Electric vehicles fast charger location-routing problem under ambient temperature

Abstract

This study investigates how the location-routing decisions of the electric vehicle (EV) DC Fast Charging (DCFC) charging stations are impacted by the ambient temperature.Electric vehicles are expected to contribute significantly to the delivery mission of logistic companies in the future. In an EV delivery logistics network equipped with DCFC stations, this study investigates how the location strategy of DCFC charging stations and the routing plan of a fleet of EVs are impacted by the ambient temperature. We formulated this problem as a mixed-integer linear programming model that captures the realistic charging behavior of the DCFC’s in association with the ambient temperature and their subsequent impact on the EV charging station location and routing decisions. Two innovative heuristics are proposed to solve this challenging model in a realistic test setting, namely, the two-phase Tabu Search-modified Clarke and Wright algorithm and the Sweep-based Iterative Greedy Adaptive Large Neighborhood algorithm. We use Fargo city in North Dakota as a testbed to visualize and validate the algorithm performances. The results clearly indicate that the EV DCFC charging station location decisions are highly sensitive to the ambient temperature, the charging time, and the initial state-of-charge. The results provide numerous managerial insights for decision-makers to efficiently design and manage the DCFC EV logistic network for cities that suffer from high-temperature fluctuations.

Appendix

A1. Variable fixing and valid inequalities

To improve the computational performance of model $\mathbf [EV-L]$, the following variable fixing and valid inequalities are introduced. We begin by introducing the variable fixing techniques first.

• The electric vehicle $e\in E$ is not able to traverse the arc between the nodes $n\in N$ and $n^{'}\in N$ if the respective traveling distance, i.e., $d_{ij}$ is greater than the maximum distance that it can travel by a fully charged battery, namely, $d_{max}$.

  $$\begin{aligned} Y_{nn^{'}e}= & {} 0 \forall n\in N, n^{'} \in N, e\in E | d_{nn^{'}}>d_{max} \end{aligned}$$

  (34)

• The electric vehicle $e\in E$ is not able to traverse the arc between the nodes $n \in N$ and $n^{'} \in N$ if the sum of the demand of costumers in respective customer nodes exceeds the weight capacity of the EV.

  $$\begin{aligned} Y_{nn^{'}e}= & {} 0 \forall n\in N, n^{'} \in N, e\in E | w_{n}+w_{n^{'}} > k_{e} \end{aligned}$$

  (35)

In addition to the above-mentioned variable fixing techniques, the following valid inequalities are introduced.

• In our study, we assume that none of the EVs can travel more than $d_{trip}$ each day. To capture this constraint, we add the following valid inequalities as a lazy constraint to model $\mathbf [EV-L]$.

  $$\begin{aligned} \sum _{n\in N}\sum _{n^{'}\in N} d_{nn^{'}} Y_{nn^{'}e}\le & {} d_{trip} \forall e \in E \end{aligned}$$

  (36)

• To further tighten the proposed model $\mathbf [EV-L]$, first, we approximate a lower bound, namely, $N^{total}_{LB}$, for the number of the EVs that are required to satisfy the customer demand. The lower bound on the number of EVs depends on two other factors, namely, the total weight associated with the requests of the costumers and maximum trip distance that each EV can traverse. Hence, in order to calculate $N^{total}_{LB}$, first, we find out the minimum number of EVs based on the freight limitation, $N^{freight}_{LB}$. To do so, we use a well-known bin packing problem (Martello and Toth 1990) given by (37)-(41). Within this formulation, $\{Z_{e}|\forall e \in E \}$ denotes if EV $e\in E$ is used or not, and $\{H_{ie}|\forall i \in I, e \in E \}$ denotes if costumer i is served by EV e.

  $$\begin{aligned} Minimize N^{freight}_{LB} = \sum _{e\in E} Z_e \end{aligned}$$

  (37)

  subject to

  $$\begin{aligned} \sum _{i\in I} w_{i} H_{ie}\le & {} k_{e} Z_{e} \forall e \in E \end{aligned}$$

  (38)
  $$\begin{aligned} \sum _{e\in E} w_{i} H_{ie}= & {} 1 \forall i \in I \end{aligned}$$

  (39)
  $$\begin{aligned} Z_{e}\in & {} \{0,1\} \forall e \in E \end{aligned}$$

  (40)
  $$\begin{aligned} H_{ie}\in & {} \{0,1\} \forall i \in I, e \in E \end{aligned}$$

  (41)

  The next lower bound on the number of EVs is based upon the maximum length of the trip, $N^{trip}_{LB}$, which utilizes the concept of the minimal spanning tree. To do so, given the feasible arcs in the network, we create a minimal spanning tree for the network consisting of the depot node and customer nodes, i.e., $I \cup \{o\}$. The total weight of this graph, where the weight is the traveling distance between vertices of the graph, provides us with an estimated minimum overall traveling distance of $d_{est}$. Hence, the second lower bound is computed as follow:

  $$\begin{aligned} N^{trip}_{LB}= & {} \lceil {\frac{d_{est}}{d_{trip}}}\rceil \end{aligned}$$

  (42)

  Having introduced these two lower bounds on the minimum number of required EVs, we use the best among them in the MILP settings, as shown below:

  $$\begin{aligned} N^{total}_{LB}= & {} max \{N^{freight}_{LB}, N^{trip}_{LB} \} \end{aligned}$$

  (43)

  Finally, to tighten the solution space of model $\mathbf [EV-L]$, we add the following valid inequality as a lazy constraint.

  $$\begin{aligned} \sum _{e\in e}\sum _{n^{'}\in N} Y_{on^{'}e} \ge N^{total}_{LB} \end{aligned}$$

  (44)