Optimal Electric Bus Charging Scheduling with Multiple Vehicle and Charger Types Considering Compatibility

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The study introduces a novel joint optimization approach that integrates charging infrastructure, fleet composition, and charging scheduling while considering vehicle-charger compatibility.  Below, I provide detailed feedback on the strengths of your work, as well as suggestions for improvement.

Strengths:
The paper presents a unique approach by integrating charging infrastructure planning, fleet composition, and charging scheduling.

• The MIP model and CG algorithm are robust and validated through comparison with CPLEX.
• The real-world case study in Nanjing, China, significantly strengthens the practical applicability of the research.
• The findings provide useful recommendations for transit agencies, particularly regarding heterogeneous fleets and vehicle-charger compatibility.

Areas for Minor Improvement:

• The paper lacks a discussion on the time complexity of the CG algorithm. Please provide a brief analysis of time complexity and computational scalability, especially for large-scale implementations.
• The study assumes constant energy consumption rates, which may not reflect real-world variability. Consider adding a short discussion on stochastic modeling techniques (e.g., Monte Carlo simulations, robust optimization).
• Some chargers in Table 9 are underutilized. While the paper discusses vehicle-charger compatibility optimization, please clarify why certain chargers remain less used and whether further optimization could improve utilization.
• The study does not discuss the impact of large-scale BEB charging on the power grid. A brief mention of demand-side management strategies (e.g., smart charging, load balancing) would improve the real-world applicability.
• Some mathematical notations (e.g., binary decision variables, charging decision variables) are dense. Consider adding a notation table to improve readability.

Author Response

Reviewer#1

 

General comments

 

1. The study introduces a novel joint optimization approach that integrates charging infrastructure, fleet composition, and charging scheduling while considering vehicle-charger compatibility.  Below, I provide detailed feedback on the strengths of your work, as well as suggestions for improvement.

 

Thank you for your valuable feedback and constructive suggestions. We sincerely appreciate your recognition of our work. Based on your suggestions, we have carefully revised the manuscript to enhance its clarity and rigor. Below, we provide detailed responses to each of your comments and outline the specific modifications made in the paper.

 

Specific comments

 

1. Strengths: The paper presents a unique approach by integrating charging infrastructure planning, fleet composition, and charging scheduling. The MIP model and CG algorithm are robust and validated through comparison with CPLEX.

 

Thank you for your recognition of our model and methodology. We sincerely appreciate your positive feedback on our work, which encourages us and reinforces the significance of our approach.

 

2. The real-world case study in Nanjing, China, significantly strengthens the practical applicability of the research. The findings provide useful recommendations for transit agencies, particularly regarding heterogeneous fleets and vehicle-charger compatibility.

 

Thank you for your recognition of the significance and practical implications of our research. We sincerely appreciate your positive feedback on our work, which encourages us and reinforces the significance of our approach.

 

3. 3. Areas for Minor Improvement: The paper lacks a discussion on the time complexity of the CG algorithm. Please provide a brief analysis of time complexity and computational scalability, especially for large-scale implementations.

 

Thank you for your valuable suggestions. We appreciate your insightful comment regarding the time complexity and computational scalability of the CG algorithm.

To address this, we have compared the optimization results and computational time of CG with the traditional commercial solver CPLEX. The results demonstrate that CG not only achieves competitive optimization performance but also significantly reduces computational time, especially as the problem scale increases. For instance, when the problem size involves 12 vehicles, the CG algorithm obtains a similar solution to CPLEX in just 17 minutes, whereas CPLEX requires 360 minutes. Furthermore, when the number of vehicles increases to 37, the CG algorithm takes 58.16 minutes to achieve a solution of 43110.85, while CPLEX requires 360 minutes to reach a suboptimal result of 54859.1.

This comparison highlights that by decomposing the optimization model into a master problem and multiple subproblems, CG effectively reduces computational complexity and improves efficiency. As problem size increases, while the overall computation time also increases, the advantage of the CG algorithm over CPLEX becomes even more pronounced. In our experiments, constrained by data availability, we conducted tests on 343 service trips. However, based on the observed trends in the results, CG remains effective for larger-scale problems, maintaining a good balance between computational efficiency and solution quality. We have provided additional explanations on the time complexity and computational scalability of the CG algorithm on page 14, lines 344-355.

 

“CPLEX achieves better results when the problem scale is small, specifically when the number of vehicles does not exceed 12. However, as the number of vehicles increases, the CG algorithm consistently yields superior results. In terms of computational time, the CG algorithm also demonstrates a clear advantage by obtaining high-quality solutions more efficiently. For instance, when the problem involves 37 vehicles, the CG algorithm takes only 58.16 minutes to achieve a solution of 43110.85, whereas CPLEX requires 360 minutes to obtain a suboptimal result of 54859.1. This comparison highlights that the CG algorithm not only enhances solution quality but also significantly reduces computational time, particularly for large-scale BEB charging scheduling problems. As the problem size increases, while the total solving time naturally rises, the gap in performance between CG and CPLEX becomes even more pronounced, further reinforcing the scalability and practicality of the CG approach.”

 

4. 4. The study assumes constant energy consumption rates, which may not reflect real-world variability. Consider adding a short discussion on stochastic modeling techniques (e.g., Monte Carlo simulations, robust optimization).

 

Thank you for your insightful suggestions. The energy consumption rate of BEBs is influenced by various factors, such as vehicle acceleration and deceleration, air conditioning usage, and road conditions. These factors introduce significant variability and complexity into the BEB scheduling process, making precise modeling a challenging task.

To maintain computational tractability, many existing studies [5,11,30] assume a constant energy consumption rate when optimizing electric bus scheduling. Similarly, our study focuses on the joint optimization of charging infrastructure, fleet composition, and charging scheduling while considering multiple types of vehicles and chargers, as well as vehicle-charger compatibility. Given the inherent complexity of this problem, incorporating a stochastic energy consumption model would further increase computational burden and problem complexity.

To extend this research, we have added a discussion on the potential incorporation of stochastic energy consumption models, including the applicability of Monte Carlo simulations and robust optimization techniques. This discussion has been included to highlight future research directions and the possible enhancements of the proposed approach.

 

“In future work, this study can be extended in the following directions: (i) The charging scheduling can be combined with a vehicle scheduling scheme, in which the service trips are allocated to different vehicles. This can make full use of different types of vehicles. (ii) This study assumes that the energy consumption rates of vehicles are constant. However, in real-world scenarios, energy consumption is influenced by multiple stochastic factors, such as variations in passenger load, road gradient, and environmental conditions. Considering these uncertainties, developing a robust electric bus scheduling strategy would be a valuable extension. Future research could explore stochastic modeling approaches, such as Monte Carlo simulations and robust optimization techniques, to account for these variations and enhance the adaptability of the scheduling model.”

 

5. 5. The study assumes constant energy consumption rates, which may not reflect real-world variability. Some chargers in Table 9 are underutilized. While the paper discusses vehicle-charger compatibility optimization, please clarify why certain chargers remain less used and whether further optimization could improve utilization.

 

Thank you for your insightful comments. We have provided additional explanations on page 15, lines 365-372. The charger utilization rate is calculated as the ratio of chargers’ utilization time to its available time. Since our study focuses on optimizing electric bus charging scheduling while considering fixed vehicle routes, buses can only charge when they are at the depot. During the available time, buses are actively operating on their assigned service trips, limiting the overall utilization of the chargers.

Despite this constraint, our results demonstrate that many chargers optimized by the CG algorithm achieve utilization rates exceeding 50%, indicating a relatively high level of efficiency. However, we acknowledge that some chargers, such as Charger 1 with a utilization rate of 25.28%, are underutilized. This occurs because, at certain times, vehicles require immediate energy replenishment to meet operational demands, and other chargers may already be occupied, necessitating the availability of additional chargers. Conversely, during other charging intervals, the total number of chargers may exceed the immediate demand, resulting in lower utilization for some chargers.

In general, reducing the number of chargers in the scheduling plan can lead to higher individual charger utilization rates. However, this must be balanced with vehicle scheduling constraints to ensure operational feasibility. A potential way to further enhance utilization would be to adopt a coordinated optimization approach that integrates vehicle routing with charging scheduling. This aspect has been discussed in the conclusion section as a possible future research direction.

 

“Furthermore, the occupancy rate of chargers (utilization time over available time) is higher in CG. The occupancy rate of each charger, shown in Table 9, indicates that CG significantly outperforms CPLEX. The results demonstrate that many chargers optimized by the CG algorithm achieve utilization rates exceeding 50%, indicating a relatively high level of efficiency. However, we acknowledge that some chargers, such as Charger 1 with a utilization rate of 25.28%, are underutilized. This occurs because, at certain times, vehicles require immediate energy replenishment to meet operational demands, and other chargers may already be occupied, necessitating the availability of additional chargers.”

 

6. 6. The study does not discuss the impact of large-scale BEB charging on the power grid. A brief mention of demand-side management strategies (e.g., smart charging, load balancing) would improve the real-world applicability.

 

Thank you for your valuable suggestions. We have supplemented our discussion by incorporating demand-side management strategies on page 21, lines 462-475, including smart charging and load balancing, which can help mitigate grid impact by optimizing charging schedules and reducing peak loads. These discussions have been added to improve the real-world applicability of our study.

 

“Based on the above results, this paper can provide the following policy implications about BEB charging scheduling and charging infrastructure planning. Firstly, an electric bus network is composed of various elements such as BEBs, and charging infrastructure, which is featured as a typical complex system. Therefore, operators need to jointly optimize electric bus charging infrastructure, fleet composition, and charging schedule to maximize the benefits. In practice, considering the coexistence of multiple types of vehicles and chargers, compatibility is also an important factor that should be considered. Additionally, incorporating demand-side management strategies, such as smart charging and load balancing, can help mitigate the impact of large-scale BEB charging on the power grid. By dynamically adjusting charging schedules based on real-time electricity demand and grid capacity, operators can reduce peak loads and enhance grid stability. Moreover, integrating ToU electricity pricing with a partial charging strategy offers an effective approach to reducing charging costs while ensuring sufficient battery levels for scheduled operations.”

 

7. 7. Some mathematical notations (e.g., binary decision variables, charging decision variables) are dense. Consider adding a notation table to improve readability.

 

Thank you for your valuable suggestions. To improve readability, we have added a notation table on page 6, lines 205-206, which includes sets, parameters, and variables used in the proposed model. We hope this addition enhances the clarity and accessibility of our mathematical formulations.

 

“Table 2. Sets, parameters, and variables in the proposed model.

Sets	Description	Sets	Description
	Set of charger types,		Set of vehicle types,
	Set of vehicles,		Set of time intervals,
	Set of trips serviced by vehicle k,		Set of chargers that match the vehicle type
Parameters	Description	Parameters	Description
	Daily purchase cost of charger type , CNY		Daily vehicle costs of type , CNY
	Charging amount of bus k at time t, kWh		Sets
	Length of a unit time interval, min		Electricity price at time t, CNY/ kWh
	The nth trip serviced by vehicle k		Departure time of trip
	Arrival time of trip		The first available charging time after trip
	The last available charging time before the next trip , or 1440 for the last trip		Electricity amount when vehicle k departures from the original stop of trip
	Electricity amount when vehicle k arrives at the destination stop of trip		Battery capacity of vehicle type u
	The last interval in the research period		Energy consumption rate of vehicle type , kWh/km
	Length of trip, km		Operation time of trip, minute
	Minimum SoC that vehicles need to satisfy during scheduling processes		Maximum SoC that vehicles need to satisfy during scheduling processes
	Parameter denotes that if the bus is located at the charging station after trip,; otherwise,
Variables	Description	Decision Variables	Description
	Total costs of the electric bus systems, CNY		Number of chargers of type
	Purchase costs of chargers, CNY		Binary variable where  if the type of bus k is u and  if not
	Electric bus fleet composition costs, CNY		Binary variable where  if bus k is charged at time t by a charger of type s and  if not
	Electricity costs, CNY		Binary auxiliary variable, represents the linearization of

”

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper proposes a mixed integer programming model to jointly optimize electric bus (BEB) charging infrastructure, fleet composition, and charging schedules, incorporating partial charging, charging continuity, and vehicle-charger compatibility. Overall, it is well prepared and provides a concrete contribution to the related topic. But there are still some places that need further revisions before qualifying for this journal publication. The detailed comments are given as follows.

1. In some cases, the use of abbreviations could be more consistent. For example, "BEB" is used frequently, but in a few places, "battery electric bus" is written out in full. Please check the entire manuscript to ensure that abbreviations are used consistently from the beginning to the end of the paper.
2. Some figures lack detailed captions. For example, Figure 5 shows BEB fleet composition and vehicle operation schedule, but the caption does not explain what the different elements in the figure represent in detail.
3. The comprehensiveness and depth of the literature review on charging scheduling technologies could be improved. For instance, A joint model of infrastructure planning and smart charging strategies for shared electric vehicles; An advantageous charging/discharging scheduling of electric vehicles in a PV energy enhanced power distribution grid.
4. There is a lack of step-by-step explanation for some complex equations. For example, the linearization equations (19 - 22) are presented without sufficient explanation of how they are derived and why they are necessary.
5. The CG algorithm (Figure 3) lacks implementation details (e.g., initialization, subproblem solving). The case study focuses on two lines in Nanjing; scalability to larger networks is unaddressed.
6. The analysis omits critical parameters (e.g., variations in trip demand, charger downtime). Please expand sensitivity tests to include stochastic factors (e.g., weather impacts on energy consumption).
7. The novelty could be further emphasized. The paper could do more to highlight how its contributions are distinct from existing studies, especially in terms of the practical applications and potential impact on the electric bus industry.

Author Response

Reviewer#2

 

General comments

 

1. The paper proposes a mixed integer programming model to jointly optimize electric bus (BEB) charging infrastructure, fleet composition, and charging schedules, incorporating partial charging, charging continuity, and vehicle-charger compatibility. Overall, it is well prepared and provides a concrete contribution to the related topic. But there are still some places that need further revisions before qualifying for this journal publication. The detailed comments are given as follows.

 

Thanks for the comments. We have made modifications in the paper.

 

Specific comments

 

1. In some cases, the use of abbreviations could be more consistent. For example, "BEB" is used frequently, but in a few places, "battery electric bus" is written out in full. Please check the entire manuscript to ensure that abbreviations are used consistently from the beginning to the end of the paper.

 

Thank you for your valuable suggestion. We have carefully revised the manuscript to ensure that the abbreviation "BEB" is used consistently throughout the paper. Additionally, we have conducted a thorough review of the entire manuscript to verify the uniformity of all abbreviations. We appreciate your constructive feedback.

 

2. Some figures lack detailed captions. For example, Figure 5 shows BEB fleet composition and vehicle operation schedule, but the caption does not explain what the different elements in the figure represent in detail.

 

Thank you for your valuable suggestions. BEB fleet composition and vehicle operation schedule are shown in Figure 5.The x-axis represents the operational time, while the y-axis corresponds to the vehicle ID. The rectangular blocks indicate the operational periods of each vehicle. For instance, a block spanning 8:00–8:56 signifies that the vehicle is executing a scheduled service trip within this time window. Different colors represent different vehicle types: light blue corresponds to Type A, green to Type B, and gray to Type C. We have carefully reviewed all figures to ensure clarity and completeness. Thank you again for your constructive feedback.

 

“The numerical results highlight the effectiveness of the proposed model and the adopted strategy to achieve the optimal configuration for BEB in transit operation. BEB fleet composition and vehicle operation schedule are shown in Figure 5.”

 

3. 3. The comprehensiveness and depth of the literature review on charging scheduling technologies could be improved. For instance, A joint model of infrastructure planning and smart charging strategies for shared electric vehicles; An advantageous charging/discharging scheduling of electric vehicles in a PV energy enhanced power distribution grid.

 

Thank you for your valuable suggestions. We have incorporated additional relevant studies, particularly those related to emerging charging scheduling technologies, to provide a more comprehensive overview. These additions can be found on page 3, lines 121-131. We sincerely appreciate your insightful feedback, which has helped improve the completeness of our literature review.

 

“Several emerging charging scheduling technologies for electric vehicles (EVs) have also been explored by researchers. Liu et al. [26] proposed a data-driven model for optimizing shared EVs infrastructure deployment and operation, incorporating TOU tariffs and Vehicle-to-Grid (V2G) technology. The study found that implementing TOU and V2G strategies can reduce costs by 17.93% and 34.97%, respectively. Das et al. [27] proposed a two-stage charging and discharging scheduling model with intelligent EV routing. Case study results demonstrated that the proposed model effectively improves peak-to-average ratios (PARs) to 1.151–1.196, surpassing the base case PAR of 1.2. Liu et al. [28] proposed a multi-objective EV charging scheduling model to minimize grid peak-valley load difference and EV charging costs. The results showed the model can reduce grid impact and costs while highlighting the influence of EV users’ charging behavior.”

 

“26. Liu, J., Yang, X., Zhuge, C. A joint model of infrastructure planning and smart charging strategies for shared electric vehicles. Green Energy and Intelligent Transportation 2024, 3(4), 100168.

27. Das, P., Kayal, P. An advantageous charging/discharging scheduling of electric vehicles in a PV energy enhanced power distribution grid. Green Energy and Intelligent Transportation 2024, 3(2), 100170.
28. Liu, L., Zhou, K. Electric vehicle charging scheduling considering urgent demand under different charging modes. Energy 2022, 249, 123714.”

 

4. 4. There is a lack of step-by-step explanation for some complex equations. For example, the linearization equations (19 - 22) are presented without sufficient explanation of how they are derived and why they are necessary.

 

Thank you for your insightful comments.

Constraint (19)  ensures that vehicles can only be charged once per charging interval to minimize battery degradation. The binary auxiliary variable  is introduced to linearize. This constraint is designed to enforce charging continuity, preventing scenarios where a vehicle undergoes multiple charging activities within the same interval (e.g., an on-off charging pattern like 10101), as illustrated in Figure 2.

However, since Constraint (19) inherently involves a quadratic term, it is nonlinear and requires further linearization. Constraints (20)–(22) are commonly used techniques to linearize quadratic terms, ensuring the model remains computationally efficient and solvable.

We have added further clarifications regarding the derivation and necessity of these equations in the revised manuscript on page 9, lines 259-266.

 

“Constraint (19) represents that vehicles can only be charged once per charging interval to reduce battery loss. This constraint is designed to enforce charging continuity, preventing scenarios where a vehicle undergoes multiple charging activities within the same interval (e.g., an on-off charging pattern like 10101), as illustrated in Figure 2. However, since Constraint (19) inherently involves a quadratic term, it is nonlinear and requires further linearization; Constraints (20)-(22) represent the linearization of , which are commonly used techniques to linearize quadratic terms, ensuring the model remains computationally efficient and solvable;”

 

5. 5. The CG algorithm (Figure 3) lacks implementation details (e.g., initialization, subproblem solving). The case study focuses on two lines in Nanjing; scalability to larger networks is unaddressed.

 

Thank you for your insightful comments.

To improve the clarity of the CG algorithm (Figure 3), we have supplemented the manuscript with additional implementation details. The initial solution is generated by constructing a set of infeasible solutions where vehicles do not charge but incur high costs. This serves as a baseline for the column generation process. Each vehicle subproblem is formulated independently, incorporating vehicle type and charging plan while ensuring energy demand constraints are satisfied. Given the structure of the subproblem, it can be efficiently solved using CPLEX, as it primarily involves determining the optimal charging schedule for a single vehicle within the given constraints. These details have been added to the revised manuscript on page 9, lines 281-285.

For the scalability of the CG algorithm for large-scale networks. To address this, we have compared the optimization results and computational time of CG with the traditional commercial solver CPLEX. The results demonstrate that CG not only achieves competitive optimization performance but also significantly reduces computational time, especially as the problem scale increases. For instance, when the problem size involves 12 vehicles, the CG algorithm obtains a similar solution to CPLEX in just 17 minutes, whereas CPLEX requires 360 minutes. Furthermore, when the number of vehicles increases to 37, the CG algorithm takes 58.16 minutes to achieve a solution of 43110.85, while CPLEX requires 360 minutes to reach a suboptimal result of 54859.1.

This comparison highlights that by decomposing the optimization model into a master problem and multiple subproblems, CG effectively reduces computational complexity and improves efficiency. More importantly, as the problem size increases, CG’s efficiency advantage over CPLEX becomes even more pronounced, reinforcing its scalability. In our experiments, constrained by data availability, we conducted tests on 343 service trips. However, based on the observed trends in the results, CG remains effective for larger-scale problems, maintaining a good balance between computational efficiency and solution quality. We have provided additional explanations on the computational scalability of the CG algorithm on page 14, lines 344-355.

 

“CPLEX achieves better results when the problem scale is small, specifically when the number of vehicles does not exceed 12. However, as the number of vehicles increases, the CG algorithm consistently yields superior results. In terms of computational time, the CG algorithm also demonstrates a clear advantage by obtaining high-quality solutions more efficiently. For instance, when the problem involves 37 vehicles, the CG algorithm takes only 58.16 minutes to achieve a solution of 43110.85, whereas CPLEX requires 360 minutes to obtain a suboptimal result of 54859.1. This comparison highlights that the CG algorithm not only enhances solution quality but also significantly reduces computational time, particularly for large-scale BEB charging scheduling problems. As the problem size increases, while the total solving time naturally rises, the gap in performance between CG and CPLEX becomes even more pronounced, further reinforcing the scalability and practicality of the CG approach.”

 

6. 6. The analysis omits critical parameters (e.g., variations in trip demand, charger downtime). Please expand sensitivity tests to include stochastic factors (e.g., weather impacts on energy consumption).

 

Thank you for your insightful suggestions.

Regarding charger downtime, we acknowledge its potential impact; however, it varies significantly across different charging stations and operational settings, often depending on station-specific policies and maintenance schedules. In our model, charger downtime was not explicitly considered as it has a relatively minor effect on optimization results. If necessary, it can be incorporated into the model by adjusting the available charging time slots, which would not introduce significant complexity to the optimization process.

The energy consumption rate of BEBs is influenced by various factors, such as vehicle acceleration and deceleration, air conditioning usage, and road conditions. These factors introduce significant variability and complexity into the BEB scheduling process, making precise modeling a challenging task.

To maintain computational tractability, many existing studies [5,11,30] assume a constant energy consumption rate when optimizing electric bus scheduling. Similarly, our study focuses on the joint optimization of charging infrastructure, fleet composition, and charging scheduling while considering multiple types of vehicles and chargers, as well as vehicle-charger compatibility. Given the inherent complexity of this problem, incorporating a stochastic energy consumption model would further increase computational burden and problem complexity.

To extend this research, we have added a discussion on the potential incorporation of stochastic energy consumption models. This discussion has been included to highlight future research directions and the possible enhancements of the proposed approach.

 

“In future work, this study can be extended in the following directions: (i) The charging scheduling can be combined with a vehicle scheduling scheme, in which the service trips are allocated to different vehicles. This can make full use of different types of vehicles. (ii) This study assumes that the energy consumption rates of vehicles are constant. However, in real-world scenarios, energy consumption is influenced by multiple stochastic factors, such as variations in passenger load, road gradient, and environmental conditions. Considering these uncertainties, developing a robust electric bus scheduling strategy would be a valuable extension. Future research could explore stochastic modeling approaches, such as Monte Carlo simulations and robust optimization techniques, to account for these variations and enhance the adaptability of the scheduling model.”

 

7. 7. The novelty could be further emphasized. The paper could do more to highlight how its contributions are distinct from existing studies, especially in terms of the practical applications and potential impact on the electric bus industry.

 

Thank you for your valuable suggestions. To better highlight the novelty and distinct contributions of our study, we have revised the manuscript on page 2, lines 58-74.

 

“While BEB charging scheduling has been extensively studied, existing research lacks an integrated optimization framework that jointly considers: (1) Fleet composition: the selection of BEBs with different battery capacities and charging power, which significantly impacts scheduling flexibility and cost.(2): Charging infrastructure deployment : ensuring that the right types and numbers of chargers are allocated efficiently. (3) Vehicle-charger compatibility: not all vehicles can charge at every station, making compatibility a key factor in optimizing system efficiency. In real-world BEB transit operations, agencies often operate mixed fleets with vehicles from multiple manufacturers, each having different charging power levels and battery capacities. This prevents the use of a one-size-fits-all charging strategy. Despite its practical importance, no prior study has jointly optimized BEB fleet composition, charging station deployment, and vehicle-charger compatibility in a single model.

This study proposes a comprehensive joint optimization framework for BEB fleet composition, charging infrastructure deployment, and charging scheduling. Specifically, our work incorporates key real-world constraints, including partial charging strategies, time of use (ToU) electricity pricing, and vehicle-charger compatibility. Then, the proposed method is further verified by a transit network in Nanjing, China.”

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The manuscript focuses on the charging scheduling problem of battery electric buses (BEB), especially the compatibility issue between multiple types of vehicles and chargers. The proposed partial charging strategy, charging continuity, and optimization of vehicle-charger compatibility can effectively reduce system costs and improve the utilization rate of charging facilities, demonstrating practical application value.

In view of the research content of the manuscript, the following modifications are suggested:

1. Although the partial charging strategies proposed in the manuscript can effectively reduce electricity costs, they do not fully consider the impact on battery life. The charging continuity constraints mentioned in the text are helpful for extending battery life, but the trade-off relationship between them and the partial charging strategies is not discussed in detail. Frequent partial charging may have a negative impact on the battery's state of health, especially in cases of deep discharge and shallow charging. Moreover, vehicles do not need to be charged during every operational break when fully charged. It is suggested that the authors explore the impact of charging after full discharge and partial charging during operational breaks on battery life.

2. Although the CG algorithm demonstrated satisfactory performance in case studies, the manuscript fails to address its scalability within large-scale networks. As the bus network expands, the complexity and computational requirements of the model may increase substantially, raising concerns about whether the CG algorithm can sustain its high efficiency. Further verification is therefore warranted.

3. Constraint conditions 6 and 7 assume that the vehicle is fully charged at the beginning and end of operation, and the charging strategy does not consider the match between the vehicle full charge time and the number of charging piles during non-operation hours, so the optimization results cannot guarantee the realization of constraint conditions 6 and 7.

4. The partial charging state of the vehicle in Figure 10 has been maintained at about 0.25 for a period of time. According to the energy consumption rate given in the manuscript, this power cannot guarantee the vehicle to operate a round trip (about 30km), so the accuracy of its optimization results needs to be further improved.

Author Response

Reviewer#3

 

General comments

 

1. The manuscript focuses on the charging scheduling problem of battery electric buses (BEB), especially the compatibility issue between multiple types of vehicles and chargers. The proposed partial charging strategy, charging continuity, and optimization of vehicle-charger compatibility can effectively reduce system costs and improve the utilization rate of charging facilities, demonstrating practical application value. In view of the research content of the manuscript, the following modifications are suggested:

 

Thank you for your valuable feedback and constructive suggestions. Below, we provide detailed responses to each of your comments and outline the specific modifications made in the paper.

 

Specific comments

 

1. Although the partial charging strategies proposed in the manuscript can effectively reduce electricity costs, they do not fully consider the impact on battery life. The charging continuity constraints mentioned in the text are helpful for extending battery life, but the trade-off relationship between them and the partial charging strategies is not discussed in detail. Frequent partial charging may have a negative impact on the battery's state of health, especially in cases of deep discharge and shallow charging. Moreover, vehicles do not need to be charged during every operational break when fully charged. It is suggested that the authors explore the impact of charging after full discharge and partial charging during operational breaks on battery life.

 

Thanks for the comments. Battery life is closely related to factors such as the number of charge-discharge cycles and the depth of discharge, and many previous studies have explored battery degradation mechanisms in BEB scheduling.

However, BEB scheduling involves multiple interdependent layers, including charging infrastructure, bus fleet management, and charging scheduling. This constitutes a complex computational and optimization process. Given the inherent complexity of jointly optimizing these aspects, our study focuses on addressing the scheduling problem in an integrated manner while considering multiple vehicle and charger types as well as compatibility constraints. This integrated approach is both practically significant and computationally challenging, which is why we did not explicitly model battery degradation.

Regarding partial charging strategies, we emphasize that charging is only performed during available operational breaks when beneficial, taking into account time of use electricity pricing. It is not necessary to charge at every available break, allowing for more flexible scheduling. This approach has already seen widespread application in real-world BEB operations. Additionally, to mitigate battery degradation, we incorporate charging continuity constraints, which help limit the extent of degradation without significantly increasing model complexity or computational difficulty.

While charging only after full discharge could reduce the number of charge-discharge cycles, it would also limit the ability to leverage time of use pricing advantages. Moreover, such an approach would require a larger number of operational vehicles, leading to a substantial increase in fixed capital investment. Given these trade-offs, our study adopts partial charging strategies with charging continuity constraints to balance battery lifespan, operational costs, model complexity, and solution feasibility, aiming for a well-rounded and practical scheduling solution.

 

2. Although the CG algorithm demonstrated satisfactory performance in case studies, the manuscript fails to address its scalability within large-scale networks. As the bus network expands, the complexity and computational requirements of the model may increase substantially, raising concerns about whether the CG algorithm can sustain its high efficiency. Further verification is therefore warranted.

 

Thank you for your valuable suggestions. We acknowledge the importance of evaluating the scalability of the CG algorithm for large-scale networks. To address this, we have compared the optimization results and computational time of CG with the traditional commercial solver CPLEX. The results demonstrate that CG not only achieves competitive optimization performance but also significantly reduces computational time, especially as the problem scale increases. For instance, when the problem size involves 12 vehicles, the CG algorithm obtains a similar solution to CPLEX in just 17 minutes, whereas CPLEX requires 360 minutes. Furthermore, when the number of vehicles increases to 37, the CG algorithm takes 58.16 minutes to achieve a solution of 43110.85, while CPLEX requires 360 minutes to reach a suboptimal result of 54859.1.

This comparison highlights that by decomposing the optimization model into a master problem and multiple subproblems, CG effectively reduces computational complexity and improves efficiency. More importantly, as the problem size increases, CG’s efficiency advantage over CPLEX becomes even more pronounced, reinforcing its scalability. In our experiments, constrained by data availability, we conducted tests on 343 service trips. However, based on the observed trends in the results, CG remains effective for larger-scale problems, maintaining a good balance between computational efficiency and solution quality. We have provided additional explanations on the computational scalability of the CG algorithm on page 14, lines 344-355.

 

“CPLEX achieves better results when the problem scale is small, specifically when the number of vehicles does not exceed 12. However, as the number of vehicles increases, the CG algorithm consistently yields superior results. In terms of computational time, the CG algorithm also demonstrates a clear advantage by obtaining high-quality solutions more efficiently. For instance, when the problem involves 37 vehicles, the CG algorithm takes only 58.16 minutes to achieve a solution of 43110.85, whereas CPLEX requires 360 minutes to obtain a suboptimal result of 54859.1. This comparison highlights that the CG algorithm not only enhances solution quality but also significantly reduces computational time, particularly for large-scale BEB charging scheduling problems. As the problem size increases, while the total solving time naturally rises, the gap in performance between CG and CPLEX becomes even more pronounced, further reinforcing the scalability and practicality of the CG approach.”

 

3. Constraint conditions 6 and 7 assume that the vehicle is fully charged at the beginning and end of operation, and the charging strategy does not consider the match between the vehicle full charge time and the number of charging piles during non-operation hours, so the optimization results cannot guarantee the realization of constraint conditions 6 and 7.

 

Thank you for your insightful comments. Constraint (6) ensures that vehicles are charged to their maximum SOC before starting operational task trips. In our model, the maximum SOC is set to 95%, which is an input condition and can be easily achieved.

Constraint (7) ensures that vehicles are charged to the maximum SOC after completing all service trips. Our study covers a 24-hour period, with a 1-minute time resolution (i.e., 1440 time intervals), allowing sufficient time for vehicles to reach the maximum SOC. Additionally, the number of chargers is a decision variable in the model, meaning the optimization process determines the required number of chargers while ensuring compliance with Constraint (7). Thus, the obtained solution inherently satisfies this constraint.

Furthermore, Constraints (8)–(26) collectively ensure that vehicle operations meet all scheduling requirements. The experimental results further validate this, as shown in Figure 6, which illustrates the daily variation of the BEB fleet’s SOC. From this figure, it is evident that Constraints (6) and (7) are successfully satisfied.

 “

Figure 6. Daily variation of BEB fleet SoC. ”

 

4. The partial charging state of the vehicle in Figure 10 has been maintained at about 0.25 for a period of time. According to the energy consumption rate given in the manuscript, this power cannot guarantee the vehicle to operate a round trip (about 30km), so the accuracy of its optimization results needs to be further improved.

 

Thank you for your thoughtful comments. In Figure 10, the final service trip for Vehicle 10 is trip 328, which occurs at time 949 with an energy consumption of 28 kWh, heading away from the depot. The corresponding decline in the SOC curve represents this discharge activity but does not include the return trip’s energy consumption.

Since this is the vehicle’s final trip, it must return to the depot afterward. In our model, we do not explicitly account for energy consumption during non-revenue return trips, as these are considered empty runs without passengers and typically have significantly lower energy consumption than regular service trips. In real-world operations, vehicles generally retain sufficient charge to return to the depot even if this segment is not explicitly modeled.

From an algorithmic perspective, incorporating the energy consumption of the return trip is straightforward. A simple modification can be made by adding a return-to-depot trip at the end, which would not introduce any additional computational complexity to the optimization framework. However, the primary focus of this study is to jointly optimize the BEB fleet, charging schedules, and charging infrastructure to improve overall operational efficiency, while also considering multiple vehicle and charger types and their compatibility constraints. This integrated approach fills a research gap and provides valuable practical insights.

 

Author Response File: Author Response.pdf