Design of Electric Bus Transit Routes with Charging Stations under Demand Uncertainty

Abstract

This paper investigates the design problem of an electric bus (E-bus) route with charging stations to smooth the operations between E-bus service and charging. The design variables include the locations of E-bus stops, number of charging piles at charging stations, fare, and headway. A mathematical programming model is proposed to maximize social welfare in consideration of the uncertain charging demand at charging stations. The model solution algorithm is also designed. The model and algorithm are demonstrated on the E-bus route 931 in the city of Suzhou, China. The results of the case studies show that (i) the right number of stops on a bus route can contribute to the highest social welfare; (ii) the pile–bus ratio decreases with the increase of E-bus fleet size, thereby improving the E-bus charging efficiency at charging stations; and (iii) deploying charging stations at one end of a bus route can achieve a shorter waiting time for E-bus compared with deployment at two ends.

1. Introduction

Replacing traditional oil-fueled buses with electric buses (E-buses) is an inevitable trend for a more sustainable environment. The E-bus industry has entered a rapidly developing period. China mainland had 420,000 pure E-buses, accounting for 60% of total bus volume by the end of 2021 [1]. However, the charging facility planning on bus routes, specifically the proportion of charging piles to E-buses, varies greatly across cities or bus routes. The absence of a theoretical model results in inconsistent pile–bus ratios, for example, one pile per bus, one pile for four buses, or one pile for six buses, leading to un-reasonable charging pile configurations in stations [2]. Therefore, optimizing the bus route and its charging stations during the widespread adoption of E-buses is crucial for the development of charging infrastructure.

The main charging mode for electric buses (E-buses) is wired charging, which has two options: plug-in charging (e.g., at charging stations or bus terminals) [3,4,5] and battery-swapping charging [6,7,8]. Although less common, static and dynamic wireless charging [9,10,11] is also a viable option. The most widespread form of charging is plug-in charging, which is performed at charging stations near terminal stops [4,5]. Installing plug-in charging stations along bus routes is crucial for the widespread adoption of E-buses.

Previous research has focused on optimizing charging infrastructure planning. For example, Xylia et al. [12] proposed a linear programming model to optimize the location of charging stations in Stockholm. Liu and Song [13] introduced a mathematical programming model to jointly optimize the placement of power transfers and E-bus battery capacity, taking into account electricity consumption and travel time uncertainties. Lin et al. [14] developed a multistage programming model to optimize both the locations of charging stations and the strategies for planning the E-bus charging infrastructure, considering the temporal aspect of infrastructure planning over the long term. However, most previous studies have focused primarily on the optimization of charging station locations, disregarding the effect of time-varying electricity prices on the E-bus charging schedule [12,15]. The location optimization models aim to minimize the cost of the charging system but do not consider E-bus service and charging schedules, which cannot minimize the overall cost of construction and operation. Therefore, joint optimization models are necessary to minimize the total cost of construction and operation.

The field of E-bus charging infrastructure planning and scheduling is still ongoing, with various studies focused on different aspects. Rogge et al. [16] proposed a method to optimize the fleet size and schedules of service and charging to ensure E-bus service and charging requirements. Wang et al. [17] analyzed patterns of E-bus service and charging from a streaming dataset, and then designed a dynamic electricity pricing model to minimize the total cost of E-bus service and charging. Houbbadi et al. [18] formulated the problem of E-bus overnight charging to determine the charging strategy that minimizes battery aging cost. An [19] created an integer programming model and a Lagrangian Relaxation algorithm to optimize E-bus fleet size and battery-swapping station locations under uncertainty in charging demand and time-varying electricity prices. Liu and Ceder [20] developed an integer programming model to minimize E-bus queues and charging station pile numbers and created lexicographic and adjusted max-flow algorithms to solve the model. Zhang et al. [21] proposed a programming model to minimize passenger wait time and operation costs by optimizing E-bus service schedules and short-turn strategies on main and short-turn lines. To minimize total costs for the E-bus fleet, charging and maintenance at the charging station, and charging infrastructure, Zhang et al. [22] presented a bilevel programming model, where the upper- and lower-level models optimize the number of piles and E-bus charging schedule, respectively. The model was solved using a column generation algorithm combined with a Tabu search approach. Despite the existing literature on charging station locations and strategies, the E-bus route length, number of bus stops, fleet size, service headway, etc., are often ignored. Hence, joint optimization of E-bus route service and charging infrastructure deployment is necessary.

Recent studies in the field of electric bus and charging infrastructure integration have considered various technical, economic, and operational factors to optimize the deployment of electric bus charging stations. The joint design model of electric bus routes and charging stations has also been explored in some studies. Iliopoulou and Kepaptsoglou [23] presented a mathematical model to optimize the deployment of charging stations along transit routes for electric buses, taking into account factors such as charging time, bus travel time, and station capacity to achieve a sustainable and efficient electric bus system. Zhang, Zhao, and Song [24] proposed an integrated optimization model that considers both transit route network design and charging station planning for battery electric buses. The aim was to minimize the total cost of charging infrastructure and bus travel time while considering charging requirements and bus travel times. Liu et al. [25] presented a multi-objective optimization algorithm that balances the cost of charging infrastructure and bus travel time to achieve a more efficient and sustainable electric bus system. Given that electric buses take a long time to charge, there will be queues at charging stations, affecting bus scheduling. Tzamakos, Iliopoulou, and Kepaptsoglou [26] focus on optimizing the location and capacity of charging stations while considering potential waiting times. Uslu and Kaya [27] presented a mathematical model to optimize the location and capacity of charging stations, aiming to minimize the total waiting time. Iliopoulou and Kepaptsoglou [28] proposed a robust optimization model that considers uncertainty in the electric transit route network design problem, ensuring a resilient and flexible electric bus system. Due to charging failure, bus anchoring, and other factors, the bus service can be unreliable. Alwesabi et al. [9] presented a robust optimization model for the deployment of dynamic wireless charging infrastructure, considering sources of uncertainty such as demand fluctuations, charging station failures, and electric bus breakdowns. He, Liu, and Song [29] proposed an integrated optimization model that balances the cost of charging infrastructure, charging efficiency, and bus travel time to achieve a more efficient and sustainable electric bus system. In conclusion, these studies emphasize the importance of considering various factors and uncertainties in the integration of electric buses and charging infrastructure.

There are three research gaps to be filled. First, a charging station consists of charging piles, which directly supply the charging service for E-buses. The number of charging piles is affected by the E-bus parking area and the power input at the charging stations. Most previous literature does not address charging piles, leading to a lack of a comprehensive charging station planning scheme that meets the needs of charging infrastructure construction. Second, the shorter service headway and the longer route increase the E-bus fleet and charging piles, resulting in higher electricity consumption. Thus, the locations and size of charging stations, the E-bus charging strategy, the number of bus stops, and service headway should be optimized in a model framework. Third, the E-bus waiting times at charging stations increase with the E-bus service time along a route, thus reducing the fleet turnover rate and thus requiring a larger E-bus fleet. The queue service process at the charging station needs to be analyzed.

Our paper focuses on the design of E-bus routes with charging stations in the face of demand uncertainty and aims to make the following main contributions to the literature:

• We develop a social welfare maximization model to optimize the E-bus charging infrastructure, charging schedules, and a bus service design (the combination of the number of bus stops, station location (or spacing), headway, and fare). Passenger travel demand between any pair of bus stops on an E-bus route and the limited driving ranges are both explicitly considered.
• The queue service process during charging operations at charging stations is analyzed to explicitly investigate the expected E-bus queue and waiting times.
• A heuristic algorithm is proposed to determine the optimal E-bus charging stations and bus route design. The impact of charging station setup (single or double end of the bus route) on social welfare, queues, and waiting times at charging stations is analyzed in this paper.

We organize the paper as follows. Section 2 makes the modeling assumptions and formulates the required E-bus service, charging constraints, and social welfare maximization. Then, we develop a heuristic algorithm to solve the decision variables for the E-bus route and charging station in Section 3. This is followed by an example illustrating the model application in Section 4 and some conclusions in Section 5.

2. Model Formulation

2.1. Assumptions

We make the following basic assumptions to facilitate formulating the E-bus route with charging stations.

A1.

We assume that the arrival of E-buses at each charging station follows a Poisson process due to the influence of traffic conditions and passenger flow on the speed and dwell time of E-buses at stops. [30]. The E-bus charging time is assumed to follow an exponential distribution because the E-bus remaining power is affected by traffic conditions. The significant variation in the energy consumption and arrival at charging stations of E-buses cannot be ignored [13,31,32,33]. Such assumptions are also made by Benoliel et al. [34] and Liu et al. [35].

A2.

We assume that the relationship between passenger demands and the quality of E-bus route service is linear. Such an assumption is similar to [36,37].

A3.

We assume the passengers board E-buses at the bus stops closest to their current location. The assumption has also been widely accepted by Wirasinghe et al. [38], Kuah et al. [39], Chien et al. [40], and Li et al. [41].

A4.

The battery charge and consumption are proportional to the charging time and the mileage, respectively.

To facilitate the model formulation, we first list all notations used in the modeling process, as shown in

Table 1. Notation and parameters in this formulation.

Notation	Detailed Definition
Subscripts and function
N	The number of E-bus stops
f	The fare
$g(x)$	Passenger density at a distance x from the terminal stop
$q(x,s)$	Passenger density at stop s at distance x
$u_s\left(x\right)$	Passenger access time from location x to stop s
W	Social welfare
G	The sum of the passenger surplus
π	E-bus operator profit
$G\left(x,s\right)$	The surplus of passengers at location x boarding from stop s
$G_s$	The passenger surplus at stop s
R	Total revenue from passengers (Operation revenue)
C	Total construction and operation cost for the E-bus route and charging stations
$\gamma$	Lagrange multipliers
Input parameters
B	The length of the route
Ds	Distance between stop s and the terminal stop
S	The s-th E-bus stop
$w_s$	Expected passenger dwelling time at stop s
$V_b$	Gap between the E-bus maximum speed $V_{b\max }$ and the actual speed $V_{b1}$
$V_{b\max }$	E-bus maximum speed
$V_{b1}$	E-bus actual expected speed
H	E-bus service headway
F	E-bus fleet size
w1, w2	Expected waiting times at charging stations on two ends of the bus route
Θ	E-bus delay on the route, including the E-bus dwelling times at stops and travel time at the route
T0	Waiting time at terminal stops
ζ	The number of waits at terminal stops
T11, T12	E-bus travel time on the route and total dwell times at the stops 1 to N
β0	Expected E-bus dwell time at each stop
${\mu }_e$	Unit electricity cost for unit travel distance
${\Delta }_0$	Fixed cost
η	Value of waiting times at both charging stations
S	E-bus maximum range
Decision variables
$l_s$	Demarcation point between stop s and s + 1
$L_s$	Distance between the passenger demarcation point $l_s$ and terminal stop
x	Distance from terminal stop
$e_w$	Sensitivity parameters for dwelling time
$e_a$	Sensitivity parameters for access time
$e_f$	Sensitivity parameters for fare
$e_v$	Sensitivity parameters for speed gap
$Q_s$	The number of passengers at each stop
$C_b$	Total cost consists of the E-bus cost
$C_o$	Operation cost
$C_e$	Charging station and pile costs
$C_w$	Sojourn cost at charging stations
Vt	E-bus speed
$G_s$	The passenger surplus at stop s
λ	The arrival rate of E-bus
μ	Charging pile service E-bus rates
ci	The number of charging piles
ξi	E-bus charging proportions at two ends of the route
$P_n^i$	Probability of n E-buses at charging station i
${\rho }_i$	Service intensity at charging station i

.

2.2. Passenger Demand

Consider an E-bus route consisting of N stops, with a total length of B, as shown in Figure 1. Let $D_s$ denote the distance between stop s and the terminal stop. Let l_s represent the demarcation point between stop s and s+1, $L_s$ is the distance between the passenger demarcation point $l_s$ and terminal stop, and $g(x)$ the passenger density at a distance x from the terminal stop. By A3, the demarcation point $l_s$ is the midpoint of the route segment (s, s + 1). Thus, we have

$$L_s=\frac{D_s+D_{s+1}}{2},\forall s=1,2,\dots N-1$$

(1)

where $D_1=0,L_N=D_N$. For convenience, let L₀ be zero.

Let $q(x,s)$ represent the passenger density at stop s at distance x, which can be written by A2 as follows.

$$q\left(x,s\right)=g(x)\left(1-e_a{u}_s\left(x\right)-e_w{w}_s-e_{f}f-e_v{V}_b\right),\forall x\in \left[0,B\right]$$

(2)

where $u_s\left(x\right)$ is the passenger access time from location x to stop s; $w_s$ the expected passenger dwelling time at stop s, f the fare; $V_b$ the speed gap between the E-bus maximum speed $V_{b\max }$ and the actual expected speed $V_{b1}$; and $e_w$, $e_a$,$e_f$, and $e_v$ the sensitivity to the dwelling time, access time, fare, and speed gap, respectively.

The following condition should be satisfied to guarantee the passenger demand cannot be less than zero:

$$0\le 1-e_a{u}_s\left(x\right)-e_w{w}_s-e_{f}f-e_v{V}_b\le 1,\forall x\in \left[L_{s-1},L_s\right]$$

(3)

We now specify the components of Equation (2). The access time $u_s\left(x\right)$ is the time the passengers walk from location x to stop s. Thus, we have

$$u_s\left(x\right)=\{\begin{array}{l}\{\left(D_s-x\right)/V_a,\forall x\le D_s\\ \{\left(x-D_s\right)/V_a,\forall x>D_s\end{array}$$

(4)

According to [37], the expected passenger dwell time at stop s, $w_s$, can be written as

$$w_s=H/2$$

(5)

where H denotes the E-bus service headway.

The number of stops can affect actual E-bus speed. Suppose the number of stops β is fixed, then the E-buses speed limit is $V_{b\max }$. Then, we can obtain the actual E-bus speed as follows:

$$V_{b1}=\frac{B{V}_{b\max }}{B+\beta N{V}_{b\max }}$$

(6)

Hence, the speed gap can be written as

$$V_b=\frac{\beta N{V}_{b\max }^2}{B+\beta N{V}_{b\max }}$$

(7)

The number of passengers at each stop $Q_s$ is

$$Q_s={\displaystyle {\int }_{L_{s-1}}^{L_s}q\left(x,s\right)}dx,\forall s=1,2,\dots ,N$$

(8)

Substituting Equations (2)–(7) into Equation (8), $Q_s$ can then be rewritten as

$$Q_s=\left(1-e_w\alpha H-e_{f}f-e_v{V}_b\right){\displaystyle {\int }_{L_{s-1}}^{L_s}g\left(x\right)}dx-\frac{e_a}{V_a}\left({\displaystyle {\int }_{L_{s-1}}^{D_s}g\left(x\right)\left(D_s-x\right)}dx+{\displaystyle {\int }_{D_s}^{L_s}g\left(x\right)\left(x-D_s\right)}dx\right)$$

(9)

2.3. Social Welfare

2.3.1. Passenger Surplus

As a public good, the E-bus service should maximize its social welfare, which is the sum of the passenger surplus, G, and the E-bus operator profit, π. Thus, we have

$$W=G+\pi$$

(10)

The passenger surplus is the gap between the costs the passengers would voluntarily pay and actually pay. The cost voluntarily paid by the passengers is integral of the inverse function of demand density over demand density. We can convert the demand density function of Equation (2) into the inverse of the demand density function in terms of fare. This yields

$$q^{-1}\left(x,s\right)=\frac{1}{e_f}\left(1-e_a{u}_s\left(x\right)-e_w{w}_s-e_v{V}_b\right)-\frac{q\left(x,s\right)}{e_{f}g\left(x\right)}$$

(11)

The surplus of passengers at location x boarding from stop s, $G\left(x,s\right)$, can be denoted by

$$\begin{array}{ll}G(x,s)& {\displaystyle ={\int }_0^{q\left(x,s\right)}\left[\frac{1}{e_f}\left(1-e_a{u}_s\left(x\right)-e_w{w}_s-e_v{V}_b\right)-\frac{z}{e_{f}g\left(x\right)}\right]}dz-q\left(x,s\right)f\\ & =\frac{g\left(x\right)}{2e_f}{\left(1-e_a{u}_s\left(x\right)-e_w{w}_s-e_{f}f-e_v{V}_b\right)}^2\end{array}$$

(12)

Hence, the passenger surplus $G_s$ at stop s is

$$G_s={\displaystyle {\int }_{L_{s-1}}^{L_s}G\left(x,s\right)dx},\forall x\in \left[L_{s-1},L_s\right]$$

(13)

Consequently, we can obtain the passenger surplus of the E-bus route:

$$G={\displaystyle \sum _{s=1}^N{G}_s}$$

(14)

2.3.2. Operator Profit

The operator profit π can be defined as the total revenue from passengers, R, minus the total construction and operation cost for the E-bus route and charging stations, C. Hence,

$$\pi =R-C$$

(15)

The operation revenue R is the total fare paid by all passengers boarding at each stop. Thus, we have

$$R={\displaystyle \sum _{s=1}^{N}f{Q}_s}$$

(16)

The total cost consists of the E-bus cost $C_b$, operation cost $C_o$, charging station and pile costs $C_e$, and sojourn cost at charging stations $C_w$. Thus, we have

$$C=C_b+C_o+C_e+C_w$$

(17)

The E-bus cost $C_b$ is the cost of the E-bus fleet. Thus, we have

$$C_b=c_b\cdot F$$

(18)

where F is E-bus fleet size, which is the quotient that sum of the E-bus travel time along the route and the expected waiting times divided by service headway. Thus, we have

$$F=\frac{\Theta +w_1+w_2}{H}$$

(19)

where w₁ and w₂ are the expected waiting times at charging stations on two ends of the bus route; Θ is the E-bus delay at the route, including the dwelling time at the terminal stops, travel time at the route, and E-bus standing time at stops, which can be denoted as

$$\Theta =\zeta T_0+2(T_{11}+T_{12})$$

(20)

where T₀ denotes the waiting time at terminal stops, and ζ the number of waits at terminal stops; T₁₁ and T₁₂ are the E-bus travel time on the route and total dwell times at the stops 1 to N, which can be written as T₁₁ = D_N/V_t and T₁₂ = β₀N. Here, V_t is the E-bus speed, and β₀ is the expected E-bus dwell time at each stop. The number “2” denotes that the E-bus runs back and forth on the route.

The E-buses consuming electricity incurs operation costs $C_o$. By assumption A4, we have

$$C_o=2{\mu }_e{D}_N$$

(21)

where ${\mu }_e$ denotes the unit electricity cost for unit travel distance.

Charging station cost C_e consists of fixed and variable costs. Thus,

$$C_e=c\Delta +{\Delta }_0$$

(22)

where c and Δ are the number and unit price of charging piles, and ${\Delta }_0$ is the fixed cost.

E-bus sojourn cost $C_w$ is the product of the total waiting time and the value of waiting times η at both charging stations. Therefore,

$$C_w=\eta (w_1+w_2)$$

(23)

The E-bus arrival rate at the charging station reflects that they need to charge after traveling their maximum range. Thus, we have

$$\lambda =\frac{2D_N}{SH}$$

(24)

where S is the E-bus maximum range.

Once running out of power, the E-buses come to charging stations to recharge, with the arrival rate of E-bus at the charging station λ, and a charging pile serves E-buses at the rate of μ. Both the charging demands and services follow the Poisson and exponential distributions because the E-buses operate and charge in uncertainty, according to A1. The source in this model is finite because only a limited number of E-buses F on the bus route can generate charging demand. Once all E-buses run out of power, no new demands for charging can occur. Thus, the charging service processes at the charging stations can be treated as a machine servicing queuing model.

The charging station locations directly affect the E-bus service efficiency and queue lengths. The charging station deployments have two cases, where the charging piles are deployed at a single end and two ends of the E-bus route, respectively. Let c_i (i = 1, 2; c = c₁ + c₂, and c_i = 0, 1, 2, …) and ξ_i (i = 1, 2; and ξ_i = c_i/c) denote the number of charging piles and the E-bus charging proportions at two ends of the route. If either c₁ or c₂ is equal to zero, then a charging station is built only at one end of the bus route; otherwise, two ends construct a charging station. Then, the E-bus arrival rate at each charging station is ${\lambda }_i=2{\xi }_i{D}_N/(SH)$ (i = 1, 2). The probability of n E-buses sojourning at charging station i can be written as

$$P_0^i=\frac{1}{F!}\frac{1}{{\displaystyle \sum _{k=0}^{c_i}\frac{1}{k!\left(F-k\right)!}{\left(\frac{c_i{\rho }_i}{F}\right)}^k+\frac{c_i{}^{c_i}}{c_i!}{\displaystyle \sum _{k=c_i+1}^F\frac{1}{\left(F-k\right)!}{\left(\frac{{\rho }_i}{F}\right)}^k}}}$$

(25)

$$P_n^i=\{\begin{array}{l}\frac{F!}{\left(F-n\right)!n!}{\left(\frac{{\lambda }_i}{\mu }\right)}^n{P}_0^i,1\le n\le c_i\\ \frac{F!}{\left(F-n\right)!c_i!c_i{}^{n-c_i}}{\left(\frac{{\lambda }_i}{\mu }\right)}^n{P}_0^i,c_i\le n\le F\end{array}$$

(26)

where n is the number of E-buses at each charging station; $P_n^i$ indicates the probability of n E-buses at charging station i; ${\rho }_i$ is service intensity at charging station i; and ${\rho }_i={\xi }_i{\lambda }_i/c_i\mu$.

Based on Equations (16)–(24), the total cost is

$$C=c_b\cdot F+{\Delta }_0+c\Delta +2{\mu }_e{D}_N+\eta {\displaystyle \sum _{i=1}^2{w}_i}$$

(27)

where we can yield $w_i={\displaystyle \sum _{n=1}^{F}n}{P}_n^i/{\lambda }_i\left(F-{\displaystyle \sum _{n=1}^{F}n}{P}_n^i\right)$ by using Little’s formula.

Substituting Equations (14)–(16) and (27) into Equation (10) yields social welfare of the E-bus route and charging stations as follows:

$$W={\displaystyle \sum _{s=1}^N{G}_s}+{\displaystyle \sum _{s=1}^{N}f\cdot Q_s}-\left(c_b\cdot F+{\Delta }_0+c\Delta +2{\mu }_e{D}_N+\eta W_s\right)$$

2.4. Social Welfare Model

The previous analysis can facilitate the formulation of the social welfare maximization model:

$$W\left(D_s,H,f\right)={\displaystyle \sum _{s=1}^N{G}_s}+{\displaystyle \sum _{s=1}^{N}f\cdot Q_s}-\left(c_b\cdot F+{\Delta }_0+c\Delta +2{\mu }_e{D}_N+\eta W_s\right)$$

(28)

$$\mathrm{s}.\mathrm{t}.\hspace{1em}\hspace{1em}H\le \mathrm{K}/{\displaystyle \sum _{s=1}^N{Q}_s}$$

(29)

$$L_s^i\le \overline{L^i}$$

(30)

$$\max \left\{w_1,w_2\right\}\le H$$

(31)

$$2D_N/(H{V}_{b\max })+{\displaystyle \sum _{i=1}^2{L}_s^i}=F$$

(32)

where K is the rated passenger capacity for an E-bus, and constraint (29) ensures that the E-bus service should satisfy the passenger demand. Let $L_s^i={\displaystyle \sum _{n=1}^{F}n{p}_n^i}$ denote the expected number of E-bus at charging station i; constraint (30) guarantees that the expected number of E-bus is less than the parking capacity of charging station i. Constraint (31) ensures that the larger E-bus waiting time at the charging station should be less than the route service headway, avoiding the E-buses bunching at the charging stations and no E-buses existing on the route. Moreover, all E-buses exist on the route and at both charging stations, and constraint (32) is the principle of E-buses conservation.

3. Solution Algorithm

This section proposes an algorithm to solve the optimal E-bus route and charging station designing problem. By setting the partial derivatives of the Lagrangian function with respect to decision variables equal to zero yields the following system of Equation (33), from which the optimal solutions, including the station location, service headway, number of charging piles at each charging station, and fare, can be solved. The procedure of the heuristic algorithm is summarized as follows.

$$\{\begin{array}{l}{\displaystyle \sum _{i=s-1}^{s+1}\frac{\partial G_i}{\partial D_s}}-\delta \left(2{\mu }_e-\frac{2{\gamma }_4}{H{V}_{\max }}\right)=0,\\ H=\sqrt{\frac{\left({\gamma }_1\mathrm{K}{V}_{\max }-2{\gamma }_4{D}_N\right)e_f}{V_{\max }\left(e_f(f-{\gamma }_1){\displaystyle {\int }_{Ls}^{Ls-1}q(x,s)dx}-{\gamma }_3{e}_f+\alpha e_w{Q}_s\right)}},\\ f-{\gamma }_1=0,\\ {\gamma }_1\left(\frac{\mathrm{K}}{H}-{\displaystyle \sum _{s=1}^N{Q}_s}\right)=0,\\ {\gamma }_3\left(\max \{w_1,w_2\}-H\right)=0,\\ {\gamma }_1\ge 0,{\gamma }_2\ge 0,{\gamma }_3\ge 0,{\gamma }_4\ge 0\end{array}$$

(33)

Step 1. Set the initial Lagrange multipliers ${\gamma }_1^{(0)}$, ${\gamma }_2^{(0)}$, ${\gamma }_3^{(0)}$, and ${\gamma }_4^{(0)}$ to zero and choose initial values for $D_s^{\left(0\right)}$, $H^{\left(0\right)}$, and $f^{\left(0\right)}$, then calculate $Q_s^{\left(0\right)}$ by Equations (9) and (13) and set j = 1.

Step 2. Update the design variables and Lagrange multipliers.

Step 2.1. Update $H^{\left(j\right)}$ with fixed $D_s^{\left(j-1\right)}$ and $f^{\left(j-1\right)}$. Check whether $H^{\left(j\right)}$ satisfies Equations (29)–(32), then $H^{\left(j\right)}$ is set at the associated bound if it does not satisfy the constraints.

Step 2.2. Update $f^{\left(j\right)}$ with fixed $D_s^{\left(j-1\right)}$ and $H^{\left(j\right)}$. Check whether $f^{\left(j\right)}$ satisfies Equation (3), then $f^{\left(j\right)}$ is set at the associated bound if it does not satisfy the constraints.

Step 2.3. Update $D_s^{\left(j\right)}$ with fixed $H^{\left(j\right)}$ and $f^{\left(j\right)}$. Check whether $D_s^{\left(j\right)}$ satisfies Equations (3) and (32), then $D_s^{\left(j\right)}$ are set at the values of associated bounds if they do not satisfy the constraints.

Step 3. Update $Q_s^{\left(j\right)}$ and $G_s^{\left(j\right)}$ by Equations (9) and (13), then update Equation (28). The value of $W^{\left(j\right)}$ can be calculated by Equation (28).

Step 4. Terminate the algorithm and output the optimal solution $\left\{D^{*},H^{*},f^{*}\right\}$, $W^{*}$, and F by Equation (21) if values of $W^{\left(j\right)}$ are sufficiently similar; otherwise, set j = j + 1, and proceed to Step 2.

The above algorithm can find the solution of continuous variables, given the fixed E-bus stops and charging piles. The presence of two integer variables makes the mixed integer programming problem challenging to solve. However, the numbers of stops and charging piles on a bus route are limited. Comparing the resulting social welfare for the different numbers of stops and charging piles can find the optimal E-bus stops and charging piles. Moreover, Step 2 sequentially updates the f, H, D_s, and Q_s while fixing the other variables. After updating each decision variable, one should check the associated constraints to guarantee that solutions at each iteration can always meet all constraints.

4. Numerical Experiment

4.1. Description of the Test Case

This section uses the 931 E-bus route in the city of Suzhou to demonstrate the model effectiveness and the algorithm and illustrates the contributions of this paper. As shown in Figure 2, the bus route consists of 23 stops on a 12.4 km route between the Xinzhuang hub and Nanhuang Bridge. The passenger density, referring to that at the end of 2022 in Suzhou, is set at 1124 pax/km.

Table 2. Notation and the value.

Notation	Definition	Value
$e_a$	Sensitivity to passenger access time (1/h)	0.98
$e_w$	Sensitivity to E-bus dwelling time at each stop (1/h)	0.98
$e_f$	Sensitivity to E-bus fare (1/¥)	0.098
$V_a$	Passenger walking speed (km/h)	4
$\beta$	Expected passenger boarding and alighting time at each stop (s)	30
$c_b$	E-bus price (¥/veh)	2,000,000
${\mu }_e$	E-bus electricity cost per unit mileage (¥/km)	2
${\Delta }_0$	Fixed cost of charging station (¥)	5,000,000
$\Delta$	Price of each charging pile (¥/item)	100,000
$\eta$	Value of E-bus waiting time at charging station (¥/ h)	20
$S$	E-bus maximum range (km)	100
$\mu$	Service rate of charging station (veh/h)	0.5
K	Rated passenger capacity for E-bus (pax/veh)	70
$\overline{L^i}$	Parking capacity at charging station (veh)	20
$V_{b\max }$	E-bus speed limit (km/h)	40

lists the other parameters.

The proposed solution algorithm was coded in Matlab and run on a PC computer with a 2.6 GHz CPU and 16G RAM. In the numerical experiment, we see that all the cases can be solved to very tight convergence precision (set as ${10}^{-6}$) of less than 24 min because the algorithm in Section 3, Steps 1 to 4, has to be re-executed 900 times, i.e., the combinations of the product of 30 E-bus and 30 stations. To the best of our knowledge, there is no prior literature reporting computational results for solving the nonlinear mixed intenger programming model to optimize so many decision variables in the E-bus and charging station design problem with so many combinations. Therefore, we can state that our solution algorithm is a promising approach to obtain good qualtiy solutions in a short computation time.

4.2. Result Analysis

4.2.1. Sensitivity of the Number of Stops

This section investigates the sensitivity of the number of stops to social welfare. We look for optimal numbers of stops by comparing the resulting values of social welfare for different E-bus fleet sizes and charging pile deployments. Figure 3 shows that the optimal social welfare increases first and then decreases with the E-bus stops, reaching the social welfare cap of ¥ 21,972 with the associated fare of 0.63 yuan, headway of 4.3 min, fleet of 27 E-buses, and charging piles of 7 at each charging station. Figure 3 demonstrates that the optimal number of stops along a bus route can lead to the highest overall social benefit. Previous studies have not found a relationship between the number of electric bus stops and social welfare or its maximum value. This finding is a significant advancement, in our understanding.

Figure 3 also shows that the increase in the number of bus stops means that E-buses must make more stops, which extends their operating time and increases the time cost for passengers. On the other hand, this requires a larger fleet of E-buses, leading to increased public transportation investment. As a result, an excessive number of bus stops can result in a decrease in overall social welfare.

Figure 4 displays the relationship between the optimal fare, E-bus fleet size, service headway, and the number of charging piles at each charging station with the number of stops on the route. In the figure, the fleet size and the number of charging piles increase, and the service headway reduces slightly with the number of stops on the route. The fleet size increases faster than charging piles. Additionally, as the fare is the Lagrange multiplier with capacity constraints, defined in Equation (33), some optimal E-bus fares are 0 when the capacity constraint is inactive but positive when it is active. Figure 4 shows there are 0 fares for routes with 14, 18, and 20 stops. It is advisable to avoid designing bus routes with fares of 0.

4.2.2. Effects of Pile-Bus Ratio

The planning of charging piles for E-buses is arbitrary due to the lack of design specifications. There are various pile–bus ratios, e.g., a pile to one E-bus or a pile to ten E-buses, resulting in a mismatch between charging piles and E-buses. Therefore, this section discusses the effect of the pile–bus ratio on charging waiting time.

Figure 5 represents the relationship between the effect of pile–bus ratio (F/c) on charging waiting time. The figure shows that as the pile–bus ratio increases, the rate of queue length reduction is reduced, and more E-buses lead to a shorter waiting time in queue, given the identical pile–bus ratio. The reason is that the total electricity consumed by the E-bus fleet and the number of charging times are fixed; the arrival rate of charging E-buses decreases with fleet size, thus shortening the E-bus waiting times at the charging stations. According to Figure 6, when the E-bus fleet size is greater than 30, 25, 20, and 15, the waiting time tends to be stable if the pile–bus ratio is larger than 1/6, 1/5, 1/4, and 1/3, respectively. Thus, we can derive the pile–bus ratio for recommendation as follows: if the E-bus fleet size is greater than 30, 25, 20, and 15, then the pile–bus ratio should be 1/6, 1/5, 1/4, and 1/3, respectively. Therefore, the pile–bus ratio decreases with the E-bus fleet size, improving the charging station efficiency.

4.2.3. Effects of Charging at Single or Two Ends

This section investigates the effects of charging at single or two ends of the bus route. Figure 6 plots the behaviors of E-bus queue lengths with the number of charging piles, fleet size, and departure intervals for charging at a single end or two ends. From the three figures, it can be seen that the queue length for charging at one end is consistently shorter than that for charging at two ends, regardless of the number of charging piles, fleet size, or departure intervals. Under the assumption of identical fares, numbers and locations of stations, and passenger demand, deploying all charging piles at one end of an E-bus route (with only one charging station) results in shorter waiting times and queue lengths compared with deploying charging piles at two ends. This is because the charging stations deployed at one end facilitate the charging piles to be shared, thus causing a shorter waiting time and queue length. This insight provides the opportunity to deploy charging piles at the end of the route with sufficient land, thereby satisfying the charging demand for E-buses while minimizing excessive land use in urban areas.

5. Conclusions

In this paper, we proposed a mathematical programming model to optimize the design variables for E-bus service and charging in an E-bus route. An algorithm was developed to optimize the design variables, such as the number of stops, fares, service headway, fleet size, and charging piles. The paper analyzed the effects of the number of bus stops, pile–buses ratio, and charging stations at only a single end and at two ends. We demonstrated the effectiveness of the proposed model on route 931 in Suzhou. Some important insights can be found. (i) The right number of stops on a bus route can contribute to achieving the most social welfare. (ii) The pile–bus ratio decreases with the increase of E-bus fleet size, improving the E-bus charging efficiency at charging stations. (iii) Deploying charging stations at one end of a bus route can achieve a shorter waiting time for E-bus than deploying stations at two ends. These insights can support the decision-making on the planning of the E-bus route with charging stations.

The proposed model of this paper focused on a single E-bus route, and some important elements in the E-bus service system were ignored and should be investigated in future studies. First, the proposed model did not involve the E-bus network, which involves the locations of charging stations and the allocation of charging piles. Thus, one of our future research directions is to optimize the locations of charging stations and the allocation of charging piles in an E-bus network. Second, the proposed paper assumed the passenger demand at stations over time (i.e., the arrival rates of passengers) are constant, and the dynamics of the passenger flow within the E-bus route are ignored. The dynamic passenger demand may affect the E-bus electricity consumption on the route, leading to the peak and off-peak charging demand and then affecting the deployment of charging infrastructure. Extending the analysis from a static travel demand pattern to a dynamic one is an important research direction for future studies. Third, battery technology for electric vehicles is improving in terms of charging times, power density, longevity, safety, and cost. Such factors may affect the planning of E-bus routes and networks and charging infrastructure, which can be another research direction for further study.