1. Introduction

2. Literature Review

3. Electric Vehicle Routing Problem

3.2. Matematical Model

We consider an electric vehicle routing problem with time windows. There are one depot, multiple vehicles, and multiple customer sites. The model for the problem is as follows.

Sets

$V=\left\{0, 1, 2, \dots , n, n+1, \dots , n+p\right\}$: (0: depot node, 1 ~ $n+p$: nodes)

$F=\left\{n+1, \dots , n+p\right\}$: Charging stations

$F\prime$: Dummy charging stations

$N=V\backslash \left\{\left\{0\right\}{\displaystyle \cup }F\right\}$: Customer sites

$A=\left\{\left(i,j\right)| i, j\in V, i\ne j\right\}$: Arcs

$M=\left\{1,\dots ,m\right\}$: Vehicles

$V^{+}\left(i\right)=\left\{j\in V| \left(i,j\right)\in A\right\}$: Successor nodes of node $i$

$V^{-}\left(i\right)=\left\{j\in V| \left(j,i\right)\in A\right\}$: Predecessor nodes of node $i$

Parameters

$\left[a_i, b_i\right]$: Time window for node $i$

$s_i$: Service time for node $i$

$q_i$: Demand for node $i$

$w_i$: Difficulty of using the charging station $i$

$d_{i,j}$: Travel time from node $i$ to node $j$

$\mathsf{\tau }$: Charging time per unit energy

$\mathsf{\delta }$: Energy consumption per unit distance

$\mathrm{K}$: Capacity of a vehicle

$\mathrm{Q}$: Capacity of energy(electricity) for a vehicle

$L$: Large constant number

Variables

$x_{i,j}^m$: Binary variable, 1 if arc $\left(i, j\right)\in A$ belongs to the optimal routes by vehicle $m$, 0 otherwise

$t_i^m$: Arrival time of vehicle $m$ at node $i$ (beginning of service)

$y_i^m$: Energy level of vehicle $m$ at node $i$

$z_i$: Binary variable, 1 if charging station $i$ is used, 0 otherwise

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

The objective function (1) aims to minimize total travel distances and difficulty of visiting charging stations. The second term plays a role as the availability. If a charging station is easy to access, the value of $w_i$ will have small value. That is the state of each charging station. Constraint (2) ensures that each customer site is served once by one vehicle. Constraint (3) presents that each charging station can be used at most once. Constraints (4) and (6) enforces that each vehicle has to leave and return to the depot. Constraint (5) presents that once a vehicle enters a node, the vehicle has to leave the node except the depot. Capacity of each vehicle is in constraint (7). Constraints (8) and (15) are time windows constraints for customer nodes. Constraint (9) denotes the constraint for time of charging. Constraints (10) and (11) present energy consumption through the arc. Constraint (12) forces each vehicle has to visit a charging station once. Constraints (13) and (14) ensure a charging station is open only if a vehicle travels the charging station. Constraint (16) is for binary variables.

If $w_i$ has large value, the charging station $i$ will have low possibility to be chosen by a vehicle. Thus, depending on the value, we can control the availability of the charging station. The model assumes that the capacity of the battery is limited, so each vehicle has to visit a charging station during a delivery tour. In the problem of two time periods, we solve the problem of electric vehicle routing with the above EVRP model. Depending on the locations of customer sites and available charging stations, decisions will be made if the vehicles visit the charging stations in the first time period or not with the solutions from the model. The solution of the model provides the optimal strategy for the electric vehicle routing problem.

4. Solution Method

In Figure 3, the problem is solved with two phases. At first, patterns of states for charging stations are estimated to present the availability of charging station. Each charging station has its own pattern of state of occupation. Collecting the data from charging stations, the patterns are determined by analyzing the data. Once the pattern is found, the pattern can be seen as the availability of charging station. In the second phase, we solve the electric vehicle routing problem of the model proposed in the previous section. The patterns become the parameters in this routing problem. The solution presents the total travel costs with states of charging stations. The solutions for time periods can be used to find the optimal strategy.

(17)

(18)

(19)

5. Experimental Results

(20)

(21)

(22)

(23)

(24)

6. Conclusions

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