Bilevel optimization for the deployment of refuelling stations for electric vehicles on road networks (2501.19000v1)

Abstract: This work consists of a procedure to optimally select, among a group of candidate sites where gas stations were already located, a sufficient number of charging points in order to guarantee that an electric vehicle can make its journey without a problem of energy autonomy and that each selected charging station has another one that serves as useful support in case of failure (reinforced coverage service). For this purpose, we propose a bilevel model that, in a former level, minimizes the number of refuelling points necessary to guarantee a reinforced service coverage for all users who transit from their origin to destination and, as a second level, maximize the volume of demand that can be satisfied subject to budgetary restrictions. With the first of the objectives we are addressing the typical requirement of the administration, which consists of guaranteeing the viability of the solutions, and the second of the objectives is a criterion typically used by the private sector initiative, compatible with the profit maximization.

• The paper proposes a bilevel optimization model for optimal electric vehicle (EV) charging station deployment, balancing minimum stations for reinforced coverage with maximum demand satisfaction.
• It reformulates the bilevel problem into two single-level exact models and develops a heuristic algorithm, comparing their performance on minimizing installation nodes versus maximizing user coverage.
• Computational results show the heuristic effectively balances public and private objectives, offering a practical approach for decision-makers deploying EV charging infrastructure on road networks.

This paper addresses the optimal deployment of EV charging stations on road networks, considering both the need for energy autonomy and reinforced coverage in case of station failure. The authors propose a bilevel optimization model to minimize the number of refueling points while maximizing demand satisfaction, reflecting both public and private sector objectives.

The paper begins by emphasizing the importance of transitioning to a carbon-neutral economy, highlighting the EU's goal of achieving net-zero emissions by 2050. It notes the increasing adoption of EVs and the critical role of charging infrastructure availability in facilitating this shift. The authors point out the disparity in charging infrastructure across EU member states and underscore the need for strategic deployment plans. They aim to select optimal locations from existing gas stations for EV charging points, ensuring energy autonomy and redundancy.

The authors formulate a bilevel model to achieve these objectives. The upper level minimizes the number of refueling points needed to guarantee reinforced service coverage, aligning with administrative requirements for viable solutions. The lower level maximizes the volume of demand served, subject to budgetary constraints, catering to the private sector's profit maximization goals.

The paper reviews existing literature on EV charging station location problems, categorizing them into theoretical models and empirical applications. It discusses various approaches, including the flow refuelling-location model (FRLM), capacitated flow refuelling location model (CFRLM), and set covering problems. The authors cite works that address the sizing problem of EV charging stations, optimization procedures employing exact and heuristic methods, and bilevel modeling approaches for different objectives. They position their work as a more developed version of a previous paper, where the two-level model is transformed into a single-level model to compare it with a heuristic based on the knapsack and conditional coverage models.

The model development section details the input data, including the origin-destination demand matrix $(d_{ij})$, shortest path matrix $(\Gamma_{ij})$, and distance matrix $T = (t_{ij})$.

$d_{ij}$: origin-destination demand matrix, where $i,j \in V$ and $V$ represents the set of gas stations or cities.

$\Gamma_{ij}$: shortest path matrix between nodes $i$ and $j$, where $\Gamma_{ij} = \{i, v_1, v_2, ..., v_k, j\}$ and $v_1, v_2, ..., v_k$ are intermediate nodes.

$T$: distance matrix between pairs of nodes through the shortest path, with elements $t_{ij}$ representing the distance between nodes $i$ and $j$.

Other inputs to the model include $q_k$ denoting the capacity of node $k \in V$ to install charging stations, and $p_k$ the unit price depending on site $k$. The model also incorporates integer variables $x_k$ representing the number of charging facilities installed at point $k \in V$, and binary variables $y_l$ indicating whether point $l \in V$ is selected to open at least one charging facility.

The authors define $R$-dense graphs and introduce the Conditional Covering Problem (CCP) to minimize the installations needed for reinforced coverage. They define a parameter matrix $B = (b_{kl})$ where $b_{kl} \in \{0,1\}$, to represent the conditional coverage between pairs of selected nodes.

$b_{kl} = 1$ if $t_{kl} \le R$ and $k \ne l$, where $t_{kl}$ is the travel time between points $k$ and $l$, and $R$ is a pre-set threshold. Otherwise, $b_{kl} = 0$.

They then formulate the CCP as an integer programming model. Additionally, they employ a knapsack problem to maximize profit, subject to a budget $P$.

The bilevel optimization model combines these elements, minimizing the number of refueling points in the upper level and maximizing demand volume in the lower level. Constraints ensure reinforced coverage, budget limitations, and the relationship between the leader's and follower's decisions. The authors acknowledge the NP-hard nature of the knapsack problem and bilevel optimization, proposing a heuristic algorithm to determine charging point locations and quantities.

A case paper is presented using a network of 27 existing gas stations in southern Spain. The authors assign values to parameters such as unit cost ($p_k = 1$), budget ($P = 20$), capacity ($q_k = 5$), and attractiveness ($\omega_k$) for each node. They also set the solution graph as 1-dense ($R = 1$). The heuristic algorithm is applied to this scenario, and the results are compared with the solution obtained from an exact model.

The authors reformulate the bilevel problem as a single-level optimization problem using primal-dual optimality conditions, which converts the follower problem into its dual. By incorporating the strong duality condition, they create a single-level model suitable for commercial solvers. This single-level reformulation is expressed by the following equations:

Minimize:

$\sum_{l\in V} y_l$

Subject to:

$\sum_{l\in V,l\not= k} b_{kl} y_l \ge 1$, $\forall k \in V$

$\sum_{k\in V} x_k p_k \le P$

$y_k \le x_k$, $\forall k \in V$

$\frac{x_k}{q_k} \le y_k$; $\frac{x_k}{q_k} \le \sum_{l\in B_k} y_l$, $\forall k \in V$

$$p_k \lambda + \beta_k + \frac{\alpha_k}{q_k} + \gamma_k + \frac{\delta_k}{q_k} \ge \omega_k$$, $\forall k \in V$

$$P\lambda + \sum_{k\in V} q_k\beta_k + \sum_{k\in V} (\sum_{l\in B_k} y_l)\alpha_k + \sum_{k\in V}y_k\delta_k + \sum_{k\in V}y_k\gamma_k = \sum_{k\in V} \omega_k x_k$$

Where:

• $y_l$ is a binary variable that takes the value 1 if we select point $l\in V$ to open at least one charging facility, and 0 otherwise.
• $b_{kl}$ is a binary parameter that takes the value 1 if $t_{kl} \le R$ and $k \ne l$, and 0 otherwise.
• $x_k$ is an integer variable that indicates the number of charging facilities installed at point $k\in V$.
• $p_k$ is the unit price depending on site $k$.
• $P$ is the total budget.
• $q_k$ is the capacity of each node $k\in V$ to install charging stations.
• $\lambda$, $\beta_k$, $\alpha_k$, $\gamma_k$, and $\delta_k$ are dual variables.
• $\omega_k$ is a weight used to quantify the attractiveness of locating a service at point $k$.

In a second approach, the authors exchange the hierarchy of the criteria, maximizing demand coverage as the primary objective and minimizing the number of installations as the secondary objective. This leads to a different single-level model, which is expressed by the following equations:

Maximize:

$\sum_{k\in V} \omega_k \cdot x_k$

Subject to:

$\sum_{k\in V} x_k p_k \leq P$

$\frac{x_k}{q_k}\leq y_k$; $\frac{x_k}{q_k} \leq \sum_{l\in B_k} y_l$, $\forall k \in V$

$$\sum_{l\in V,l\not= k} b_{kl} y_l \geq 1, \hspace*{2.5cm} \forall k\in V$$

$y_k \leq x_k, \hspace*{3.8cm}\forall k \in V$

$$\sum_{l\in B_k} \alpha_k +\beta_k \leq 1,\hspace*{2.4cm} \forall k \in V$$

$$\sum_{k\in V}\alpha_k+\sum_{k\in V}\beta_k x_k = \sum_{k\in V} y_k$$

Where:

• $y_l$ is a binary variable that takes the value 1 if we select point $l\in V$ to open at least one charging facility, and 0 otherwise.
• $b_{kl}$ is a binary parameter that takes the value 1 if $t_{kl} \le R$ and $k \ne l$, and 0 otherwise.
• $x_k$ is an integer variable that indicates the number of charging facilities installed at point $k\in V$.
• $p_k$ is the unit price depending on site $k$.
• $P$ is the total budget.
• $q_k$ is the capacity of each node $k\in V$ to install charging stations.
• $\alpha_k$ and $\beta_k$ are dual variables.
• $\omega_k$ is a weight used to quantify the attractiveness of locating a service at point $k$.

The paper presents computational results from 25 experiments conducted on the 27-node network, comparing the heuristic solutions with those obtained from the two single-level models (E1 and E2). The results indicate that the heuristic provides a good balance between minimizing the number of nodes and maximizing attractiveness. Further experiments on a larger network with 57 nodes demonstrate the scalability and effectiveness of the proposed methodologies. The first exact model (E1) yields a minimum number of nodes but does not give good attractiveness results. The second exact model (E2) yields a maximum user population covered. The heuristic (H) gives solutions that are closer to the optimal value of the second model.

In conclusion, the authors propose a methodology for optimally deploying EV charging stations, considering both coverage and demand satisfaction. The bilevel model and heuristic algorithm offer a practical approach to addressing this complex problem, balancing public and private sector objectives. The comparison of different solution methodologies provides insights into their strengths and weaknesses, guiding decision-makers in selecting the most appropriate approach for their specific context.