Overnight charging scheduling of battery electric buses with uncertain charging time

Abstract

With the rapid development of battery electric buses (BEBs) in urban public traffic, it arises the problem of BEB charging scheduling, which aims to supply electric power for all the BEBs to meet the bus timetable in the smallest cost. Practical experience reports that both weather temperature and accumulative battery using time have a non-negligible impact on battery charging efficiency, and bring about the uncertainty of charging time of a battery. It may cause a negative influence to the departure schedule of the BEBs. Motivated by the above observation, this work investigates a BEB charging scheduling problem with uncertain charging time. The objective is to minimize the expected total charging cost, which consists of in-service cost, energy consumption cost and penalty cost due to over-low charging. We first prove the strong NP-hardness of the considered problem. A stochastic linear programming model is then established. A scenario-reduction based enhanced sample average approximation approach and an improved genetic algorithm are proposed to solve large-scale instances of the considered problem. Numerical experiments and comparisons with adapted previous algorithms are conducted to demonstrate the effectiveness of the proposed approaches.

Appendix A Calculation of the optimal charging level \(H^*_{j}(\omega )\) of one BEB

In this work, the theoretical optimal charging level of BEB $j \in {\mathcal {N}}$ can be calculated in the following two cases. Denote by ${\mathcal {C}}_j(\omega )$ the total charging cost contributed by BEB j in event $\omega$.

Case 1: $a_j\ge 80$.

$t_{jw}(\omega ) = 2\cdot \rho _j(\omega )\cdot w$ ($j\in {{\mathcal {N}}}$ and $\omega \in \Omega$), and then ${\mathcal {C}}_j(\omega )$ can be formulated as below.

$$\begin{aligned}&{\mathcal {C}}_j(\omega )=\left( 2\rho _j(\omega )\cdot p^u\cdot (c^e+c^s)-2\cdot c^p\cdot (ub-a_j) \right) \cdot H_j(\omega )\\&\qquad +c^p\cdot (H_j(\omega ))^2+c^p\cdot (ub-a_j)^2, \qquad {j\in {{\mathcal {N}}}}, \omega \in \Omega \end{aligned}$$

Case 2: $a_j< 80$.

In this case, we formulate ${\mathcal {C}}_j(\omega )$ ($j\in {{\mathcal {N}}}$ and $\omega \in \Omega$) in two sub-cases where $l_j\ge 80$ and where $l_j< 80$.

Case 2.1: $l_j\ge 80$.

As $H_j(\omega )+a_j(\omega )\ge 80$ in this case, we know that

$$\begin{aligned}&{\mathcal {C}}_j(\omega )=\rho _j(\omega )\cdot p^u\cdot (c^e+c^s)\cdot (2\cdot H_j(\omega )-80+a_j)\\&\qquad +c^p\cdot \left( ub-(H_j(\omega )+a_j)\right) ^2. \end{aligned}$$

Case 2.2: $l_j< 80$. In this case, the expression of formula ${\mathcal {C}}_j(\omega )$ depends on the value of $H_j(\omega )$. Specifically, we reformulate ${\mathcal {C}}_j(\omega )$ according to the value of $H_j(\omega )$ as below.

$$\begin{aligned} {\mathcal {C}}_j(\omega )=\left\{ \begin{array}{ll} \rho _j(\omega )\cdot p^u\cdot (c^e+c^s)\cdot H_j(\omega )\\ +c^p\cdot (ub-H_j(\omega )-a_j)^2, if \ H_j(\omega )+a_j \le 80;\\ \rho _j(\omega )\cdot p^u\cdot (c^e+c^s)\cdot (2\cdot H_j(\omega )-80+a_j) \\ +c^p\cdot \left( ub-(H_j(\omega )+a_j)\right) ^2, if \ H_j(\omega )+a_j > 80. \end{array}\right. \end{aligned}$$

Observation A.1

In any case, we can find the theoretical optimal charging level of BEB $j \in {\mathcal {N}}$ within interval $[\max \{l_j-a_j,0\},\min \{ub-a_j, \tau _{d_j-r_j}(\omega )\}]$ by calling the univariate quadratic function solver. Note that $\tau _{d_j-r_j}(\omega )$ denotes the maximum charging level that BEB j can achieve in the time period, i.e., time $r_j$ to time $d_j-r_j-(t^{in}+t^{off})-2\cdot \min \limits _{i\in {\mathcal {M}}}\{\frac{dis_{ij}}{v}\}$). Besides, no matter what solver is used, the obtained numerical result needs to be rounded because the charging level is set to be a positive integer.