiation data are taken as average or typical. There is no analysis of confidence intervals or the impact of forecast errors on financial metrics.
2. The economics of batteries is not considered in sufficient depth: Although the authors provide battery costs and sensitivity analysis, there is no detailed comparison with alternative investments, as well as an assessment of the residual value at the end of the service life.
3. In the context of constantly changing energy policy (e.g. possible permission for partial export), no scenarios are presented that take these changes into account.
4. The paper does not calibrate or verify the simulation model on real data (e.g. comparing predicted and actual savings).
5. In my opinion, the justification for MCDA weights looks subjective: It would be useful to use the questionnaire method or the analytic hierarchy process method to more justify the choice of weights.
Recommendations
1. "operates" is written as "perates" (line 284) - a typo.
2. "Chigh = ct | ct > Q75(t)" - incorrect formatting: it should be clarified that this is a logical filtering, not an equation.
3. "sizing methodology matters" is repeated several times in the headings and text. It is recommended to diversify the vocabulary to improve readability.
4. “To do this we develop…” correct “To do this, we develop…”
5. “Combined PV+Battery systems offer a middle ground, balancing the stability of PV generation with the price-dependent nature of battery arbitrage.” Split into two sentences
6. Tables 2–4 are overloaded with data. It is worth highlighting the most significant indicators (e.g. in bold) and/or using color gradation for visualization.
7. Highly recommended. Expand the discussion on regulatory aspects and prospects. And also add a comparison with alternative investments (e.g. energy efficiency, heat pumps, etc.).

Comments on the Quality of English Language

Recommendations
1. "operates" is written as "perates" (line 284) - a typo.
2. "Chigh = ct | ct > Q75(t)" - incorrect formatting: it should be clarified that this is a logical filtering, not an equation.
3. "sizing methodology matters" is repeated several times in the headings and text. It is recommended to diversify the vocabulary to improve readability.
4. “To do this we develop…” correct “To do this, we develop…”
5. “Combined PV+Battery systems offer a middle ground, balancing the stability of PV generation with the price-dependent nature of battery arbitrage.” Split into two sentences
6. Tables 2–4 are overloaded with data. It is worth highlighting the most significant indicators (e.g. in bold) and/or using color gradation for visualization.
7. Highly recommended. Expand the discussion on regulatory aspects and prospects. And also add a comparison with alternative investments (e.g. energy efficiency, heat pumps, etc.).

Author Response

Comment 1: There is no assessment of the uncertainty of the input data parameters: For example, consumption and solar radiation data are taken as average or typical. There is no analysis of confidence intervals or the impact of forecast errors on financial metrics.

Response: We thank the reviewer for this important point. Our current analysis is deterministic, which is a limitation. We have now acknowledged this in the new "Limitations and Future Work" section and have proposed stochastic modeling as a key area for future research.

The new section includes the following point:

•  "Second, our analysis is deterministic, using historical data for consumption, irradiation, and prices. Future work could incorporate stochastic modeling to account for uncertainty in these variables and assess the impact of forecast errors on financial outcomes."

Comment 2: The economics of batteries is not considered in sufficient depth: Although the authors provide battery costs and sensitivity analysis, there is no detailed comparison with alternative investments, as well as an assessment of the residual value at the end of the service life.

Response: We agree that this would provide a more complete picture for homeowners. We have added this as a point for future research in the "Limitations and Future Work" section. We have also clarified in the manuscript that we conservatively assume a zero residual value for the battery.

The new section now states:

• "Third, the economic analysis of batteries could be expanded. A comparative analysis against alternative investments, such as energy efficiency measures (e.g., insulation, heat pumps), would provide a more holistic view for homeowners. We also acknowledge that our model assumes a zero residual value for the battery at the end of its service life, which is a conservative assumption."

Comment 3: In the context of constantly changing energy policy (e.g. possible permission for partial export), no scenarios are presented that take these changes into account.

Response: This is a very relevant point. Our study is based on the current regulatory framework. We have now explicitly mentioned the need to explore alternative policy scenarios as an area for future work.

The "Limitations and Future Work" section now includes:

•  "Finally, our study is based on the current regulatory framework in Estonia. Future research should explore alternative policy scenarios, such as the introduction of partial grid export allowances or feed-in tariffs, to assess how such changes would impact the economic viability of these systems."

Comment 4: The paper does not calibrate or verify the simulation model on real data (e.g. comparing predicted and actual savings).

Response: We agree that empirical validation is a critical step. We have acknowledged this limitation in the "Limitations and Future Work" section.

The text now includes:

•  "The simulation model was also not calibrated against real world performance data from specific installations, which remains an important step for future validation studies."

Comment 5: In my opinion, the justification for MCDA weights looks subjective: It would be useful to use the questionnaire method or the analytic hierarchy process method to more justify the choice of weights.

Response: We thank the reviewer for this constructive suggestion. We agree that the weights are subjective and have added a new paragraph ("Weighting subjectivity") in Section 4.1 to address this. 

The new paragraph :

• "It is important to acknowledge that the choice of weights in the MCDA is inherently subjective and reflects a specific set of priorities. The weights used here represent a typical homeowner's preference for a quick payback and strong investment efficiency. A different set of priorities (e.g., maximizing long term absolute profit) would result in a different weighting scheme and could alter the final rankings. A formal sensitivity analysis of these weights or the use of methods like the Analytic Hierarchy Process (AHP) to derive them more objectively would be a valuable extension for future work but falls outside the scope of this study."

Comment 6: Tables 2–4 are overloaded with data. It is worth highlighting the most significant indicators (e.g. in bold) and/or using color gradation for visualization.

Response: We thank the reviewer for this practical suggestion. We have revised Tables 2, 3, and 4 in the manuscript, using bold formatting to highlight the key performance indicators as recommended. 

Comment 7: Highly recommended. Expand the discussion on regulatory aspects and prospects. And also add a comparison with alternative investments (e.g. energy efficiency, heat pumps, etc.).

Response: We thank the reviewer for this recommendation to broaden the study's context. We have expanded the "Limitations and Future Work" section to address these points directly.

The new text includes the following additions:

• "Third, the economic analysis of batteries could be expanded. A comparative analysis against alternative investments, such as energy efficiency measures (e.g., insulation, heat pumps), would provide a more holistic view for homeowners..."
•  "Finally, our study is based on the current regulatory framework in Estonia. Future research should explore alternative policy scenarios, such as the introduction of partial grid export allowances or feed-in tariffs, to assess how such changes would impact the economic viability of these systems."

Recommendations 1-7 (Typos, Formatting, etc.): Thank you for these specific recommendations. We have addressed them as follows:
1.  Typo "perates": Corrected.
2. Formatting of `Chigh`: Rephrased in Section 3.2 to be a textual description.
3. Vocabulary for "sizing methodology matters": The point in the Conclusions has been revised for clarity.
4. Comma in "To do this, we...": Corrected.
5. Splitting long sentence: The sentence in Section 4.3 has been split for readability.
6. Table readability: We have revised Tables 2, 3, and 4 to bold the key indicators to improve readability.
7. Expand discussion: This has been addressed via the new "Limitations and Future Work" section.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Thanks for addressing my comments. 

Reviewer 3 Report

Comments and Suggestions for Authors

no