Optimizing Electric Bus Charging Infrastructure: A Bi-Level Mathematical Model for Strategic Station Location and Off-Board Charger Allocation in Transportation Network

Abstract

This study presented a novel bi-level mathematical model for designing charging infrastructure in an interstate electric bus transportation network, specifically addressing long-haul operations. To the best of our knowledge, no existing study integrates charging station locations with the number of off-board chargers while simultaneously optimizing their allocation and charging schedules. The proposed model fills this gap by formulating an exact algorithm using a mixed-integer linear programming (MILP). The first-level model determines the optimal placement and number of charging stations. The second-level model optimizes the number of off-board chargers, charger allocation, and bus charging schedules. This ensures operational efficiency and integration of decisions between both levels. The experiments and sensitivity analysis were conducted on a real case study of an interstate bus network in Thailand. The results provided valuable insights for policymakers and transportation planners in designing cost-effective and efficient electric bus transportation systems. The proposed model provides a practical framework for developing eco-friendly transportation networks, encouraging sustainability, and supporting the broader adoption of electric buses.

1. Introduction

Transportation activities contribute to approximately one-quarter of global CO₂ emissions, significantly due to fossil fuel-powered vehicles. As urbanization accelerates, the resulting traffic congestion and environmental impacts highlight the critical role of the transportation sector in climate change. In response, governments worldwide are promoting the adoption of electric vehicles (EVs) as a key strategy to achieve net-zero emissions, reduce pollution, and foster sustainable communities. Among these, electric buses (EBs) are increasingly recognized as essential for transitioning toward eco-friendly public transportation systems.

The transition to EBs introduces significant challenges in public transportation planning, particularly in infrastructure investment and operational development. Ensuring efficient and reliable EB systems requires addressing strategic, tactical, and operational issues [1]. While EBs have gained attraction globally, particularly in Europe and major cities in China, operational challenges persist for long-haul interstate routes. Limited travel range necessitates on-route recharging, which can disrupt schedules and reduce passenger satisfaction. These challenges are even more pronounced in developing countries, where cost-effective solutions are crucial for widespread adoption.

This research was motivated by Thailand’s commitment to achieving its net-zero emissions goals, reflecting the country’s focus on sustainable development. The adoption of EVs in Thailand, such as electric scooters, cars, and delivery trucks, has steadily increased. However, the transition to EBs has been limited or slow to develop. A significant barrier is the high infrastructure cost, particularly the investment required for DC fast-charging stations to support large vehicles. This challenge is compounded by budget constraints in developing countries, highlighting the need for careful planning and strategic resource allocation. While the environmental and social benefits of EBs are widely acknowledged, this study focuses on the economic aspect, particularly investment and infrastructure planning, which are crucial for large-scale EB adoption. Although economic feasibility is the primary concern, optimizing charging infrastructure also yields significant environmental benefits by lowering energy consumption, minimizing emissions, and reducing unnecessary detours for charging. Strategic infrastructure investment facilitates a smoother transition to fully electrified bus fleets, supporting long-term sustainability goals. Policymakers and planners must establish cost-effective charging infrastructure that meets the operational demands while considering real-world constraints.

Research on EB technologies and planning processes has gradually gained attention in recent years, focusing on various aspects such as fleet replacement, charging infrastructure, and operational strategies. Studies on fleet replacement addressed battery sizing, investment costs, and infrastructure compatibility [2,3,4,5]. Infrastructure-related research explored cost evaluation, technology selection, charger sizing, and lifecycle battery replacement [3,5,6,7]. Charging infrastructure placement studies emphasized optimizing charging station locations [8,9,10,11,12,13]. Iliopoulou and Kepaptsoglou [14] simultaneously optimized transit route design and wireless charging placement, balancing passenger convenience with operator costs. While most studies focus on short-haul bus networks, challenges unique to long-haul networks have received limited attention. Recent research on optimizing charging station locations emphasized the need for scalable and cost-effective solutions for long-haul networks [15,16]. Uslu and Kaya [15] relied on a single-server queuing model that oversimplifies real-world operations, whereas Yalçın et al. [16] addressed budget constraints limiting full electrification. These limitations reduced their applicability to fully electrifying long-haul EBs.

For operational strategies, existing research primarily addressed charging schedules to optimize costs based on time-of-use electricity prices and power loads [17,18,19,20,21]. Other works optimized the number of stationary chargers [22] and charger deployment strategies [23]. While some studies integrated charging placement and scheduling [24,25,26,27,28,29], they often relied on simplifying assumptions, such as immediate station availability or limited charging locations. Stumpe et al. [24] limited charging to transit and terminal stops specified in the timetable. Hu et al. [25] developed a joint optimization model that included opportunity fast charging but assumed no queuing delays. Similarly, Olsen and Kliewer [27] integrated charging station location planning with vehicle scheduling. However, their model was restricted to single-depot systems, making it less suitable for long-haul operations.

Despite growing research on EB infrastructure, existing models often address charging station placement, charger deployment, or charging scheduling as separate optimization problems. Few studies integrate these aspects into a unified framework. While various studies have explored these factors, most are designed for urban transit or short-haul networks. To the best of our knowledge, few studies address the unique challenges of long-haul interstate EB systems. Existing research rarely integrates key decisions such as charging station placement, charger allocation, and charging scheduling into a single framework. Specifically, no study explicitly optimizes these decisions simultaneously for a long-haul network, where challenges include longer travel distances, partial charging strategies, and scheduling complexities.

This study addresses these gaps by proposing a bi-level mathematical model, formulated as a mixed-integer linear programming (MILP), for the optimal design of EB infrastructure. The focus is on long-haul EBs involving multi-depot operations and fixed routes covering diverse origins and destinations. The model explicitly optimizes key decisions, including charging station locations, the number of off-board chargers, charger allocation, charging times, and charging schedules. Unlike previous studies, the proposed model simultaneously considers infrastructure planning and charging operations, providing a more comprehensive decision-making framework. It is designed to ensure operational efficiency while adhering to predefined schedules and accounting for real-world constraints, such as limited charging infrastructure, partial charging strategies, and waiting time restrictions to prevent excessive delays. Furthermore, this study analyzes cost and operational trade-offs to provide insights for policymakers and transportation planners in strategic infrastructure planning.

It is important to note that the charging stations analyzed in this study are DC fast-charging stations, potentially located at selected existing en-route bus stops within the network. Additionally, an off-board charger refers to an external device installed at a station that directly supplies DC power to a vehicle’s battery. One charging station can comprise multiple off-board chargers based on demand. The primary objective of the model is to determine the optimal locations for charging stations and the required number of off-board chargers to ensure operational efficiency while adhering to the original schedules. The proposed model facilitates the strategic planning for cost-effective charging infrastructure, supporting the broader adoption of EBs in long-haul transportation.

The contributions of this study are summarized as follows.

• This study introduces a novel bi-level mathematical model for the optimal design of EB infrastructure in long-haul transport systems. It explicitly optimizes key decisions, including charging station locations, the number of off-board chargers, off-board charger allocation, charging times, and charging schedules.
• The experimental analysis examines key parameters to provide decision-makers with valuable insights for strategic planning.
• The model offers a practical framework for policymakers and planners to design efficient charging infrastructure. It helps promote sustainable public transportation and supports the broader adoption of EBs.

This paper is organized as follows: Section 2 reviews related studies. Section 3 provides a detailed problem description. The mathematical model, including its framework and formulation, is presented in Section 4. Section 5 discusses the experimental results, and Section 6 provides the conclusion of the study.

2. Literature Review

Studies on EB charging infrastructure optimization have addressed challenges such as charging station placement, bus operations, and charging schedules, but these are often tackled independently due to their complexity. Kunith et al. [8] were among the pioneers in studying charging station placement for an EB network. They developed MILP models to optimize the number of charging stations and required battery capacity for each bus line. Kunith et al. [9] later enhanced their model by incorporating battery costs into the objective function and tested it on a subnetwork of the Berlin bus system. These models proved effective for short-haul city networks, relying on charging during dwell times.

Vasilovsky et al. [11] proposed a mathematical model to optimize charging station locations. The model was validated using a simulation with AnyLogic and daily operational data from the city of Zilina. However, this study focused exclusively on terminal stops and transit stops as potential charging station locations. Iliopoulou and Kepaptsoglou [14] introduced a bi-level optimization framework that integrated transit route design with wireless charging placement. The upper level generated and evaluated candidate route sets, while the lower level optimized the placement of wireless charging infrastructure at either terminal or en-route stops. Othman et al. [30] employed a heuristic integrated with a Voronoi diagram to optimize fast charging station placement in urban transit networks. Their zone-based strategy effectively determined the number of charging stations, the number of bus stops per station, available charging slots, and charging time, followed by allocating each charging station to its corresponding subregion within the zone. Chen et al. [12] focused on optimal placement of charging stations to enhance voltage profiles, reduce power losses, and minimize installation costs. A novel metaheuristic was developed to enhance convergence speed and accuracy. Esmaeilnejad et al. [31] optimized en-route charging station locations and charging durations using a two-stage stochastic programming model. The objective was to minimize passenger waiting times and reduce operational and capital costs. The model incorporated weather-induced uncertainties in ridership and battery performance and was tested on Calgary’s bus routes. The study found that allowing charging at en-route stations not only reduced travel costs and driver work time due to deadhead trips but also decreased passenger waiting times.

Most existing studies focus on optimizing charging infrastructure for short-haul city bus networks, while only a few recent studies have addressed the unique challenges of long-haul EB networks. These challenges include, but are not limited to, range limitations due to greater distances, sparse charging locations in rural areas, longer charging times affecting schedules, higher investment and operational costs, and the need to minimize passenger disruptions during extended journeys. Yalçın et al. [16] proposed a multi-objective model to optimize charging station locations and EB deployment for intercity networks. The model considered budget constraints that limit full electrification. It aimed to maximize the number of EBs and minimize the number of charging stations using a weighted sum method. Uslu and Kaya [15] developed a MILP model to optimize charging station locations and capacities for intercity networks. The model incorporated queueing theory with a 1 h waiting time constraint and was applied to a case study of intercity bus networks in Turkey using real-world data from coach companies. However, the reliance on a single-server queuing model was noted as a limitation, as it potentially oversimplifies real-world charging operations.

In terms of operational strategies at charging stations, some studies analyze the charging scheduling problem across various configurations. Liu and Ceder [22] examined the scheduling problem for battery-electric transit vehicles. Their study focused on determining the minimum number of stationary chargers at transit terminals, without considering charging station locations. McCabe & Ban [26] proposed two models: BEB-OCL for optimizing charger locations and BEB-BRP for revising bus schedules. The key decisions included charger placement, number of chargers, charging schedules, and backup buses. The study considered charging at both en-route and terminal stations. En-route stations provided 15 min charging, with backup buses deployed if the charging was insufficient. All buses were assumed to be charged overnight at terminal stations to prepare for the next day. However, the limited en-route charging time necessitated the use of backup buses, potentially increasing costs for EBs and driver work time.

In addition to the charging scheduling problem mentioned above, Yao et al. [32] addressed the electric vehicle scheduling problem for multiple vehicle types. They developed an optimization model to minimize annual costs, including bus and charger purchases, deadheading, and timetabled operations. A heuristic approach was proposed to optimize recharging trips and vehicle type substitutions. Similarly, He et al. [21] proposed an optimization model for charging scheduling in fast charging battery EB systems. However, both studies focused on optimizing schedules based on existing charging stations rather than incorporating charging station location optimization into their models. To address inefficiencies in conventional charging approaches, Zeng et al. [33] proposed a bus replacement strategy to optimize electric bus scheduling and charging infrastructure. Instead of waiting to recharge at terminals, low-battery buses are swapped with fully charged standby buses, ensuring continuous operation and reducing passenger delays. While effective, this approach may require additional standby buses. Subsequently, Tzamakos et al. [34] and Lacombe et al. [35] studied a transit system where all bus lines shared terminal stops. While these studies considered charging schedules, they primarily focused on overnight charging before the next day’s operations. This reliance on overnight terminal charging may be practical for urban transportation systems with relatively short driving ranges but could pose challenges for long-haul bus operations that require more flexible and continuous charging solutions.

While there are interdependencies between designing and operating EB charging infrastructure, only a few studies have addressed these challenges simultaneously. Li et al. [36] and Lu et al. [37] proposed joint models to optimize station locations and bus timetable scheduling for mixed fleets of electric and fuel-based buses. Li et al. [36] focused on refueling stations at terminal stops, whereas Lu et al. [37] considered transit stops. Similarly, Cui et al. [38] developed a joint optimization model for vehicle and recharging scheduling for mixed fleets under limited charger availability. However, these studies primarily optimized scheduling and station placement but did not incorporate detailed battery charging schedules, which are critical for efficient EB operations. Research on planning charging infrastructure and scheduling for fully electrified bus systems includes studies by Rogge et al. [23] and Yao et al. [32], which focused on depot charging points. In contrast, Häll et al. [39] examined opportunity charging points, including en-route, terminal, and transit stations. These studies integrated charging schedules into bus timetables while considering station installation. However, Cui et al. [38] and Häll et al. [39] focused on optimizing EB routes and selecting charging points from existing stations, without considering the strategic installation of additional charging stations across the network.

In addition, Hu et al. [25], Koháni and Babčan [28], and Gkiotsalitis et al. [29] developed joint optimization models integrating charging station location decisions and EB charging schedules. Hu et al. [25] focused on opportunity fast charging while considering uncertainties. The model aimed to minimize total costs and ensure reliable operations by placing charging stations at selected en-route bus stops. However, it assumed immediate charger availability without queuing, which may not reflect real-world conditions. Similarly, Koháni and Babčan [28] applied a VNS metaheuristic to optimize charging station locations and charging scheduling while minimizing infrastructure costs. However, their model is designed for short-haul bus operations, prioritizing charging stations at terminals and depots. Gkiotsalitis et al. [29] proposed a mixed-integer linear programming (MILP) model that optimizes charging station locations by considering both slow and fast chargers and using day-ahead scheduling to minimize deadheading costs. While this model effectively reduces operational expenses, it relies on average demand estimates for scheduling, which may not fully capture real-time variations in charging needs. Beyond joint optimization, some studies have examined charging station placement and scheduling constraints. Stumpe et al. [24] presented an approach to optimize charging station locations and vehicle scheduling for EBs, restricting charging to transit and terminal stops based on predetermined waiting times. Olsen and Kliewer [27] proposed a VNS-based approach to optimize charging station locations and vehicle scheduling. Their method demonstrated cost efficiency. However, it was limited to a single-depot system, reducing its scalability for long-haul interstate operations.

Table 1. Key characteristics of selected literature on electric bus planning relevant to this study.

Authors	Problem Characteristics
	Year	Charging Station Location	Possible Locations for Charging *	No. Chargers	Partial Charging	Bus Timetable	Long-Haul Route	Charging Scheduling	Cost Analysis
Kunith et al. [9]	2017	✓	N-E		✓				✓
Iliopoulou and Kepaptsoglou [14]	2019	✓	N-E		✓				✓
Vasilovsky et al. [11]	2019	✓	T-E
He, et al. [21]	2020		N-E		✓	✓		✓	✓
Liu and Ceder [22]	2020		E	✓	✓	✓			✓
Othman et al. [30]	2020	✓	E						✓
Yao et al. [32]	2020					✓		✓	✓
Chen et al. [12]	2021	✓	E						✓
Stumpe et al. [24]	2021	✓	T-E		✓	✓			✓
Uslu and Kaya [15]	2021	✓	N		✓	✓	✓		✓
Hu et al. [25]	2022	✓	N		✓	✓		✓	✓
Olsen and Kliewer [27]	2022	✓	E		✓	✓			✓
Tzamakos et al. [31]	2022	✓	N-E		✓	✓		✓
Esmaeilnejad et al. [31]	2023	✓	N		✓	✓			✓
McCabe and Ban [26]	2023	✓	E	✓	✓	✓		✓	✓
Yalçın et al. [16]	2023	✓	N				✓		✓
Zeng et al. [33]	2023		E			✓			✓
Cui et al. [38]	2023		T			✓		✓	✓
Koháni and Babčan [28]	2024	✓	E		✓	✓		✓
Lacombe et al. [35]	2024	✓	E			✓		✓
Gkiotsalitis et al.	2025	✓	T		✓	✓		✓	✓
This paper	-	✓	N	✓	✓	✓	✓	✓	✓

* N: en-route charging location, T: transit center charging location, E: terminal charging location.

summarizes key literature addressing the major challenges in EB planning relevant to this study. The symbol ✓ denotes the key characteristics of existing studies on EB planning. This structured comparison further highlights the contributions of this research in advancing both infrastructure design and operational planning for EBs. While extensive research exists on charging infrastructure design, few studies integrate charging station locations with the number of off-board chargers required to support long-haul interstate EB systems. Moreover, no existing research simultaneously optimizes the number of off-board chargers, their allocation, and charging schedules within a unified framework. Most prior studies treat these factors independently, limiting their applicability to real-world long-haul networks. Additionally, many existing models rely on unrealistic assumptions regarding off-route charging locations and no queuing constraints. Unlike previous studies, this research explicitly combines charging station placement, off-board charger allocation, and charging schedules into a single framework. The proposed model jointly considers strategic-level infrastructure planning and operational-level scheduling, providing a more realistic approach. In particular, this paper addresses existing research gaps by proposing an exact algorithm, formulated as a mixed-integer linear programming (MILP) model, to optimize charging station locations, off-board charger allocation, and charging schedules for interstate bus travel.

3. Problem Descriptions

This study addresses the challenges of transitioning a conventional long-haul interstate bus network to an EB system. It focuses on the optimal design of infrastructure, including location planning and charging facilities.

Similarly to a conventional interstate bus network, an interstate EB transportation network spans large geographic areas, connecting multiple cities and towns with several stops along the routes. The network involves long-haul EBs operating from multiple depots on fixed routes that cover diverse origins and destinations. Each bus adheres to a specific departure schedule, follows a predetermined route, and stops at multiple locations to pick up and drop off passengers. In this study, it is assumed that each bus departs from its origin with a fully charged battery. Then, it is fully recharged upon arrival at its destination terminal using a slow DC charger before starting its next trip. Importantly, all buses must complete their routes within defined timeframes. It should also be noted that routes can vary significantly in length, from short trips between neighboring states to long-haul journeys crossing multiple state lines. Additionally, multiple EBs may share common stops within the network.

The transition to an EB system involves several complex decisions. One of the most critical decisions in transportation network planning is determining the optimal placement of charging stations to effectively support all buses in the network. The charging stations analyzed in this study are DC fast-charging stations, potentially located at selected existing en-route bus stops within the network. Given the limited driving range of EBs, it is essential to optimize the amount of charge each bus receives at these stations to ensure they reach their destinations with sufficient battery state-of-charge. To address these challenges, this study adopts a partial charging strategy. This approach minimizes unnecessary charging time by allowing buses to receive only the amount of charge needed to complete their routes. Key decisions also include determining the optimal number of off-board chargers required at each charging station. An off-board charger, an external device that supplies DC power to a vehicle’s battery, may be installed in multiple amounts at a single station. This requires balancing sufficient charger availability to meet demand while avoiding unnecessary costs. Another critical decision is scheduling the charging of EBs at the stations while accounting for waiting time constraints. Effective scheduling ensures that buses adhere to their scheduled arrival times at destinations.

To tackle these complexities, this study proposes a bi-level mathematical model to optimize the design of an EB transportation network, with a focus on charging infrastructure. The objectives are to identify the optimal locations for charging stations and determine the minimum number of off-board chargers required within the network. While the environmental and social benefits of EBs are widely acknowledged, this approach aims to minimize the charging infrastructure cost associated with transitioning from conventional buses to EBs. Detailed explanations of the proposed model are provided in the following section.

4. Mathematical Model

This section presents the framework, assumptions, and formulations of the proposed mathematical model.

4.1. Model Framework

The framework of the proposed bi-level mathematical model is illustrated in Figure 1. The first-level model aims to determine the minimum number and optimal locations of DC fast-charging stations needed to support all buses in the transportation network. To achieve this, the model allocates EBs to charging stations while tracking their battery state-of-charge (SOC) at each stop. It also calculates the required charging times for each bus to ensure they reach their final destinations on schedule with sufficient battery levels.

Once the locations of charging stations and related decisions are determined, the second-level model optimizes the number of off-board chargers needed at each station. The solutions from the first-level model, including station locations, bus allocations, and required charging times, serve as inputs for this stage.

The bi-level structure reflects the sequential nature of charging infrastructure planning. The first level focuses on the strategic placement of charging stations and bus allocation, ensuring that all buses can complete their routes. The second level refines these decisions by determining the required number of chargers at each station while considering bus arrival schedules, queueing constraints, and charging times.

The interaction between these two levels is crucial. The first-level model identifies where charging stations should be located and how frequently they are used. The second-level model ensures these locations can efficiently accommodate bus demand without excessive infrastructure investment. This structured approach balances network-wide planning with station-level efficiency, leading to a cost-effective and operationally viable charging network.

In particular, key solutions from the first-level model that serve as input parameters for the second-level model include charging station locations, bus allocations to each station, required charging times, and the total number of charging occurrences per station. Understanding charging occurrences is crucial for determining the demand for off-board chargers. If a station has only one charging occurrence, a single charger is sufficient. However, if multiple buses require charging at the same station, additional chargers may be needed to prevent excessive waiting times. For this reason, only charging stations with at least two charging occurrences are considered in the second-level model for charger allocation.

The second-level model aims to minimize the total number of off-board chargers across all stations while ensuring efficient charging schedules. It allocates chargers to buses, taking into account arrival times and queueing constraints. If bus waiting times exceed the maximum allowable limit, additional chargers are assigned; otherwise, a single charger can be shared among multiple buses. Additionally, the model determines the complete charging time for each bus at every charging occurrence.

By integrating both levels, the proposed bi-level model provides data-driven insights for optimal charging station design. The combined results enable cost minimization while ensuring reliable operations, making it an effective framework for transitioning to an electric bus network.

4.2. Model Assumptions

In this study, a homogeneous fleet is assumed for modeling purposes. Long-haul interstate buses are generally of similar size to accommodate a large number of passengers comfortably over extended distances while ensuring adequate luggage storage capacity. This assumption reflects the standardized bus sizes currently used in many interstate and long-haul transportation systems, including those in Thailand and similar regions. It is also important to note that the proposed model focuses on the design and optimization of the charging infrastructure, assuming a stable and robust grid system without compromising the power quality of the utility grid. Furthermore, this study does not incorporate any renewable energy sources.

Accordingly, the first- and second-level models are formulated with the following assumptions.

• All EBs are homogeneous vehicles with the same battery capacity.
• All EBs are assumed to travel at a constant speed.
• Each bus departs from its origin with a fully charged battery.
• All en-route charging stations are standardized as DC fast-charging, providing a consistent and uniform charging rate across all stations.
• Each charging station has an unlimited power supply.
• Breakdowns are not considered, and pre-emption is not allowed.
• All parameters are assumed to be known in advance.

4.3. The First-Level Mathematical Model

The primary goal of the first-level model is to identify the optimal locations for DC fast-charging stations at existing en-route bus stops. Additionally, the model determines the allocation of EBs to charging stations, as well as the required charging times and quantities for each EB under a partial charging policy. The objective is to minimize the total number of charging stations while ensuring all buses reach their destinations on time with sufficient battery state-of-charge.

The model is formulated as a mixed-integer linear programming (MILP) problem using the following sets, parameters, and decision variables.

• Sets:

$I$	: Set of EBs; $I=\{1, 2, 3, \dots , q)$
$S$	: Set of bus stops; $S=\left\{1, 2, 3,\dots ,r\right\}$
$K$	: Set of stop sequences; $K=\left\{1, 2, 3,\dots ,c\right\}$

• Parameters:

$A_{i,j}^k$	= $\left\{\begin{array}{c}1 \mathrm{i}\mathrm{f} \mathrm{E}\mathrm{B} i \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{s} \mathrm{t}\mathrm{o} \mathrm{b}\mathrm{u}\mathrm{s} \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p} j \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p} \mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} k \left(i\in I, j\in S, k\in K\right)\hfill \\ 0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}\right.$
$d_{j,l}$	: Distance between bus stops j and l $\left(j,l\in S\right)$
$t_{j,l}$	: Travel time between bus stops j and l $\left(j,l\in S\right)$
$st\_or{g}_{i,j}$	: Scheduled departure time of EB i from the origin j $\left(i\in I, j\in S\right)$
$T_i^{max}$	: Maximum allowable time for bus route i to reach its destination $\left(i\in I\right)$
$B^{cap}$	: Battery capacity of the EB
$e_{j,l}$	: Energy consumption rate of the EB between bus stops j and l $\left(j,l\in S\right)$
$g$	: Charging rate at a DC fast-charging station
$v_i^k$	: Required transfer time for passengers of EB i at stop sequence k                                                        $\left(i\in I, k\in K\right)$
$c{v}_{max}$	: Maximum allowable charging time at each charging station

• Decision Variables:

$X_{i,j}$	=$\left\{\begin{array}{c}1 \mathrm{i}\mathrm{f} \mathrm{E}\mathrm{B} i \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{d} \mathrm{a}\mathrm{t} \mathrm{b}\mathrm{u}\mathrm{s} \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p} j \left(i\in I, j\in S\right)\hfill \\ 0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}\right.$
$Y_j$	=$\left\{\begin{array}{c}1 \mathrm{i}\mathrm{f} \mathrm{s}\mathrm{t}\mathrm{o}\mathrm{p} j \mathrm{i}\mathrm{s} \mathrm{s}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{a}\mathrm{s} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \left(j\in S\right)\hfill \\ 0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}\right.$
$B_{i,j}$	: Battery state-of-charge of EB i upon the arrival at stop j $\left(i\in I, j\in S\right)$
$P{R}_{i,j}$	: Battery state-of-charge of EB i upon the departure from stop j $\left(i\in I,j\in S\right)$
$c{v}_{i,j}$	: Charging time of EB i at stop j $\left(i\in I, j\in S\right)$
$s{t}_{i,j}^k$	: Arrival time of EB i at stop j in stop sequence k $\left(i\in I, j\in S, k\in K\right)$
$N_j$	: The total number of charging occurrences at stop j $\left(j\in S\right)$

• Objective function:

$$\mathrm{Minimize} Z={\sum }_j^r{Y}_j$$

(1)

• Subjected to:

$A_{i,j}^1{B}_{i,j}\le B^{cap}$	$\forall i\in I and \forall j\in S$	(2)
$B_{i,l}\le B_{i,j}+c{v}_{i,j}g – A_{i,j}^{k-1}{A}_{i,l}^k{e}_{j,l}{d}_{j,l}+B^{cap}(1-A_{i,j}^{k-1}{A}_{i,l}^k)$	$\begin{array}{c}\forall i\in I,\forall j,l\in S, j \ne 1\\ and \forall k\in K, k\ne 1\end{array}$	(3)
$B_{i,j}+c{v}_{i,j}g\le B^{cap}$	$\forall i\in I and \forall j\in S$	(4)
$c{v}_{i,j}g\le B^{cap}{X}_{i,j}$	$\forall i\in I and \forall j\in S$	(5)
$B_{ij}+c{v}_{ij}g=P{R}_{ij}$	$\forall i\in I and \forall j\in S$	(6)
$c{v}_{i,j}\le c{v}_{max}\cdot X_{i,j}$	$\forall i\in I and \forall j\in S$	(7)
$s{t}_{i,j}^1=st\_or{g}_{i,j}$	$\forall i\in I,\forall j\in S$	(8)
$s{t}_{i,j}^{k-1}+A_{i,j}^{k-1}{A}_{i,l}^k{t}_{j,l}+v_i^{k-1}\left(1-X_{i,j}\right)+c{v}_{i,j}\le s{t}_{i,l}^k$	$\begin{array}{c}\forall i\in I,\forall j,l\in S, j \ne 1\\ and \forall k\in K, k\ne 1\end{array}$	(9)
$s{t}_{i,j}^k{A}_{i,j}^k\le T_i^{max}$	$\forall i\in I,\forall j\in S and \forall k\in K$	(10)
$N_j\ge {\sum }_i^q{X}_{ij}$	$\forall j\in S$	(11)
$X_{i,j}\le Y_j$	$\forall i\in I and \forall j\in S$	(12)

Equation (1) represents the objective function, aiming to minimize the number of charging stations. Constraint (2) indicates that the battery state-of-charge of any EB at the origin point or the first stop (k = 1) must be less than or equal to the battery capacity. Constraints (3)–(5) ensure that the battery state-of-charge of the EB upon arriving at any stop is greater than or equal to its state-of-charge at the subsequent stop, and the battery state-of-charge at any stop must remain less than the battery’s full capacity. Constraint (6) states that the battery state-of-charge of the EB upon departure from any stop must be equal to its state-of-charge upon arrival at that stop plus the charging quantity (if charging occurs). Constraint (7) guarantees that, when an EB is charged, the charging time does not exceed the maximum allowable charging time. Constraint (8) specifies the departure time of an EB from its origin or initial stop (k = 1), ensuring the alignment with the scheduled departure time. Constraint (9) determines the arrival time of an EB at each stop sequence along its route. Constraint (10) ensures that the arrival time of an EB at each stop sequence does not exceed the maximum allowable time to reach its destination. Constraint (11) determines the total number of charging occurrences (if any) at each bus stop. Constraint (12) assures the relationship between the charging decision and the location decision of the charging station.

As described in the previous section, the following key solutions obtained from the first-level model are employed in the second-level model.

• The location of each charging station, represented as $Y_j$ (with a value of 1 and corresponding $N_j$ > 1) in the first-level model, is used to define the set of charging occurrences, M, in the second-level model.
• The number of charging occurrences at station j, represented as $N_j$, in the first-level model, is used to set the data metric for bus allocation to stations in the second-level model.
• The allocation of EB i at bus stop j, represented as $X_{i,j}$ in the first-level model, is used to determine $n_i$ in the second-level model.
• The charging time of EB i at bus stop j, represented as $c{v}_{i,j}$ in the first-level model, is replaced by parameter $P_{i,m}^f$, which denotes the charging time of EB i on off-board charger f during the corresponding charging occurrence m, in the second-level model.
• The arrival time of an EB during its first charging occurrence, as tracked in the first-level model, is set as the parameter ${\widehat{s}}_i$ in the second-level model.

4.4. The Second-Level Mathematical Model

The second-level model determines the required number of off-board chargers at each charging station and schedules the charging of EBs while accounting for waiting time constraints. Only stations with two or more charging occurrences are considered in this model. Initially, the number of candidate off-board chargers at each charging station equals the total number of EBs. The objective is to minimize the total number of off-board chargers across all stations.

The model is formulated as a mixed-integer linear programming (MILP) problem using the following sets, parameters, and decision variables.

• Sets:

$I$	: Set of EBs; $I=\{1, 2, 3, \dots , q)$
$M$	: Set of charging occurrences; $M=\{1, 2, 3, \dots , g)$
$F$	: Set of off-board chargers; $F=\{1, 2, 3, \dots , b)$

• Parameters:

$n_i$	: The total number of charging occurrences for EB i $\left(i\in I\right)$
$P_{i,m}^f$	: Charging time of EB i during charging occurrence m on an off-board charger $f$, $\left(i\in I, m\in M, f\in F\right)$
$E_{i,m}^f$	$=\left\{\begin{array}{c}1, \mathrm{if} \mathrm{EB} i \mathrm{during} \mathrm{charging} \mathrm{occurrence} m \mathrm{is} \mathrm{eligible} \mathrm{to} \mathrm{access} \mathrm{an} \mathrm{off}-\mathrm{board}\hfill \\ \mathrm{charger} f(i\in I,m\in M,f\in F)\hfill \\ 0, \mathrm{otherwise}\hfill \end{array}\right.$
${\widehat{s}}_i$	: Arrival time at the first charging occurrence for EB i $\left(i\in N\right)$
$t_{i,m-1,m}$	: The total time for EB i, encompassing both travel time and passenger transfer time, between the charging occurrences m − 1 and m
L	: A large number
$WT$	: Maximum allowable waiting time for an EB at a charging station

• Decision variables:

$X_{i,m}^f$	= $\left\{\begin{array}{c}1, \mathrm{if} \mathrm{EB} i \mathrm{during} \mathrm{charging} \mathrm{occurrence} m \mathrm{is} \mathrm{charged} \mathrm{on} \mathrm{off}-\mathrm{board} \mathrm{charger} f\hfill \\ (i\in I,m\in M,f\in F)\hfill \\ 0, \mathrm{otherwise}\hfill \end{array}\right.$
$Y_{i,m,w,t}^f$	= $\left\{\begin{array}{c}1, \mathrm{if} \mathrm{EB} i \mathrm{during} \mathrm{charging} \mathrm{occurrence} m \mathrm{is} \mathrm{charged} \mathrm{before} \mathrm{EB} w \mathrm{during}\hfill \\ \mathrm{charging} \mathrm{occurrence} t \mathrm{on} \mathrm{off}-\mathrm{board} \mathrm{charger} f (i,w\in I,m,t\in M,f\in F)\hfill \\ 0, \mathrm{otherwise}\hfill \end{array}\right.$
$N^f$	= $\left\{\begin{array}{c}1, \mathrm{i}\mathrm{f} \mathrm{o}\mathrm{f}\mathrm{f}-\mathrm{b}\mathrm{o}\mathrm{a}\mathrm{r}\mathrm{d} \mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{r} f \mathrm{i}\mathrm{s} \mathrm{u}\mathrm{s}\mathrm{e}\mathrm{d} \left(f\in F\right)\hfill \\ 0, \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\hfill \end{array}\right.$
$S_{i,m}$	: Adjusted arrival time of EB $i$ at charging occurrence $m \left(i\in I, m\in M\right)$
$C_{i,m}$	: Charging completion time of EB i at charging occurrence m, $\left(i\in I, m\in M\right)$

• Objective function:

$$\mathrm{Minimize} Z={\sum }_f^b{N}_f$$

(13)

• Subjected to:

${\sum }_{f\in F}{X}_{i,m}^f=1$	$\forall i\in I and \forall m\in M, m\le n_i$	(14)
$S_{i,1}={\widehat{s}}_i$	$\forall i\in I$	(15)
$C_{i,m-1}+t_{i,m-1,m}=S_{i,m}$	$\begin{array}{c}\forall i\in I and \forall m\in M, m\le n_i and m\ne 1\hfill \end{array}$	(16)
$S_{i,m}+{\sum }_{f\in F}{P}_{i,m}^f{X}_{i,m}^f\le C_{i,m}$	$\forall i\in I and \forall m\in M, m\le n_i$	(17)
$C_{w,t}-C_{i,m}+L\left(1-Y_{i,m,w,t}^f\right)+L\left(1-X_{i,m}^f\right)+L\left(1-X_{w,t}^f\right)+WT\ge P_{i,m}^f$	$\begin{array}{c}\forall i,w\in I, i\ne w,\\ \forall m,t\in M, m,t\le n_i, and\\ \forall f\in F\end{array}$	(18)
$C_{i,m}-C_{w,t}+L\left(Y_{i,m,w,t}^f\right)+L\left(1-X_{i,m}^f\right)+L\left(1-X_{w,t}^f\right)+WT\ge P_{w,t}^f$	$\begin{array}{c}\forall i,w\in I, i\ne w,\\ \forall m,t\in M, m,t\le n_i, and\\ \forall f\in F\end{array}$	(19)
$X_{i,m}^f\le E_{i,m}^f$	$\forall i\in I, \forall m\in M,and \forall f\in F$	(20)
$N^f\ge X_{i,m}^f$	$\forall i\in I, \forall m\in M,and \forall f\in F$	(21)

Equation (13) represents the objective function aimed at minimizing the total number of off-board chargers. Equation (14) indicates that each charging occurrence of an EB can be performed by only one off-board charger at a time. Equation (15) specifies the arrival time at the initial charging station for each EB, as derived from the first-level model. Equations (16) and (17) determine the arrival and completion times of an EB at any charging occurrence. Equations (18) and (19) determine the charging sequence for two EBs using the same off-board charger. These constraints ensure that each bus waits within the maximum allowable time and prevent simultaneous charging on the same charger. If the subsequent bus cannot wait within the specified timeframe, an additional charger is required. Equation (20) assures the accessibility of the EB to the off-board charger corresponding to its charging occurrence. Equation (21) defines the usage of the off-board charger when it is selected for charging.

5. Computational Experiment

5.1. Test Instance and Parameter Setting

In this section, numerical experiments were conducted using data derived from the current routes of several interstate bus companies in Thailand. The focus was on highway routes connecting Bangkok to the northern region and routes within the northern region. Each instance, defined by specific bus routes and stops, represents a subset of these routes. Partial bus timetables were used to avoid computational infeasibility that would arise from incorporating complete schedules. The experiment aimed to analyze diverse network structures, ranging from small to large transportation networks.

In addition, the experiment varied two key parameters, the bus battery capacity ($B^{cap}$) and the maximum allowable charging time at each charging station ($c{v}_{max}$), to create various scenarios. The results from these scenario combinations were analyzed to understand the impact of variables on the overall design of the charging infrastructure.

Table 2. Values of parameters used in the experiment.

Parameters	Value
$B^{cap}$	300, 350, 400, and 450 kWh
$c{v}_{max}$	20, 25, 30 and 40 min
$e_{j,l}$	1.2 kWh/km
$g$	7.5 kW/min
$v_i^k$	10 min
WT	15 min

summarizes the parameters used in the experiment. The $B^{cap}$ value of EB ranged from 300 to 450 kWh, corresponding to driving ranges of approximately 250 to 375 km. The $c{v}_{max}$ value varied from 20 to 40 min, while the maximum allowable waiting time for charging at each station (WT) is set at 15 min. For simplicity in the test instances, the charging rate at the station ($g$) were kept constant. Similarly, the energy consumption rate between bus stops ($e_{j,l}$), and the passenger transfer time at each stop ($v_i^k$) were treated as average values across the transportation network.

5.2. Experimental Results on Test Instances

In this study, the model was implemented using LINGO optimization software version 20.0. The programs were executed on Intel^® CoreTM i9-12900K CPU 3.20 GHz with 32 GB of RAM.

Table 3. Optimal solutions by Lingo Optimization Solver.

Instance	Network		${\mathit{B}}^{\mathit{c}\mathit{a}\mathit{p}}$ (kWh)	$\mathit{c}{\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$ (mins)	1st Level		2nd Level
	No. Buses	No. Bus Stops			No. Charging Stations	Time (hh:mm:ss)	No. Chargers	Time (hh:mm:ss)
1	3	13	300	25	7	00:00:03	7	00:00:18
2				30	6	00:00:03	7	00:00:11
3				40	6	00:00:02	6	00:00:08
4			350	25	5	00:00:02	5	00:00:12
5				30	5	00:00:02	5	00:00:13
6				40	5	00:00:02	5	00:00:13
7			400	25	4	00:00:02	4	00:00:09
8				30	4	00:00:02	4	00:00:10
9				40	4	00:00:02	4	00:00:13
10			450	20	6	00:00:03	6	00:00:17
11				25	4	00:00:03	4	00:00:15
12				30	3	00:00:03	4	00:00:14
13				40	3	00:00:03	3	00:00:05
14	6	16	300	25	9	00:00:03	9	00:00:22
15				30	9	00:00:03	9	00:00:21
16				40	8	00:00:03	8	00:00:17
17			350	20	10	00:00:03	10	00:00:25
18				25	8	00:00:04	10	00:00:16
19				30	7	00:00:03	9	00:00:14
20				40	7	00:00:03	8	00:00:47
21			400	20	6	00:00:05	6	00:00:44
22				25	6	00:00:06	7	00:00:30
23				30	5	00:00:05	6	00:00:36
24				40	5	00:00:06	5	00:00:33
25			450	20	5	00:00:06	5	00:00:46
26				25	4	00:00:06	4	00:00:41
27				30	4	00:00:05	5	00:00:26
28				40	4	00:00:03	4	00:00:18
29	9	18	300	25	12	00:00:09	12	00:00:16
30				30	11	00:00:11	13	00:01:53
31				40	10	00:00:07	11	00:00:05
32			350	25	12	00:00:14	13	00:00:07
33				30	11	00:00:19	12	00:00:03
34				40	9	00:00:13	13	00:00:04
35			400	25	9	00:00:07	12	00:00:01
36				30	9	00:00:12	10	00:00:22
37				40	7	00:00:08	7	00:00:01
38			450	25	9	00:00:03	11	00:00:03
39				30	8	00:00:14	9	00:00:06
40				40	6	00:00:07	7	00:00:01
41	12	21	300	40	10	00:00:16	12	00:00:15
42			350	30	11	00:00:13	12	00:00:23
43				40	9	00:00:13	11	00:00:20
44			400	25	9	00:00:09	11	00:00:17
45				30	9	00:00:17	9	00:00:16
46				40	7	00:00:10	9	00:00:15
47			450	25	9	00:00:17	9	00:00:34
48				30	8	00:00:14	8	00:00:32
49				40	6	00:00:33	7	00:00:28
50	16	25	350	30	12	00:00:42	15	2:37:42
51				40	11	00:02:15	13	2:11:56
52			400	25	10	00:00:43	13	2:27:43
53				30	10	00:00:53	13	2:17:24
54				40	8	00:00:41	12	2:10:46
55			450	25	9	00:00:53	12	3:09:35
56				30	8	00:00:56	11	2:23:57
57				40	7	00:00:39	10	2:12:49
58	19	31	400	25	12	00:01:23	14 *	>24:00:00
59				30	11	00:01:24	14 *	>24:00:00
60				40	10	00:01:16	13 *	>24:00:00
61			450	25	10	00:01:25	13 *	>24:00:00
62				30	10	00:00:35	12 *	>24:00:00
63				40	9	00:00:25	11 *	>24:00:00

* Solutions obtained after 24 h.

presents the optimal number of charging stations and off-board chargers required for the bus transportation network, as determined by the first- and second-level models, respectively. The analysis considered various problem configurations, defined by the number of buses, the number of bus stops, battery capacity, and the maximum allowable charging time at the station. It is noted that the total combinations of all configurations resulted in 96 instances; however, some configurations led to infeasible solutions. Therefore, Table 3 only presents solutions obtained from 63 instances. The computational time for each configuration is also reported.

As shown in Table 3, the first-level model successfully provides optimal charging station solutions for most instances (63 out of 96). These solutions, along with their associated decision variables, were then used as inputs for the second-level model. For smaller transportation networks, involving up to 12 buses and 21 stops, both models can easily find optimal solutions. However, for larger networks, such as those with 16 buses and 25 stops, the second-level model required significantly more computational time to find solutions. In even larger instances, with 19 buses and 31 stops, the second-level model struggled to find solutions and failed to achieve an optimal result within the 24 h computational limit. This is because the second-level model handles more complex decision variables and constraints, including the allocation and scheduling of buses to off-board chargers at all stations. It should be noted that the results reported in instances 58 - 63 were the solutions obtained after 24 h of computation. However, these solutions are not guaranteed to be optimal.

It is also observed from Table 3 that instances with a battery capacity of 300 kWh and a maximum charging duration of 20 min resulted in infeasible solutions (not shown in Table 3). This is because the combination of small battery capacities and insufficient charging time prevents buses from reaching their next stops, ultimately making it impossible to complete their routes. For smaller buses, increasing the maximum allowable charging time would be necessary to enable them to complete longer journeys. Additionally, in larger transportation networks, which likely include longer routes between stops, small buses are often unsuitable for covering the entire network. As shown in Table 3, instances involving larger transportation networks tend to achieve feasible solutions when using buses with larger battery capacities. However, the use of larger buses must be accompanied by sufficient charging times to ensure route completion.

To provide deeper insight into the solutions, Instance 2 was used as an illustrative example.

Table 4. Information on Bus Routes for Instance 2 ($B^{cap}$ = 300 kWh, $c{v}_{max}$ = 30 min).

Bus	Stop Sequences	Route Distance (km)
1	S2–S4–S6–S8–S10–S13	510.36
2	S2–S3–S5–S8–S10–S12	629.01
3	S5–S7–S8–S10–S14–S16–S18	553.08

presents the information of bus stop sequences and route distances for three buses. Figure 2 illustrates the detailed optimal solutions obtained from the proposed model.

In this example, all EBs are equipped with a battery capacity of 300 kWh. The drop-off and pick-up time for passengers is set as 10 min per station. If a bus requires recharging, the maximum allowable time at each station is 30 min. As shown in Figure 2, each bus follows its route and departs from its origin on a predetermined schedule with a fully charged battery. Each bus is required to stop at all bus stops along its route to drop off and pick up passengers and must arrive at its destination within a designated timeframe. The common stops for these three buses are S8 and S10. Due to their limited range, interstate EBs cannot reach their destinations without recharging. This study assumes that a fast-charging station can be located at any existing en-route bus stop in the network. Figure 2 identifies the optimal charging station locations at S3, S4, S6, S7, and S10. Bus 1 charges at S4, S6, and S10; Bus 2 charges at S3; and Bus 3 charges at S7 and S10. Partial charging quantities and times are also optimized to avoid unnecessary charging. Another key decision is determining the number of off-board chargers needed at each station. Charging stations at S3, S4, S6, and S7 are equipped with a single off-board charger, as only one bus charges at these locations. However, S10 is equipped with two off-board chargers to ensure that Bus 3, arriving after Bus 1, does not exceed the maximum allowable waiting time for charging. In total, the network includes five charging stations and six off-board chargers. With this configuration, the infrastructure cost is minimized to effectively support the network. The solutions in Figure 2 also indicate the battery state-of-charge and the arrival and departure times at each bus stop.

In addition, the complexity of the proposed model was analyzed to further investigate how runtime scales with problem size. Computing time was measured for different instances, and regression models were fitted to describe the relationship between input size and computational effort. Figure 3a,b illustrate the average computational time required to obtain the optimal solution for the first- and second-level models, respectively.

As shown in Figure 3a, the first-level model exhibits a quadratic growth pattern, where computing time increases polynomially with the number of bus stops. This suggests that solving larger instances requires significantly more computational resources, though the increase remains manageable within practical limits. In contrast, Figure 3b demonstrates that the second-level model follows exponential time complexity, where computing time grows rapidly as the number of candidate chargers increases. This rapid escalation indicates that solving large-scale instances becomes computationally infeasible beyond a certain threshold. The computational limit of 24 h imposes practical constraints, emphasizing the need for efficient solution approaches or heuristic methods when dealing with large problem instances.

These findings highlight the varying complexity of the two models, with the first-level model being relatively more scalable compared to the second-level model, which demands more computational resources as the problem size grows.

5.3. Real Case Study

The proposed model was applied to a subset of real-world data from the actual interstate bus timetable of one of Thailand’s largest transportation companies. The focused bus routes connect Bangkok (BKK), the capital city in central Thailand, with eight major cities in the northern region. These routes also included return trips to Bangkok, involving a total of 22 bus routes operated by 11 buses. The network spans 15 provinces, encompasses 17 bus stops, and covers an area of 147,422.30 square kilometers. The experiment incorporated the complete bus timetable for these routes based on the current operations of diesel buses.

Table 5. Interstate bus information for the case study.

Bus Route No.	Stop Sequence	Distance (km)	DT	ETA	Bus Route No.	Stop Sequence	Distance (km)	DT	ETA
1	J1-J2-J3-J5-J9-J10-J14	843.37	7:15 a.m.	06:30 a.m.	12	J14-J10-J9-J5-J3-J2-J1	843.37	9.00 a.m.	6.30 p.m.
2	J1-J2-J3-J5-J9-J10-J14	843.37	7:45 p.m.	7:00 p.m.	13	J14-J10-J9-J5-J3-J2-J1	843.37	9:00 p.m.	6:30 a.m.
3	J1-J2-J6-J7-J11-J15-J16-J17	842.03	7:30 a.m.	08:00 a.m.	14	J17-J16-J15-J11-J7-J6-J2-J1	842.03	5.00 p.m.	4.00 a.m.
4	J1-J2-J6-J7-J11-J15-J16-J17	842.03	9:00 p.m.	8:30 p.m.	15	J17-J16-J15-J11-J7-J6-J2-J1	842.03	7.00 p.m.	6.00 a.m.
5	J1-J2-J3-J5-J9-J10	809.37	8.00 p.m.	05.30 a.m.	16	J10-J9-J5-J3-J2-J1	809.37	6.45 a.m.	8.15 p.m.
6	J1-J2-J3-J5-J9	736.37	9.15 a.m.	5.45 p.m.	17	J9-J5-J3-J2-J1	736.37	7.30 a.m.	10.45 p.m.
7	J1-J2-J6-J7-J12	776.37	8.00 p.m.	11.45 a.m.	18	J12-J7-J6-J2-J1	776.37	7.00 p.m.	11.30 a.m.
8	J1-J2-J6-J7-J11-J15-J16	754.03	8:30 p.m.	04:30 a.m.	19	J16-J15-J11-J7-J6-J2-J1	754.03	7.45 p.m.	3.45 a.m.
9	J1-J2-J6-J7-J11-J15-J16	754.03	9:00 p.m.	05:00 a.m.	20	J16-J15-J11-J7-J6-J2-J1	754.03	8.00 p.m.	4.00 a.m.
10	J1-J2-J4-J6-J7-J11	629.28	10.00 p.m.	06.15 a.m.	21	J11-J7-J6-J4-J2-J1	629.28	9.45 a.m.	6.15 p.m.
11	J1-J2-J3-J5-J8-J13	991.81	8.45 p.m.	10.30 a.m.	22	J13-J8-J5-J3-J2-J1	991.81	1.00 p.m.	3.00 a.m.

DT: departure time, ETA: estimated arrival time.

provides information about the interstate bus routes used in the case study, including stop sequences, route distances, departure time, and estimated arrival time. The data shows that the distances between origins and destinations range from approximately 600 km to 990 km, requiring buses to be recharged en-route to complete their journeys.

Table 6. Experimental Results for the Case Study.

${\mathit{B}}^{\mathit{c}\mathit{a}\mathit{p}}$ (kWh)	$\mathit{c}{\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$ (mins)	1st Level		2nd Level
		No. Charging Stations	Time (hh:mm:ss)	No. Chargers	Time (hh:mm:ss)
350	20	5	00:06:08	15	4:21:45
	25	4	00:06:58	13	3:43:24
	30	4	00:01:30	13	3:21:29
	40	4	00:01:18	13	3:34:51
400	20	4	00:04:47	14	3:55:35
	25	4	00:00:52	13	3:11:26
	30	4	00:01:13	12	3:45:11
	40	3	00:00:48	12	2:56:42
450	20	4	00:01:34	10	3:44:29
	25	3	00:00:48	11	3:25:17
	30	3	00:00:57	9	2:34:53
	40	3	00:00:50	7	2:31:42

presents the optimal number of charging stations and off-board chargers required for the bus transportation network, as determined by the first- and second-level models, respectively. The optimal solutions are reported across various scenarios involving different values of $B^{cap}$ and $c{v}_{max}$.

It is important to note that the results for a battery capacity of 300 kWh are not included in the table. This is because a bus with a 300 kWh battery resulted in infeasible solutions, as it can travel only 250 km. In this case study, the maximum distance between stops is 284 km, exceeding the driving range of a 300 kWh bus. Consequently, buses with a battery capacity of only 300 kWh are unsuitable for this transportation network.

As seen in Table 6, smaller buses with lower battery capacities require more charging stations and chargers due to their limited range and the need for more frequent charging. This increases the likelihood of multiple buses using the same station simultaneously. As a result, additional chargers are required to meet demand and avoid delays. For example, buses with a battery capacity of 350 kWh require up to 5 charging stations and 15 chargers at a maximum charging time of 20 min. In contrast, the use of larger buses with higher battery capacities (e.g., 450 kWh) requires less frequent charging. This reduces the demand for charging stations and chargers within the transportation network. At a maximum charging time of 40 min, only three charging stations and seven chargers are needed for buses with a 450 kWh battery capacity. Longer battery ranges also reduce the complexity of charger allocation and charging scheduling.

Figure 4 presents a map of the transportation network used in the case study in Thailand. It highlights the optimal charging station locations and the number of chargers at each station for the scenario where $B^{cap}$ is 450 kWh and $c{v}_{max}$ is 40 min. The marked charging stations and required chargers emphasize their role in supporting the long-haul EB transportation network.

5.4. Sensitivity Analysis on a Case Study

In this section, a sensitivity analysis was conducted to evaluate the impact of three key parameters $B^{cap}$: $c{v}_{max}$, and WT, on the results of the proposed model. The parameters $B^{cap}$ and $c{v}_{max}$ belong to the first-level model, while WT is associated with the second-level model. These parameters are systematically explored to assess their influence.

The case study of a Thailand transportation company (explained in Section 5.3) was used in this analysis to evaluate investment decisions for transitioning from diesel to EBs under real-world operational conditions. The analysis considered infrastructure costs, including charging stations and chargers, as well as electric bus costs. It also examined the total system cost, combining infrastructure and bus expenses, under different parameter values.

Table 7. Cost elements of charging infrastructure and electric bus.

Parameters	Value
Construction cost of charging station (THB/station) [40]	2.24 million
Cost of off-board charger with 450 kW (THB/unit) [40]	5.09 million
Cost of bus without battery (THB/unit) [41]	16.14 million
Cost of 1 kWh battery capacity (THB/kWh) [41]	10,758.73

1 THB = 0.029 USD.

presents cost elements used in this study, as referenced in [40,41]. These include the construction cost of a charging station, the cost of an off-board charger unit, the cost of an EB without a battery, and battery capacity-related costs.

5.4.1. Impact of $B^{cap}$ and $c{v}_{max}$ on Charging Infrastructure Cost

The analysis explored the impact of $B^{cap}$ (350, 400, and 450 kWh) and $c{v}_{max}$ (20, 25, 30, and 40 min) on the number of charging stations, off-board chargers, and associated infrastructure cost. The results are presented in Figure 5.

According to Figure 5, increasing both $B^{cap}$ and $c{v}_{max}$ reduces the number of required charging stations and off-board chargers. For a given battery capacity, extending the charging time decreases the need for stations and chargers. Similarly, for a fixed maximum charging time, larger battery capacity reduces demand for stations and chargers.

These results highlight a trade-off between battery capacity, charging time, and infrastructure requirements. Smaller buses with lower battery capacities require more frequent charging, increasing the investment in charging infrastructure to avoid delays and congestion. In contrast, despite higher costs upfront, larger buses with higher battery capacity reduce infrastructure requirements and offer operational savings. Extending charging times further enhances efficiency, particularly for larger buses, making them a more cost-effective option for large-scale transportation networks.

In terms of infrastructure cost, increasing $B^{cap}$ and extending $c{v}_{max}$ result in cost reductions. In this case study, this effect is most pronounced with the largest $B^{cap}$ (450 kWh) and the longest $c{v}_{max}$ (40 min). In addition, a statistical test was conducted to further analyze the effects of individual parameters and their interactions on infrastructure cost. The main effect and interaction plots, together with ANOVA results, are presented in Figure 6.

In Figure 6a, the main effects plot shows that increasing $B^{cap}$ from 350 kWh to 450 kWh leads to a continuous reduction in infrastructure cost, with the most substantial decrease occurring between 400 kWh and 450 kWh, indicating a nonlinear trend. Conversely, $c{v}_{max}$ produces a relatively consistent, linear reduction in cost, with the lowest cost observed at 40 min.

In Figure 6b, the interaction plot reveals contrasting trends between smaller and larger battery capacities. For buses with $B^{cap}$ = 350 kWh, the lowest cost occurs at $c{v}_{max}$ = 25 min, with costs increasing as $c{v}_{max}$ rises. In contrast, for $B^{cap}$ = 450 kWh, the highest cost appears at 25 min, and cost decreases with longer charging times. This indicates that optimal $c{v}_{max}$ depends on the battery capacity, emphasizing the need for tailored strategies to minimize infrastructure cost for different $B^{cap}$ values. ANOVA analysis in Figure 6c further confirms the significance of $B^{cap} and c{v}_{max}$ at a 95% confidence level. The p-values for $B^{cap}$ (0.001) and $c{v}_{max}$ (0.034) are both below the threshold of 0.05, confirming their statistical significance. However, the F-values indicate that $B^{cap}$ (32.51) has a far greater impact on infrastructure cost compared to $c{v}_{max}$ (5.77), which has a moderate effect.

In conclusion, while both $B^{cap}$ and $c{v}_{max}$ significantly reduce infrastructure cost, battery capacity plays a dominant role. Additionally, achieving the lowest cost requires selecting an appropriate $c{v}_{max}$ based on the chosen battery capacity, as a uniform strategy may not apply across all scenarios.

5.4.2. Impact of $B^{cap}$ and $c{v}_{max}$ on Electric Bus Cost and Total System Cost

In practice, when transitioning to an EB interstate system, transportation companies must consider both infrastructure cost and EB investment. EB cost depends on factors such as battery and passenger capacity. Although higher battery capacity enables longer travel distances and reduces the need for charging stations and off-board chargers, it increases bus costs. In this study, the total system costs comprise infrastructure and EB costs, representing a company’s total investment.

This section investigated the impact of $B^{cap}$ and $c{v}_{max}$ on three cost components: infrastructure cost, EB cost, and total system cost.

Table 8. Analysis results of $B^{cap}$ and $c{v}_{max}$ on cost components: infrastructure cost, electric bus cost, and total system cost (Million THB).

${\mathit{B}}^{\mathit{c}\mathit{a}\mathit{p}}$(kWh)	$\mathit{c}{\mathit{v}}_{\mathit{m}\mathit{a}\mathit{x}}$(mins)	No. CS	No. CG	Infrastructure Cost (a)	Electric Bus Cost (b)	Total System Cost (a) + (b)
350	20	5	15	87.57	181.28	268.86
	25	4	13	75.15	181.28	256.44
	30	4	13	75.15	181.28	256.44
	40	4	13	75.15	181.28	256.44
400	20	4	14	80.24	181.82	262.07
	25	4	13	75.15	181.82	256.97
	30	4	12	70.06	181.82	251.88
	40	3	12	67.82	181.82	249.64
450	20	4	10	59.87	182.36	242.03
	25	3	11	62.73	182.36	245.09
	30	3	9	52.54	182.36	234.90
	40	3	7	42.36	182.36	224.72

summarizes the results for each cost component across the analyzed scenarios, while Figure 7 illustrates the cost proportions for each category.

As shown in Table 8 and Figure 7, EB cost consistently accounts for the largest proportion of the total system cost for all scenarios. As $B^{cap}$ increases from 350 kWh to 450 kWh, infrastructure cost decreases significantly, while EB cost rises. For instance, at $B^{cap}$ = 350 kWh and $c{v}_{max}$ = 20 min, infrastructure cost accounts for 67.43% of the total cost (87.57 million THB), while EB cost represents 32.57%. In contrast, at $B^{cap}$ = 450 kWh and $c{v}_{max}$ = 40 min, infrastructure cost drops to 18.85% (42.36 million THB), while EB cost rises to 81.15%.

The total system cost also decreases as battery capacity and charging time increase, with the lowest total system cost observed at $B^{cap}$ = 450 kWh and $c{v}_{max}$ = 40 min, reaching 224.72 million THB. This cost reduction is primarily driven by the significant decrease in infrastructure cost, despite the rising EB cost.

These findings demonstrate a clear trade-off: increasing $B^{cap}$ and $c{v}_{max}$ reduces infrastructure cost while increasing EB cost. In this case study, the total system cost trend shows that the combination of 450 kWh battery capacity and 40 min charging time is the most cost-effective solution, as it minimizes overall expenses while balancing infrastructure and bus costs.

The statistical test was conducted to analyze the effects of individual parameters and their interactions on EB cost and total system cost. The main effect and interaction plots, together with ANOVA results, are presented in Figure 8.

EB cost is primarily driven by battery capacity. As shown in the main effect plot (Figure 8a), increasing $B^{cap}$ from 350 kWh to 450 kWh leads to a sharp rise in bus cost due to the higher investment associated with larger battery capacities. In contrast, for a given battery capacity, $c{v}_{max}$, has no significant effect, as illustrated by the flat trend in the plot. In addition, the interaction plot (Figure 8b) shows that there are no interactive effects between $B^{cap}$ and $c{v}_{max}$, and only $B^{cap}$ directly influences EB cost. As a result, in Figure 8e, the F-values and p-values are represented by ‘*’, indicating that ANOVA could not compute these values because the mean square of the error is zero. This suggests that there is no variation in the error term, making statistical significance testing impossible. However, the main and interaction plots show that only battery capacity has an effect, while $c{v}_{max}$ has no measurable impact.

The total system cost combines EB cost and infrastructure cost. It decreases as both $B^{cap}$ and $c{v}_{max}$ increase. The main effect plot (Figure 8c) shows a substantial reduction in total cost when $B^{cap}$ increases from 350 kWh to 450 kWh, and when $c{v}_{max}$ increases from 20 min to 40 min. The interaction plot (Figure 8d) shows that cost reduction is most significant for larger batteries (450 kWh) with longer charging times. This indicates a complementary relationship between these two parameters. The ANOVA results (Figure 8f) confirm the significance of both $B^{cap}$ and $c{v}_{max}$ on total system cost, with p-values of 0.001 and 0.034, respectively. However, the F-value for $B^{cap}$ (29.95) is greater than for $c{v}_{max}$ (5.77), indicating that battery capacity has a more dominant influence on total system cost compared to charging time.

These findings align with the earlier analysis of infrastructure cost, where increasing $B^{cap}$ reduced infrastructure cost but raised bus costs. Both $B^{cap}$ and $c{v}_{max}$ contribute to reducing total system cost, with battery capacity playing a dominant role. Selecting an appropriate $c{v}_{max}$ based on the chosen $B^{cap}$ leads to the lowest total cost. Larger $B^{cap}$ benefits significantly from longer charging times, while smaller $B^{cap}$ shows minimal cost reductions as $c{v}_{max}$ increases.

5.4.3. Impact of WT on Number of Off-Board Chargers and Charging Infrastructure Cost

In the second-level model, the parameter WT (maximum allowable waiting time for an EB at a charging station) directly impacts the number of off-board chargers in the system, which subsequently influences infrastructure cost. This section examined the effects of WT on the number of off-board chargers and the associated infrastructure cost, using the scenario where an EB has a $B^{cap}$ of 450 kWh and a $c{v}_{max}$ of 40 min. It is noted that, in this scenario, the number of charging stations is minimized at three stations across the network. The cost of EBs was not considered in this analysis, as it remains constant.

Table 9. Analysis results of WT on the number of off-board chargers and infrastructure cost (Million THB).

WT (mins)	No. CG	Infrastructure Cost
10	8	47.45
15	7	42.36
20	5	32.17
25	4	27.08
30	4	27.08
60	3	21.99

and Figure 9 present the results of various WT values and their impact on the number of off-board chargers and infrastructure cost.

The results show that WT significantly influences the number of off-board chargers and the associated infrastructure cost. As shown in Table 9 and Figure 9, increasing WT allows buses to wait longer before charging, reducing the overall number of off-board chargers in the system. At WT = 10 min, the system requires eight off-board chargers, resulting in the highest infrastructure cost of 47.45 million THB. As WT increases, the number of chargers decreases. When WT = 60 min, only three chargers are needed, with one charger assigned to each of the three charging stations. In this scenario, the infrastructure cost drops to 21.99 million THB.

The analysis reveals that the most significant cost reductions occur at lower WT values. Increasing WT from 10 to 20 min decreases the number of chargers from eight to five, achieving a cost reduction of over 15 million THB. However, beyond WT = 25 min, the number of chargers stabilizes at four, and further cost reductions become marginal. This diminishing return indicates that while increasing WT improves cost efficiency, its impact becomes less significant as the system approaches its minimum charger requirements. It is important to note that even though higher WT can reduce costs, longer waiting times may impact service efficiency by increasing bus dwell times at stations. Thus, decision-makers and planners must consider not only cost saving but also service quality.

The results of this sensitivity analysis provide valuable guidance for decision-makers and planners in designing cost-effective charging infrastructure for EBs. By identifying key parameters, such as $B^{cap}$, $c{v}_{max}$, and WT, which influence infrastructure cost and system requirements, transportation companies can strategically plan investments and optimize the placement of charging facilities. However, some aspects of this analysis warrant further investigation. Future research should explore the trade-offs between these parameters and operational indicators, such as service reliability and passenger satisfaction. Additionally, the long-term implications of cost reductions on user experience and system performance should be assessed to ensure a balanced approach to infrastructure planning.

To strengthen the evaluation of the model’s economic feasibility, a cost comparison between diesel buses (DBs) and electric buses (EBs) was conducted, covering both purchasing costs and travel expenses. The analysis is based on the case study scenario with $B^{cap}$ = 450 kWh, $c{v}_{max}$ = 40 min, and WT = 15 min. The purchasing cost for DBs is estimated at 90.75 million THB [42], while the cost for EBs is 182.36 million THB. Fuel and electricity prices are based on average market rates in Thailand in 2024, with diesel at 31.82 THB per liter and electricity at 7.76 THB per kWh. The annual fuel cost for DBs is approximately 108.61 million THB, whereas the annual electricity cost for EBs is approximately 59.97 million THB.

While the initial investment in EBs is higher by 91.59 million THB, lower energy costs result in annual savings of 48.64 million THB. Using this cost difference, the payback period is estimated from two perspectives as follows:

• Payback period to recover the additional purchasing cost of EBs compared to DBs through energy cost savings: 1.88 years
• Payback period to recover the total purchasing cost of EBs through energy cost savings: 3.75 years

These results demonstrate that, despite the higher initial investment, EBs offer significant long-term cost savings, supporting their economic feasibility. This evaluation provides useful guidance for policymakers and transportation planners considering the transition to EBs, particularly in Thailand and other countries pursuing sustainable public transportation systems.

6. Conclusions

Research on optimizing EB technologies and planning processes has gained increasing attention due to global sustainability efforts. However, studies have primarily focused on infrastructure and operations for short-haul city bus networks and addressed individual problems in isolation. Limited research has explored long-haul interstate bus networks while integrating multiple challenges simultaneously. This study proposed a bi-level mathematical model for the optimal design of charging infrastructure in an interstate EB network, considering multi-routing systems. The first-level model focuses on identifying optimal locations for fast-charging stations, chosen from existing en-route bus stops, within the network. Additionally, it determines optimal partial charging quantities for each bus, ensuring that all buses reach their final destinations on time with sufficient battery state-of-charge. The objective is to minimize the total number of charging stations. Based on the results obtained from the first-level model, the second-level model determines the required number of off-board chargers at each charging station and schedules the charging of EBs.

The proposed model was applied to a set of instances derived from the current routes of several interstate bus companies in Thailand in order to analyze results across various network configurations. The model was further applied to a subset of real-world data from a transportation company in Thailand. The focused routes connect Bangkok (BKK), the capital city in central Thailand, with eight major cities in the northern region, and the complete bus timetable for these routes was taken into consideration. Sensitivity analysis was conducted to evaluate the impact of key parameters on charging infrastructure decisions. The results showed that $B^{cap}$ (battery capacity) and $c{v}_{max}$ (maximum allowable charging time) influence both the location of charging stations in the first-level model and the number of off-board chargers in the second-level model. A larger $B^{cap}$ reduces the number of charging stations and off-board chargers, significantly lowering infrastructure cost but increasing EB cost. While $c{v}_{max}$ had no direct effect on bus costs, selecting an appropriate $c{v}_{max}$ based on $B^{cap}$ minimizes total system cost. Longer charging times benefited larger batteries, whereas smaller batteries yielded minimal cost reductions as $c{v}_{max}$ increased. Additionally, the analysis revealed that WT (maximum allowable waiting time for charging) directly affects the number of off-board chargers in the second-level model. Allowing longer WT at charging stations reduces the number of chargers required and infrastructure costs, but its impact diminishes beyond a certain threshold. Thus, while increasing WT can enhance cost efficiency, planners must balance cost savings with potential service delays.

This study acknowledges several limitations. The model assumes a homogeneous bus capacity, excludes power grid capacity constraints at the charging station, and does not account for uncertainty factors such as vehicle breakdowns and traffic delays. Additionally, other parameters, such as charging rate, charging time, and travel time, are treated as static in the current model, whereas they may vary dynamically in real-world operations due to factors like traffic conditions, peak and off-peak demand patterns, and passenger volumes. These simplifications were made to maintain tractability; however, their variability is recognized as an important aspect for future research.

Furthermore, the mathematical model is currently limited to solving small-scale problems and may not be directly applicable to large transportation networks. Ongoing research aims to address these gaps by developing advanced algorithms to improve computational efficiency. Future work will focus on benchmarking different metaheuristic approaches to evaluate their effectiveness in enhancing scalability and enabling the model to handle more complex networks.

Beyond computational improvements, further research is needed to examine the trade-offs between cost savings and operational efficiency. This includes evaluating the impact of service reliability, passenger satisfaction, and potential scheduling disruptions due to uncertainty. Future studies could also extend the model by incorporating dynamic and stochastic elements, such as time-varying charging demand, charging rate, fluctuating electricity prices, and grid constraints. Additionally, variations in bus capacities and long-term cost reduction strategies should be examined alongside user experience and system performance to ensure a practical and balanced approach to infrastructure planning.

In conclusion, the proposed bi-level mathematical model provides a practical and adaptable framework for policymakers and transportation planners in designing efficient charging infrastructure. The model serves as a valuable foundation for optimizing charging infrastructure. It also offers valuable insights for investment planning, ensuring operational reliability and cost-effectiveness. By addressing the unique challenges of long-haul networks, the proposed model supports governments and private companies in transitioning to EBs, fostering sustainability, and promoting wider EB adoption.