**1. Introduction**

A growing number of airports are electrifying their apron vehicle fleets to meet goals for climate-neutral airports (Bopst et al. [1], Interreg CENTRAL EUROPE [2], Flughafen München GmbH [3] and Royal Schiphol Group [4]). At Stuttgart Airport, for example, 40% of apron vehicles are equipped with electric drives and apron buses are already exclusively electrically powered (Bulach et al. [5]). Conductive charging is the state-of-the-art technology for charging these vehicles. However, this technology results in long downtimes due to vehicles charging and requires large batteries. A potential option for charging the vehicle batteries is dynamic inductive charging: Vehicles are wirelessly charged while in motion on a charging track installed below the road surface. This technology can substantially reduce downtimes. In addition, the need to have special charging stations is eliminated as well as the need for human involvement to plug in the charging cable. The objective of this paper is to report on methodological questions related to the potential usage of that charging technology for airport apron vehicles. We focus on the exemplary case of passenger buses transporting passengers from and to aircraft standing at outside parking positions. Still, we are convinced that the results hold for other types of service vehicles as well.

In order to charge apron vehicles with this technology, a dynamic inductive charging infrastructure would have to be implemented on the airport apron. This infrastructure consists of two components: the Power Supply Unit (PSU) and the Inductive Transmitter Unit (ITU). The PSU provides an alternating current of the required frequency to the ITU.

**Citation:** Pöch, N.; Nozinski, I.; Broihan, J.; Helber, S. Numerical Study on Planning of Inductive Charging Infrastructures for Electric Service Vehicles on Airport Aprons. *Energies* **2022**, *15*, 6510. https:// doi.org/10.3390/en15186510

Academic Editor: Adel El-Shahat

Received: 9 August 2022 Accepted: 5 September 2022 Published: 6 September 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The ITU is installed below the road surface and charges the battery if a vehicle travels along (Panchal et al. [6]). Since the infrastructure requires high initial investments, only a fraction of the road network should be equipped with a charging track. At the same time, however, it must be spatially allocated in such a manner that the vehicles' batteries can be sufficiently charged while they are operating.

We use mathematical optimization models to formally characterize the problem of finding a spatial placement of the required infrastructure components such that the necessary capital investment is as small as possible. First models for planning inductive charging infrastructures on airport aprons have already been developed (Helber et al. [7] and Broihan et al. [8]). The standard approach to solving those models is to employ high-end commercial mixed-integer linear programming solvers such as Gurobi or CPLEX. However, Broihan et al. [8] have shown that solving real-world-sized test instances with standard solvers to proven mathematical optimality in a reasonable time is very often not possible.

This leads to the research questions addressed in this paper: Which features tend to make a particular instance of the infrastructure design problem of spatially allocating the components of the charging infrastructure on the airport apron road system hard to solve? Hard to solve in this context means that even within hours or days of computation time, it is not possible to find an infrastructure allocation that is known to be optimal in the mathematical sense of the underlying problem. If it turns out that this is indeed the case, a second question arises: Can we at least make a statement about the potential "optimality gap", i.e., indicating how far away from the optimal solution quality we can be at most? In order to answer these questions, we systematically generated a large-scale test bed of synthetic problem instances that reflect different types of real-world spatial structures of airport terminals as well as apron road networks and aircraft parking positions.

We will show that the proof of optimality takes a long time, although an admissible solution can already be found quickly. We will also investigate the influence of the problem's size and certain parameter specifications on the computation times. We show that the investments in the PSUs and ITUs, the vehicles' energy consumption and the energy intake can significantly impact the computation time.

To this end, we analyze the properties of the Dynamic Inductive Charging Problem (DICP) experimentally to determine why this problem is difficult to solve for standard solvers. In particular, we examine the problem properties that lead to high computation times. The structure of the paper is such that we first provide a brief overview of the inductive charging technology, characterize the resulting airport apron design problem and report on related literature in Section 2. In Section 3.1, we formulate the model assumptions based on the previously stated properties of airport aprons. The introduced model in Section 3.2 is a variant of the optimization model presented in Broihan et al. [8]. Section 4 describes the generation of our instances that we use in the numerical study. We describe the general instance generation process in Section 4.2 and characterize the properties of the generated instance set for the analysis in Section 4.3. The results of the numerical study are presented in Section 5. We analyze the properties of the different instances and relate them to the computation time. Section 6 summarizes the results of this paper.
