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

Abstract

Battery electric buses (BEBs) are a sustainable and environmentally friendly solution for modern transit systems, offering zero tailpipe emissions and reduced noise pollution. However, due to the relatively short driving range and limited charging resources, it is crucial to jointly optimize the BEB fleet, charging schedules, and charging infrastructure to improve operational efficiency. This study proposes a BEB charging scheduling method with multiple types of vehicles and chargers. In particular, a partial charging strategy, charging continuity, and the compatibility of vehicles with chargers are incorporated. We first formulate a mixed-integer programming model to minimize the total costs of the BEB transit system, including the purchase costs of chargers, fleet composition costs, and electricity costs. Then, a column generation (CG) algorithm is designed to solve the model, and a case study based on a real transit network in Nanjing, China, was conducted. The results verify the effectiveness of the proposed model and algorithm. The findings in this study provide practical guidance on BEB charging scheduling and promote the sustainable development of bus transit systems.

1. Introduction

Growing concerns about global warming are driving governments to take action to reduce greenhouse gas emissions. In response, the transportation sector is increasingly pursuing low-carbon and sustainable development strategies [1,2]. Among them, BEBs are gradually taking the place of diesel buses in public transit, owing to their low-carbon and zero-emission benefits. By 2020, their numbers had reached half a million, and they are expected to make up over 67% of the market by 2040 [3]. Numerous countries are implementing significant efforts to advance the electrification of public transportation [4]. For example, Singapore announced a bus fleet plan to switch all 5800 traditional buses to electric buses by 2040 [5]. Similar targets have been announced in Canada and the United States. These efforts are essential for tackling environmental issues and lowering greenhouse gas emissions from transit systems.

Although BEBs are more eco-friendly, their broader implementation remains limited by various challenges. Unlike fuel buses, BEBs are limited by relatively long charging time, and the driving ranges of BEBs are about 33.33% lower than those of diesel buses [6]. As such, it is expected that BEBs require midday charging to meet operational demands. Otherwise, the fleet size will significantly increase to serve the same demand. However, several contextual and logistical barriers exist to installing en-route charging stations [7]. For example, in China, the focus is on depot charging, which is illustrated in recent work [8,9]. In depot charging, BEBs must return to the depot for charging (i.e., there are no en-route charging stations). In this situation, it is crucial to optimally allocate the resources of depot charging stations, including the number of chargers of different types.

In recent work, charging BEBs was carried out using full and partial charging [10]. In the full charging approach, vehicles were recharged until they reached maximum capacity. However, full charging is not cost-effective, especially with peak demand and time-of-use electricity tariffs [11]. Therefore, this study proposes a partial charging strategy, in which vehicles are partially recharged during operation while considering electricity price, charger availability, and dwelling times. In addition, to reduce the number of charging and discharging activities and extend battery life, charging continuity is also incorporated, in which vehicles can be charged once in a time interval.

While BEB charging scheduling has been extensively studied, existing research lacks an integrated optimization framework that jointly considers the following: (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. The proposed method is further verified by a transit network in Nanjing, China.

Contributions

Several contributions are offered in this study. Practical contributions include the following:

(i)

A joint optimization model of charging infrastructure, fleet composition, and charging scheduling is proposed, and multiple types of vehicles and chargers are considered.

(ii)

The compatibility of vehicles and chargers is optimized to reduce the total system costs.

(iii)

A partial charging strategy, including ToU electricity price and charging continuity, is proposed, which aligns more with BEB transit operation and practice.

From a methodological perspective, the study contributes to the following:

(iv)

By model linearization, the proposed model integrates constraints of partial charging strategies, minimum charging time, and continuity requirements under a series of state-of-charge (SoC) updating functions without quadratic terms.

(v)

A column generation algorithm is designed to obtain the optimal solution, and a comparison is conducted between the CG algorithm and the CPLEX solver (Version: 12.8.0).

(vi)

Two BEB routes are used as case studies to validate the effectiveness of the proposed model and algorithm. Additionally, a sensitivity analysis is performed under various parameter combinations.

The remainder of this manuscript is structured as follows. Section 2 briefly synthesizes the relevant literature. Section 3 describes the joint optimization problem and formulates the mixed-integer model. Section 4 designs a CG algorithm to solve the model. Section 5 reports on a numerical example and compares the CG algorithm with CPLEX. Section 6 discusses the optimization results and conducts a sensitivity analysis of key parameters. Section 7 concludes the study and provides potential extensions of our work.

2. Literature Review

Bus transit electrification plays a crucial role in cutting carbon emissions and easing energy shortages [12,13,14]. Accordingly, extensive research has been devoted to BEBs, focusing on aspects such as energy consumption estimation [15,16,17], vehicle scheduling [18,19], and system configuration [20,21,22], and so forth.

Unlike fuel buses, BEBs need to be recharged during operation due to their relatively long charging time and short operation range. Considering charging scheduling, significant contributions have been achieved in the literature. For example, Wang et al. [23] developed a method to optimize charging scheduling, and a numerical example in Davis was adopted to validate the proposed method. Their findings indicate that certain charging strategies can alleviate the range anxiety of BEBs. He et al. [24] proposed a model with decision variables of charging time and power. They demonstrated that the model could reduce electricity costs compared to uncontrolled charging strategies. Zhou et al. [10] combined a battery degradation effect and partial charging to optimize charging time. The results show that bus operators should consider battery degradation in charging scheduling. Zhou et al. [25] developed a robust model for BEB systems in which charging scheduling and vehicle types are considered. These studies conclude that robust charging scheduling models can effectively prevent the problems of the low SoC and delayed departure of vehicles.

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 EV infrastructure deployment and operation, incorporating TOU tariffs and Vehicle-to-Grid (V2G) technology. The study found that implementing TOU and V2G strategies could 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. The case study results demonstrated that the proposed model effectively improved 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 could reduce grid impact and costs while highlighting the influence of EV users’ charging behavior.

In real-world practice, BEB networks are composed of several routes, which are operated by multiple operators. Therefore, having multiple brands and types of electric buses in a network is common. There are differences in the parameters of these buses, such as charging power, battery capacity, and purchase costs, which all affect the charging schedule. In this case, coordinating multiple types of vehicles and building bus fleets brings new challenges for electric bus operations. Zhang et al. [29] proposed a BEB scheduling method in which multiple vehicle types were considered. Yao et al. [30] and Zhang et al. [11] developed electric bus scheduling models to optimize vehicle types, and heuristic algorithms were designed to solve them. These studies concluded that heterogeneous fleets can fully utilize the advantages of multiple types of vehicles, thereby reducing the total costs of BEB transit systems. However, they did not consider the impacts of charging infrastructure configuration.

In a BEB network, the type and number of chargers are related to charging demand and vehicle operation schemes. Therefore, charging infrastructure, fleet composition, and charging scheduling need to be jointly considered. Rogge et al. [31] revealed that the joint optimization method is beneficial for bus operations. Wang et al. [32] revealed that the energy consumption rate and BEB fleet costs were the most significant factors influencing total costs. Considering emissions, Foda et al. [33] developed a generic model to optimize charging infrastructure, battery capacity, and charging scheduling schemes. Their findings indicated that using more parameters as decision variables enhanced its practical significance. These studies concluded that the joint optimization of charging infrastructure, bus fleets, and charging schedules is crucial for the efficient operation of BEB transit systems.

To summarize, scholars have conducted extensive research on charging scheduling and charging infrastructure optimization, but there are still some limitations. Firstly, the number of chargers is usually involved in charging infrastructure optimization, while the type of chargers is rarely considered. Secondly, few studies have included multiple types of vehicles and chargers, where it is particularly important to consider the matching of chargers and vehicles. Finally, charging scheduling is closely related to the types of vehicles and chargers. For example, charging and discharging power can both affect the charging time of vehicles. As shown in

Table 1. Recent battery electric bus literature.

Study	Charger Deployment		Fleet Composition		Partial Charging	Vehicle–Charger Compatibility
	Number	Type	Homogenous	Heterogenous
Zhang et al. [17]	✓		✓		✓
Foda et al. [33]	✓	✓	✓		✓
Zhang et al. [11]	✓			✓	✓
Chen et al. [12]	✓			✓	✓
He et al. [22]	✓			✓	✓
He et al. [24]			✓		✓
Zhou et al. [10]	✓			✓	✓
Rogge et al. [31]	✓			✓
Wang et al. [32]	✓			✓	✓
Sung et al. [34]	✓	✓		✓		✓
Liu et al. [35]			✓		✓
Ji et al. [36]			✓		✓
McCabe et al. [37]	✓		✓
Zhou et al. [38]	✓	✓	✓
Present study	✓	✓		✓	✓	✓

, there is little research on the joint consideration of these three parts [20]. The study by Sung et al. [34] is closest to our study, and they optimized heterogeneous vehicles and chargers in electric bus scheduling. However, they did not optimize the charging schedule, which is an essential factor in electric bus scheduling. In addition, these research efforts often rely on simulation methods and heuristic algorithms to obtain the optimal solution, which cannot guarantee accurate results. To fill this gap, we conduct the joint optimization of charging infrastructure, fleet composition, and charging scheduling, considering compatibility, and design a column generation algorithm to obtain optimal results.

3. Methodology

3.1. Problem Description

The joint optimization problem is formulated as follows. Given a BEB network, bus operators need to determine the charger deployment plan, fleet composition, and charging schedule. Due to their relatively short driving ranges, vehicles should be recharged during operation to complete their assigned trips. Furthermore, considering the ToU electricity price variation and short available time between trips, bus operators usually adopt a partial charging strategy to reduce electricity costs. The BEB fleet comprises multiple vehicles with varying purchase costs, battery capacities, energy consumption rates, and charging power. Chargers also have multiple types, distinguished by purchase costs and charging-rated powers. Different vehicle–charger combinations can impact BEB transit systems’ charging schedule and total costs. In addition, the matching of vehicles and chargers should be considered; that is, vehicles can only be recharged by the corresponding type of chargers. An example illustrating the joint optimization problem is shown in Figure 1.

During the charging process, charging continuity is another issue that needs to be considered. According to previous studies [10], the lifespan of a battery is related to discharging and recharging cycles and the depth of discharge (DoD). In a charging interval, multiple and discontinuous charging activities are not only unreasonable but also damage the battery life and increase the complexity of charging scheduling. Therefore, increasing the constraint of charging continuity in charge scheduling is necessary. As shown in Figure 2, we suppose that a vehicle needs to be charged for 3 min between 12:01 and 12:05. We list four scenarios, in which scenarios (1) and (3) are reasonable and commonly used charging modes. Scenarios (2) and (4) are not good charging schemes, and they should be avoided during the scheduling process to prolong battery life.

To reduce the complexity of the proposed model, we added two additional assumptions: (1) the energy consumption rate of BEBs is not influenced by stochastic factors, such as weather and road conditions, and (2) service vehicles for each trip and the timetable are known. Sets, parameters, and variables in the proposed model are shown in

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

Sets	Description	Sets	Description
$S$	Set of charger types, $s\in S$	$U$	Set of vehicle types, $u\in U$
$K$	Set of vehicles, $k\in K$	$T$	Set of time intervals, $t\in T$
$N_k$	Set of trips serviced by vehicle k, $n\in N_k$	$U\left(s\right)$	Set of chargers that match vehicle type $u$
Parameters	Description	Parameters	Description
$C_s^a$	Daily purchase cost of charger type $s$, CNY	$C_u^b$	Daily vehicle costs of type $u$, CNY
$E_k^t$	Charging amount of bus k at time t, kWh	${\epsilon }_s$	Charging power of chargers of type $s$, kW
$\Delta t$	Length of unit time interval, min	$f_t$	Electricity price at time t, CNY/kWh
$k\left(n\right)$	nth trip serviced by vehicle k	$t_{k\left(n\right)}^d$	Departure time of trip $k\left(n\right)$
$t_{k\left(n\right)}^a$	Arrival time of trip $k\left(n\right)$	$t_{k\left(n\right)}^s$	First available charging time after trip $k\left(n\right)$
$t_{k\left(n\right)}^e$	Last available charging time before next trip $k\left(n+1\right)$, or 1440 for last trip	$e_{t_{k\left(n\right)}^d}^k$	Electricity amount when vehicle k departures from original stop of trip $k\left(n\right)$
$e_{t_{k\left(n\right)}^a}^k$	Electricity amount when vehicle k arrives at destination stop of trip $k\left(n\right)$	$e_u^{\max }$	Battery capacity of vehicle type u
$T_{\max }$	Last interval in research period	${\eta }_u$	Energy consumption rate of vehicle type $u$, kWh/km
$l_{k\left(n\right)}$	Length of trip $k\left(n\right)$, km	$t_{k\left(n\right)}$	Operation time of trip $k\left(n\right)$, minute
$SO{C}_{\min }$	Minimum SoC that vehicles need to satisfy during scheduling processes	$SO{C}_{\max }$	Maximum SoC that vehicles need to satisfy during scheduling processes
$c_{k\left(n\right)}$	Parameter denotes that if bus is located at charging station after trip $k\left(n\right)$, $c_{k\left(n\right)}=1$; otherwise, $c_{k\left(n\right)}=0$, and value can be obtained before optimization
Variables	Description	Decision Variables	Description
$Z$	Total costs of BEB system, CNY	$r_s$	Number of chargers of type $s$
$Z_1$	Purchase costs of chargers, CNY	${\lambda }_{k,u}$	Binary variable where ${\lambda }_{k,u}=1$ if type of bus k is u and ${\lambda }_{k,u}=0$ if not
$Z_2$	BEB fleet composition costs, CNY	${\mu }_{k,s}^t$	Binary variable where ${\mu }_{k,s}^t=1$ if bus k is charged at time t by charger of type s and ${\mu }_{k,s}^t=0$ if not
$Z_3$	Electricity costs, CNY	$c_{k,s}^{t,t+1}$	Binary auxiliary variable, representing linearization of $\left(1-{\mu }_{k,s}^t\right){\mu }_{k,s}^{t+1}$

.

3.2. Model Development

The objective function comprises three parts: the purchase costs of chargers, BEB fleet composition costs, and electricity costs. The variables are the type and number of chargers and vehicles, as well as the charging plans of vehicles.

3.2.1. Purchase Costs of Chargers

The cost of purchasing chargers depends on both their type and quantity, as illustrated in Equation (1):

$$Z_1={\displaystyle \sum _{s\in S}{C}_s^a{r}_s}$$

(1)

where $Z_1$ denotes the purchase costs of chargers, CNY; $S$ denotes the set of charger types; $C_s^a$ is the daily purchase cost of charger type $s$, CNY, and this is estimated by dividing the acquisition costs of the chargers by the expected service life; and $r_s$ is the number of chargers of type $s$ used in the operation process.

3.2.2. Fleet Composition Costs

The fleet composition costs are equal to the sum of the number of vehicles of each type used multiplied by the unit cost, as shown in Equation (2):

$$Z_2={\displaystyle \sum _{u\in U}{\displaystyle \sum _{k\in K}{C}_u^b}}{\lambda }_{k,u}$$

(2)

where $Z_2$ denotes the fleet composition costs, CNY; $U$ denotes the set of vehicle types; $K$ denotes the set of vehicles; $C_u^b$ represents the daily vehicle costs of type $u$, CNY, and this is estimated by dividing the purchase cost of vehicles by the expected service life; and ${\lambda }_{k,u}$ is a binary variable where ${\lambda }_{k,u}=1$ if the type of bus k is u and ${\lambda }_{k,u}=0$ if not.

3.2.3. Electricity Costs

The electricity costs are related to the energy demand and ToU electricity price, and the charging amount can be calculated by Equation (3):

$$E_k^t={\displaystyle \sum _{s\in S}\frac{{\mu }_{k,s}^t{\epsilon }_s\Delta t}{60}}$$

(3)

where ${\mu }_{k,s}^t$ is a binary variable where ${\mu }_{k,s}^t=1$ if bus k is charged at time t by a charger of type s and ${\mu }_{k,s}^t=0$ if not; ${\epsilon }_s$ is the charging power of charger of type $s$, kW; and $\Delta t$ represents the length of a unit time interval, min.

The electricity costs can be calculated by Equation (4):

$$Z_3={\displaystyle \sum _{k\in K}{\displaystyle \sum _{t\in T}{E}_k^t}}{f}_t$$

(4)

where $Z_3$ denotes the electricity costs, CNY; $T$ denotes the set of time intervals.

3.2.4. Objective Function and Constraints

The objective function and constraints are listed in Equations (5)–(26):

$$Z=Z_1+Z_2+Z_3$$

(5)

$s.t.$

$$\begin{array}{cc}{e}_{t_{k\left(1\right)}^d}^k={\displaystyle \sum _{u\in U}SO{C}_{\max }{\lambda }_{k,u}{e}_u^{\max },}& \forall k\in K\end{array}$$

(6)

$$\begin{array}{cc}{e}_{t_{k\left(\left|N_k\right|\right)}^a}^k+{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t=t_{k\left(\left|N_k\right|\right)}^a}^{T_{\max }}{\mu }_{k,s}^t{\epsilon }_s\frac{\Delta t}{60}={\displaystyle \sum _{u\in U}SO{C}_{\max }{\lambda }_{k,u}{e}_u^{\max },}}}& \forall k\in K\end{array}$$

(7)

$$\begin{array}{cc}{e}_{t_{k\left(n\right)}^d}^k-e_{t_{k\left(n\right)}^a}^k={\displaystyle \sum _{u\in U}{\lambda }_{k,u}{\eta }_u}{l}_{k\left(n\right)},& \forall k\in K,\forall n\in N_k\end{array}$$

(8)

$$\begin{array}{cc}{e}_{t_{k\left(n+1\right)}^d}^k-e_{t_{k\left(n\right)}^a}^k={\displaystyle \sum _{s\in S}{\displaystyle \sum _{t=t_{k\left(n\right)}^a}^{t_{k\left(n+1\right)}^d}\frac{{\mu }_{k,s}^t{\epsilon }_s\Delta t}{60}}},& \forall k\in K,\forall n\in N_k\end{array}$$

(9)

$$\begin{array}{cc}{e}_{t_{k\left(n\right)}^d}^k\le {\displaystyle \sum _{u\in U}SO{C}_{\max }{\lambda }_{k,u}}{e}_u^{\max },& \forall k\in K,\forall n\in N_k\end{array}$$

(10)

$$\begin{array}{cc}{e}_{t_{k\left(n\right)}^a}^k\ge {\displaystyle \sum _{u\in U}SO{C}_{\min }{\lambda }_{k,u}}{e}_u^{\max },& \forall k\in K,\forall n\in N_k\end{array}$$

(11)

$$\begin{array}{cc}{\displaystyle \sum _{k\in K}{\mu }_{k,s}^t\le r_s,}& \forall s\in S,\forall t\in T\end{array}$$

(12)

$$\begin{array}{cc}{\mu }_{k,s}^t\le {\lambda }_{k,u},& \forall k\in K,\forall t\in T,\forall u\in U,\forall s\in U\left(s\right)\end{array}$$

(13)

$$\begin{array}{cc}{\mu }_{k,s}^t\le 1-{\lambda }_{k,u},& \forall k\in K,\forall t\in T,\forall u\in U,\forall s\notin U\left(s\right)\end{array}$$

(14)

$$\begin{array}{cc}{\displaystyle \sum _{u\in U}{\lambda }_{k,u}=1,}& \forall k\in K\end{array}$$

(15)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\mu }_{k,s}^t\le 1,}& \forall k\in K,\forall t\in T\end{array}$$

(16)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^a,t_{k\left(n+1\right)}^d\right]}{\mu }_{k,s}^t}=0},& \forall k\in K,\forall n\in N_k,c_{k\left(n\right)}=0\end{array}$$

(17)

$${\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^d,t_{k\left(n\right)}^a\right]}{\mu }_{k,s}^t}=0,}\begin{array}{cc}& \forall k\in K,\forall n\end{array}\in N_k$$

(18)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^s-1,t_{k\left(n\right)}^e-1\right]}{c}_{k,s}^{t,t+1}}\le 1},& \forall k\in K,\forall n\in N_k\end{array}$$

(19)

$$\begin{array}{cc}{c}_{k,s}^{t,t+1}\le \left(1-{\mu }_{k,s}^t\right),& \forall k\in K,\forall n\in N_k,\forall t\in \left[t_{k\left(n\right)}^a-1,t_{k\left(n+1\right)}^d-1\right],\forall s\in S\end{array}$$

(20)

$$\begin{array}{cc}{c}_{k,s}^{t,t+1}\le {\mu }_{k,s}^{t+1},& \forall k\in K,\forall n\in N_k,\forall t\in \left[t_{k\left(n\right)}^a-1,t_{k\left(n+1\right)}^d-1\right],\forall s\in S\end{array}$$

(21)

$$\begin{array}{cc}{c}_{k,s}^{t,t+1}\ge {\mu }_{k,s}^{t+1}-{\mu }_{k,s}^t,& \forall k\in K,\forall n\in N_k,\forall t\in \left[t_{k\left(n\right)}^a-1,t_{k\left(n+1\right)}^d-1\right],\forall s\in S\end{array}$$

(22)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^a,t_{k\left(n+1\right)}^d-2\right]}{c}_{k,s}^{t,t+1}}=0},& \forall k\in K,\forall n\in N_k,c_{k\left(n\right)}=0\end{array}$$

(23)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^d,t_{k\left(n+1\right)}^a-2\right]}{c}_{k,s}^{t,t+1}}=0},& \forall k\in K,\forall n\in N_k\end{array}$$

(24)

$$\begin{array}{cc}{c}_{k,s}^{t,t+1},{\mu }_{k,s}^t\in \left\{0,1\right\},& \forall k\in K,t\in T,s\in S\end{array}$$

(25)

$$\begin{array}{cc}{\lambda }_{k,u}\in \left\{0,1\right\},& \forall k\in K,u\in U\end{array}$$

(26)

where $Z$ denotes the total costs of BEB transit systems, CNY; $k\left(n\right)$ denotes the nth trip serviced by vehicle k; $e_{t_{k\left(n\right)}^d}^k$ and $e_{t_{k\left(n\right)}^a}^k$ denote the remaining energy when vehicle k departs from the original stop of trip $k\left(n\right)$ and arrives at the destination stop of trip $k\left(n\right)$, respectively, kWh; $t_{k\left(n\right)}^s$ denotes the first available charging time after trip $k\left(n\right)$; $t_{k\left(n\right)}^e$ denotes the last available charging time before the next trip $k\left(n+1\right)$, or 1440 for the last trip; $T_{\max }$ represents the last interval in the research period (for example, if the research period is 24 h, that is, 1440 min, and the time interval is 1 min, then $T_{\max }=1440$); $N_k$ denotes the set of trips serviced by vehicle k; $l_{k\left(n\right)}$ denotes the length of trip $k\left(n\right)$, km; $U\left(s\right)$ denotes the set of chargers that match vehicle type $u$; and $c_{k,s}^{t,t+1}$ is a binary auxiliary variable, representing the linearization of $\left(1-{\mu }_{k,s}^t\right){\mu }_{k,s}^{t+1}$.

Constraint (6) ensures that vehicles are charged to the maximum SOC before starting operational task trips; Constraint (7) ensures that vehicles are charged to the maximum SOC after completing all service trips; Constraint (8) represents the change in remaining energy during discharging processes; Constraint (9) represents the change in the electricity amount during charging processes; Constraints (10) and (11) represent the constraints on the SoC of vehicles, according to a previous study [39], so controlling the charging and discharging processes to certain SoC intervals can help extend the lifespan of batteries; Constraint (12) represents the constraint on the number of chargers; Constraints (13) and (14) represent the vehicles can only be charged by the matching chargers; Constraint (15) represents that each electric bus can only choose one vehicle type; Constraint (16) represents that the vehicles can be charged by at most one type of charger in each time interval; Constraint (17) represents that vehicles cannot be recharged when they are not located at the charging station after completing service trips; Constraint (18) represents that vehicles cannot be recharged during operation; and 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 $\left(1-{\mu }_{k,s}^t\right){\mu }_{k,s}^{t+1}$, which are commonly used techniques to linearize quadratic terms, ensuring that the model remains computationally efficient and solvable; Constraints (23) and (24) are similar to Constraints (17) and (18); Constraints (25) and (26) represent the range of variables.

4. Solution Algorithm

In this paper, the proposed model is a mixed-integer program, which can be solved by solvers, such as CPLEX. However, given a large-scale problem, variables and constraints increase exponentially as the number of trips increases. In this case, it is difficult for the solvers to obtain the optimal solutions in an acceptable length of time.

In addition, countering unexpected situations, such as abnormal weather, requires prompt decisions to rearrange electric bus scheduling schemes, which necessitates a short calculation time. In this respect, the CG algorithm is an efficient method for solving complex optimization problems, which has been widely used to address bus scheduling problems [40,41]. Therefore, we devised a CG algorithm to address the proposed model. The master problem is intended to solve the charger deployment problem, including the type and number of chargers. Each vehicle is a separate subproblem that includes the vehicle type and charging plan, in which the energy demand constraints are met. 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. 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 CG process. The algorithm flow is detailed in Figure 3.

4.1. Master Problem Formulation

The master problem is defined as follows:

$$\min \left({\displaystyle \sum _{s\in S}{C}_s^a{r}_s+{\displaystyle \sum _{k\in K}{\displaystyle \sum _{p\in P}{C}_{kp}{x}_{kp}}}}\right)$$

(27)

$s.t.$

$$\begin{array}{cc}{\displaystyle \sum _{p\in P}{x}_{kp}=1,}& \forall k\in K\end{array}$$

(28)

$$\begin{array}{cc}{\displaystyle \sum _{k\in K}{\displaystyle \sum _{p\in P}{\mu }_{k,s}^{p,t}{x}_{kp}}\le r_s,}& \forall s\in S,\forall t\in T\end{array}$$

(29)

$$\begin{array}{cc}{x}_{kp}\in \left\{0,1\right\},& \forall k\in K,\forall p\in P\end{array}$$

(30)

$$\begin{array}{cc}{\mu }_{k,s}^{p,t}\in \left\{0,1\right\},& \forall k\in K,s\in S,\forall t\in T,\forall p\in P\end{array}$$

(31)

where $C_{kp}$ denotes the costs of scheduling scheme p for bus k, CNY. $x_{kp}$ denotes a binary variable; if scheme p is selected for bus k, $x_{kp}=1$. Otherwise, $x_{kp}=0$. $P$ denotes the set of all schemes; ${\mu }_{k,s}^{p,t}=1$ when bus k is charged in scheme p at time t by chargers of type s. Otherwise, ${\mu }_{k,s}^{p,t}=0$.

Equation (27) is to minimize the costs of chargers and the selected scheduling schemes; Constraint (28) represents that each electric bus can only adopt one scheduling scheme; Constraint (29) represents that the number of vehicles charging should not exceed the number of chargers; and Constraints (30) and (31) are the ranges for the variables.

4.2. Reduced Cost Calculation

Let ${\delta }_{kp}$ be the dual variables of the constraints in Equation (28) for bus k, and let $w_s^t$ be the dual variables of the constraints in Equation (29) for charger type s and time t. For BEB k, the reduced cost ${\overline{C}}_{kp}$ of its scheme p can be calculated by Equation (32):

$${\overline{C}}_{kp}=C_{kp}-{\delta }_{kp}-{\displaystyle \sum _{t\in T}{\displaystyle \sum _{s\in S}{w}_s^t{\mu }_{k,s}^{p,t}}}$$

(32)

$C_{kp}$ includes the total fleet composition costs and electricity costs, which can be calculated by Equation (33):

$$C_{kp}={\displaystyle \sum _{u\in U}{\lambda }_{k,u}^p}{C}_u^b+{\displaystyle \sum _{t\in T}{\displaystyle \sum _{s\in S}\frac{{\mu }_{k,s}^{p,t}{\epsilon }_s{f}_t\Delta t}{60}}}$$

(33)

where ${\lambda }_{k,u}^p$ denotes a binary variable. If bus type u is selected in scheme p for bus k, ${\lambda }_{k,u}^p=1$; otherwise, ${\lambda }_{k,u}^p=0$.

4.3. Subproblems with Math Programming

The subproblems are as follows:

$$\min {\overline{C}}_{kp}=\min {\displaystyle \sum _{u\in U}{\lambda }_{k,u}^p}{C}_u^b+{\displaystyle \sum _{t\in T}{\displaystyle \sum _{s\in S}\frac{{\mu }_{k,s}^{p,t}{\epsilon }_s{f}_t\Delta t}{60}}}-{\delta }_{kp}-{\displaystyle \sum _{t\in T}{\displaystyle \sum _{s\in S}{w}_s^t{\mu }_{k,s}^{p,t}}}$$

(34)

$s.t.$

$$e_{t_{k\left(1\right)}^d}^{k,p}={\displaystyle \sum _{u\in U}SO{C}_{\max }{\lambda }_{k,u}^p{e}_u^{\max }}$$

(35)

$$e_{t_{k\left(\left|N_k\right|\right)}^a}^{k,p}+{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t=t_{k\left(\left|N_k\right|\right)}^a}^{T_{\max }}{\mu }_{k,s}^{p,t}{\epsilon }_s\frac{\Delta t}{60}={\displaystyle \sum _{u\in U}SO{C}_{\max }{\lambda }_{k,u}^p{e}_u^{\max }}}}$$

(36)

$$\begin{array}{cc}{e}_{t_{k\left(n\right)}^d}^{k,p}-e_{t_{k\left(n\right)}^a}^{k,p}={\displaystyle \sum _{u\in U}{\lambda }_{k,u}^p{\eta }_u}{l}_{k\left(n\right)},& \forall n\in N_k\end{array}$$

(37)

$$\begin{array}{cc}{e}_{t_{k\left(n+1\right)}^d}^{k,p}-e_{t_{k\left(n\right)}^a}^{k,p}={\displaystyle \sum _{s\in S}{\displaystyle \sum _{t=t_{k\left(n\right)}^a}^{t_{k\left(n+1\right)}^d}\frac{{\mu }_{k,s}^{p,t}{\epsilon }_s\Delta t}{60}}},& \forall n\in N_k\end{array}$$

(38)

$$\begin{array}{cc}{e}_{t_{k\left(n\right)}^d}^{k,p}\le {\displaystyle \sum _{u\in U}SO{C}_{\max }{\lambda }_{k,u}^p}{e}_u^{\max },& \forall n\in N_k\end{array}$$

(39)

$$\begin{array}{cc}{e}_{t_{k\left(n\right)}^a}^{k,p}\ge {\displaystyle \sum _{u\in U}SO{C}_{\min }{\lambda }_{k,u}^p}{e}_u^{\max },& \forall n\in N_k\end{array}$$

(40)

$$\begin{array}{cc}{\mu }_{k,s}^{p,t}\le {\lambda }_{k,u}^p,& \forall t\in T,\forall u\in U,\forall s\in U\left(s\right)\end{array}$$

(41)

$$\begin{array}{cc}{\mu }_{k,s}^{p,t}\le 1-{\lambda }_{k,u}^p,& \forall t\in T,\forall u\in U,\forall s\notin U\left(s\right)\end{array}$$

(42)

$${\displaystyle \sum _{u\in U}{\lambda }_{k,u}^p=1}$$

(43)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\mu }_{k,s}^{p,t}\le 1,}& \forall t\in T\end{array}$$

(44)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^a,t_{k\left(n+1\right)}^d\right]}{\mu }_{k,s}^{p,t}}=0},& \forall n\in N_k,c_{k\left(n\right)}=0\end{array}$$

(45)

$${\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^d,t_{k\left(n\right)}^a\right]}{\mu }_{k,s}^{p,t}}=0,}\begin{array}{cc}& \forall n\in N_k\end{array}$$

(46)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^s-1,t_{k\left(n\right)}^e-1\right]}{c}_{k,s,p}^{t,t+1}}\le 1},& \forall n\in N_k\end{array}$$

(47)

$$\begin{array}{cc}{c}_{k,s,p}^{t,t+1}\le \left(1-{\mu }_{k,s}^{p,t}\right),& \forall n\in N_k,\forall t\in \left[t_{k\left(n\right)}^a-1,t_{k\left(n+1\right)}^d-1\right],\forall s\in S\end{array}$$

(48)

$$\begin{array}{cc}{c}_{k,s,p}^{t,t+1}\le {\mu }_{k,s}^{p,t+1},& \forall n\in N_k,\forall t\in \left[t_{k\left(n\right)}^a-1,t_{k\left(n+1\right)}^d-1\right],\forall s\in S\end{array}$$

(49)

$$\begin{array}{cc}{c}_{k,s,p}^{t,t+1}\ge {\mu }_{k,s}^{p,t+1}-{\mu }_{k,s}^{p,t},& \forall n\in N_k,\forall t\in \left[t_{k\left(n\right)}^a-1,t_{k\left(n+1\right)}^d-1\right],\forall s\in S\end{array}$$

(50)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^a,t_{k\left(n+1\right)}^d-2\right]}{c}_{k,s,p}^{t,t+1}}=0},& \forall n\in N_k,c_{k\left(n\right)}=0\end{array}$$

(51)

$$\begin{array}{cc}{\displaystyle \sum _{s\in S}{\displaystyle \sum _{t\in \left[t_{k\left(n\right)}^d,t_{k\left(n+1\right)}^a-2\right]}{c}_{k,s,p}^{t,t+1}}=0},& \forall n\in N_k\end{array}$$

(52)

$$\begin{array}{cc}{c}_{k,s,p}^{t,t+1}\in \left\{0,1\right\},& t\in T,s\in S\end{array}$$

(53)

$$\begin{array}{cc}{\lambda }_{k,u}^p\in \left\{0,1\right\},& u\in U\end{array}$$

(54)

where $e_{t_{k\left(n\right)}^d}^k$ and $e_{t_{k\left(n\right)}^a}^k$ denote the remaining energy when vehicle k departs from the original stop of trip $k\left(n\right)$ and arrives at the destination stop of trip $k\left(n\right)$ in scheme p, kWh; $c_{k,s,p}^{t,t+1}$ is a binary auxiliary variable representing the linearization of $\left(1-{\mu }_{k,s}^{p,t}\right){\mu }_{k,s}^{p,t+1}$.

Constraints (35)–(40) are equivalent to Constraints (6)–(11), and Constraints (41)–(54) correspond to Constraints (13)–(26) in terms of meaning.

5. Case Study

Experimental Background

This study used data from lines 68 and 72 in Nanjing, China, as a case study to validate the proposed method. The lengths of these lines are 14.8 km and 16 km, respectively. Overall, the lines complete 343 service trips per day, using 37 vehicles. As shown in Figure 4, the two lines share the same depot and charging station.

The operation scheme of the bus lines is shown in

Table 3. Operation scheme of bus lines.

Trip No.	Departure Time	Operation Time (min)	Line No.	Direction	Vehicle ID
1	5:00	56	68	Inbound	19
2	5:16	56	68	Inbound	14
3	5:31	56	68	Outbound	10
4	5:31	60	72	Inbound	32
5	5:33	56	68	Inbound	17
6	5:40	56	68	Outbound	7
7	5:42	60	72	Inbound	23
…	…	…	…	…	…
343	22:00	60	72	Outbound	31

, and the experimental parameters are shown in

Table 4. Experimental parameters.

Parameter	Value	Unit
$T_{\max }$	1440	Minute
$\Delta t$	1	Minute
$SO{C}_{\max }$	95%	Percentage
$SO{C}_{\min }$	20%	Percentage

.

Furthermore, three BEB types were considered in this paper, and the parameters of vehicle types are listed in

Table 5. Parameters of BEB types.

Parameter	Type A	Type B	Type C
Battery capacity (kWh)	200	150	100
Energy consumption rate (kWh/Km)	1.875	1.6875	1.5
Daily cost (CNY) ≃ (USD 0.137)	800	700	600
Similar BEBs in market	King-Long XMQ6016G	Hager KLQ6650GEV	Yutong ZK6606BEVG4

. The chargers included three types, and the parameters are shown in

Table 6. Parameters of charger types.

Parameter	DC Fast Charging	Type I	Type II
Charging power (kW)	240	150	90
Daily cost (CNY)	3000	2500	1800
Similar chargers in market	Xi’an Boston 240 kW Chargers	EAST 150 kW chargers	Bull 90 kW Chargers

.

For the matching between chargers and vehicles, vehicle Type A can be charged by DC fast charging and Type I chargers, vehicle Type B can be charged by Type I and II chargers, and Type II chargers can charge vehicle Type C. The ToU electricity prices for different times (CNY/kWh) are given in Formula (55):

$$f\left(t\right)=\left\{\begin{array}{cc}0.3& 0\le t<8\\ 0.9& 8\le t<12,18\le t<22\\ 0.6& 12\le t<18,22\le t<24\end{array}\right.$$

(55)

6. Results

6.1. Comparison Between CPLEX and CG Algorithm

Table 7. Comparison between CPLEX and CG algorithm.

	CPLEX Results		CG Results
Number of Vehicles	Obj. (CNY)	Time (min)	Obj. (CNY)	Time (min)
3	4907.8	0.3	5509.7	3.0
6	7356.3	3.4	7844.5	5.5
9	10,828.6	73.5	11,350.0	10.9
12	16,358.9	360.0	16,846.9	17.0
15	25,765.6	360.0	20,823.8	25.3
18	33,814.3	360.0	23,383.7	31.8
21	36,564.4	360.0	28,082.0	36.2
24	39,387.8	360.0	30,517.9	39.6
27	43,046.3	360.0	33,111.1	45.0
30	45,517.2	360.0	35,507.6	50.8
33	48,951.7	360.0	38,146.5	54.7
37	54,859.1	360.0	43,110.9	58.2

compares CPLEX and the proposed CG algorithm, and the algorithms’ performance at different scales of the proposed problem is represented by the number of vehicles. The CPLEX version was 12.8.0, and the running time was set to 6 h.

CPLEX achieved better results when the problem scale was small, specifically when the number of vehicles did not exceed 12. However, as the number of vehicles increased, the CG algorithm consistently yielded superior results. In terms of computational time, the CG algorithm also demonstrated a clear advantage by obtaining high-quality solutions more efficiently. For instance, when the problem involved 37 vehicles, the CG algorithm took only 58.16 min to achieve a solution of 43,110.85, whereas CPLEX required 360 min to obtain a suboptimal result of 54,859.1. This comparison highlights that the CG algorithm not only enhanced solution quality but also significantly reduced computational time, particularly for large-scale BEB charging scheduling problems. As the problem size increased, while the total solving time naturally rose, the gap in performance between CG and CPLEX became even more pronounced, further reinforcing the scalability and practicality of the CG approach.

The optimal BEB transit system solutions of both the CPLEX and CG algorithms are shown in

Table 8. Optimization results of CPLEX and CG algorithm.

Related Indicators	CPLEX	CG Algorithm
The number of BEBs for Type A (200 kWh)	19.0	5.0
The number of BEBs for Type B (150 kWh)	12.0	19.0
The number of BEBs for Type C (100 kWh)	6.0	13.0
The number of chargers for DC fast charging	4.0	1.0
The number of chargers for Type I	0.0	2.0
The number of chargers for Type II	6.0	3.0
The overall occupancy rate of chargers (%)	27.8	49.3
The purchase cost of chargers (CNY)	22,800.0	13,400.0
Fleet composition costs (CNY)	27,200.0	25,100.0
Electricity costs (CNY)	4859.1	4610.9

. It is clear that CPLEX used more BEBs of Type A (200 kWh). Thus, the fleet composition costs were relatively higher. In addition, fewer chargers were configured based on CG, resulting in a significant reduction in the purchase costs of chargers.

Furthermore, the occupancy rate of chargers (utilization time over available time) was higher in CG. The occupancy rate of each charger, shown in

Table 9. Occupancy rate of chargers of CPLEX and CG algorithm.

CPLEX			CG Algorithm
Charger No.	Type	Occupancy Rate (%)	Charger No.	Type	Occupancy Rate (%)
1	DC	27.25	1	DC	25.28
2	DC	19.90	2	I	51.53
3	DC	21.40	3	I	51.04
4	DC	23.88	4	II	63.75
5	II	28.54	5	II	51.32
6	II	33.75	6	II	53.12
7	II	30.06	-	-	-
8	II	31.26	-	-	-
9	II	21.70	-	-	-
10	II	40.18	-	-	-

, indicates that CG significantly outperformed CPLEX. For example, the occupancy rate of Charger 1 was only 27.25%, indicating that the solution obtained by CPLEX could be significantly improved.

Furthermore, the occupancy rate of chargers (utilization time over available time) was higher in CG. The occupancy rate of each charger, shown in Table 9, indicates that CG significantly outperformed CPLEX. The results demonstrate that many chargers optimized by the CG algorithm achieved 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%, were underutilized. This occurred 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.

6.2. Numerical Results

The numerical results highlight the effectiveness of the proposed model and the adopted strategy to achieve the optimal configuration for BEBs in transit operation. The 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 executed 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.

With buses running the assigned trips, the battery SoC decreased gradually. Combined with the ToU price mechanism, buses were partially charged between two trips to satisfy the electricity requirement. The BEB fleet operated with a lower battery SoC during on-peak electricity periods to minimize electricity costs, as depicted in Figure 6.

Figure 7 shows the number of chargers used across different periods. The results show that more chargers were used during the off-peak periods (higher utilization rates), leading to a reduced electricity cost.

Table 10. Comparisons of different vehicles and chargers.

Types of Vehicles	Types of Chargers	Fleet Composition Costs (CNY)	Purchase Cost of Chargers (CNY)	Electricity Costs (CNY)	Total Costs (CNY)
A, B, C	DC fast charging	29,600	15,000	4349.60	48,949.60
A, B, C	DC fast charging, I	26,400	15,500	4365.45	46,265.45
A, B, C	DC fast charging, II	25,100	15,000	4408.65	44,508.65
A, B, C	DC fast charging, I, II	25,100	13,400	4610.85	43,110.85

highlights four optimal system configurations using different vehicle types and charger powers. It is clear that considering a heterogeneous fleet and chargers led to cost reduction, with a maximum of 11.93%. This indicates that using multiple types of vehicles and chargers in the scheduling process is beneficial for service providers.

As shown in

Table 11. Vehicle-to-charger compatibility matching.

Charger ID	Charger Type	Vehicle Type A		Vehicle Type B		Vehicle Type C
		Number of Vehicles (#)	Daily Charging Duration (min)	Number of Vehicles (#)	Daily Charging Duration (min)	Number of Vehicles (#)	Daily Charging Duration (min)
1	DC	5	364	0	0	0	0
2	Type I	2	32	15	710	0	0
3		1	4	16	731	0	0
4	Type II	0	0	11	358	9	560
5		0	0	8	247	8	492
6		0	0	8	340	12	425
Total		8	400	58	2386	29	1477

, Type I and II chargers had higher utilization rates because they could support two types of vehicles. The results indicate that the proposed vehicle-to-charger compatibility optimization contributed significantly to enhancing the utilization of resources.

Implementing a partial charging strategy also contributed significantly to reducing system costs. Compared with the full charging strategy, as illustrated in

Table 12. Comparisons of total system costs.

Charging Strategy	Fleet Composition Costs	Purchase Costs of Chargers (CNY)	Electricity Costs (CNY)	Total Costs (CNY)
Partial Charging	25,100.00	13,400.00	4610.85	43,110.85
Full Charging	25,700.00	13,400.00	5613.45	44,713.45
Variation (%)	−2.33%	−0.00%	−17.33%	−3.58%

, it reduced the total costs by 3.58%. Particularly, the electricity costs were reduced by 17.33%, and the fleet composition costs were reduced by 2.33%.

Furthermore, the full charging strategy led to longer charging duration during peak electricity prices, as highlighted in Figure 8. The optimal charging schedule based on the partial charging strategy is depicted in Figure 9. The figure demonstrates that the charging events were relatively more concentrated in off-peak and mid-peak electricity demand periods.

Vehicles 10 and 18 were further analyzed to inspect their charging behavior, depicted in Figure 10. The results indicate that the vehicles were fully charged multiple times during operation under the full charging strategy. In comparison, and acknowledging the impact of ToU electricity price, the vehicles were fully charged after completing all the service trips under the partial charging strategy. The final service trip for Vehicle 10 was trip 328, and this paper did 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.

Taken together, these results indicate the effectiveness of adopting a partial charging strategy for the optimal configuration and utilization of battery electric transit systems.

6.3. Sensitivity Analysis

We further analyzed the influences of key parameters on the system configuration. Firstly, we analyzed the influences of the charging power of Type I and II chargers on the total costs, as shown in Figure 11. As these two parameters increased, the advantages of DC fast chargers gradually diminished, and more Type I and II chargers were adopted in the optimal scheme.

In addition, we analyzed the impact of Type C vehicle battery capacity on costs, as shown in Figure 12. The cost of a Type C vehicle was the lowest, and the battery capacity had the greatest impact on results. The results show that as the battery capacity of Type C vehicles increased, more Type C vehicles were used, decreasing fleet composition costs and electricity costs.

Charging compatibility is indeed one of the main constraints for BEB charging infrastructure allocation. As such, we analyzed the different hypothetical compatibility schemes between vehicles and chargers as follows:

(1) Scheme 1, all-vehicles all-chargers: vehicle Types A, B, and C could all be charged by DC fast, Type I, and Type II chargers.

(2) Scheme 2, single-vehicle-to-charger: vehicle Type A could be charged by DC fast chargers, vehicle Type B could be charged by Type I chargers, and Type II chargers could charge vehicle Type C.

(3) Scheme 3, constraint-vehicle-to-charger: this represented the proposed solution in this study and reflected the current market constraints.

The results in Figure 13 show that as chargers were compatible with more types of vehicles, the total costs could be effectively reduced. Expanding into practice and policy, we can suggest that bus operators not only consider costs, battery capacity, and charging power in operations and planning but also consider compatibility to reduce system costs and improve charging facility utilization.

6.4. Policy Implications

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 schedules to maximize 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.

Secondly, models and algorithms complement each other, and models that are more realistic also place higher demands on algorithms. Especially in the optimization of BEB transit systems, BEBs often encounter unexpected situations, such as abnormal weather. This requires prompt decisions to rearrange BEB scheduling schemes, which necessitates a short calculation time. Therefore, in the scheduling process, policymakers need to develop efficient and fast solving algorithms.

7. Conclusions

This study proposes a mixed-integer model to optimize electric bus charging infrastructure, fleet composition, and charging schedules. Particularly, the partial charging strategy, charging continuity, and the compatibility of chargers with vehicles are incorporated. We designed a CG algorithm to solve the model and used a real case study in Nanjing to validate the proposed method. The conclusions are as follows:

(i) The numerical experiment indicated that the joint optimization method could reduce the total costs by up to 11.93%, which verifies the effectiveness of the proposed model. Compared with the full charging strategy, the total costs and electricity costs under the partial charging strategy were reduced by 3.58% and 17.33%, respectively. The results show that the partial charging strategy could only use lower-price periods and reduce the electricity costs effectively.

(ii) The proposed CG algorithm was superior to CPLEX in solving quality and time for the BEB system optimization problem. Specifically, in the solution obtained by the CG algorithm, fewer chargers were used, and the utilization rate of chargers was higher, thereby reducing costs. By decomposing the proposed model into a master problem and subproblems, the solution time was significantly reduced.

(iii) Sensitivity analysis showed that the total costs of BEB transit systems are related to battery capacity and charging power. With the increase in the charging power of Type I and II chargers and Type C vehicles, more low-cost chargers and vehicles were adopted, decreasing total costs. In addition, as chargers were compatible with more types of vehicles, the total costs could be effectively reduced. The results demonstrate the importance of considering vehicle-to-charger compatibility in planning electric bus operation.

In future work, this study can be extended in the following directions: (i) 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.