**6. Conclusions**

This paper presents an optimization model for planning an inductive charging infrastructure on airport aprons. The model is a simplified variant of the model presented by [8]. The paper aimed to investigate the problem properties that lead to high computation times when solving the model with standard solvers and the reasons behind these high computation times.

For this purpose, we systematically generated a large set of instances in three steps. At first, we created initial graphs based on real airport apron structures. Afterward, we adapted these initial graphs to the three defined instance classes small, medium and large. Finally, we created instances from each graph with different characteristics, such as the ratio of energy intake to energy consumption.

In the numerical evaluation, we showed that current commercial standard solvers such as Gurobi can find a feasible solution quickly. They can even find good and potentially very good solutions in an acceptable time. However, proving the optimality of a solution often takes very long. One reason for this is the high number of equally optimal solutions, which are, among other reasons, due to symmetries in the problem structure.

The computation times of instances resulting from graphs of similar size vary significantly and depend on the instance's attributes. Specific attributes lead to remarkably high computation times. We showed that an increasing number of service requests and PSU candidates lead to higher computation times, whereby the influence of the PSU candidates is more significant. In addition, we investigated the influence of the ratio of energy intake to energy consumption and the ratio of PSU investment to ITU investment. The energy ratio significantly determines the size of the charging infrastructure. The numerical investigations showed that, in particular, an energy ratio that leads to a medium-sized charging infrastructure also leads to high computing times, as in this case, there tend to be many equally good alternatives. The investment ratio of PSUs to ITUs influences whether an additional PSU or more ITUs must be built. Comparatively cheap PSUs lead to high computing times.

In real applications, the problem instances may be even more complex than those considered in this paper. We showed that for some instances, optimality could not be proven by standard solvers within a time limit of eight days. For this reason, it is crucial to investigate how the computation time can be reduced. One possibility is to break symmetries by extending the model with symmetry-breaking restrictions or considering undirected graphs. This study showed that the presented rather intuitive model formulation is very hard to solve. For this reason, one might search for alternative model formulations that lead to lower computation times. For example, a flow-based model formulation in which the PSUs serve as sources and the ITUs serve as sinks would be conceivable. Further, future research projects should consider the use of heuristics and customized solution approaches.

Further research could further investigate the interaction of the power dimensioning and the allocation of the charging infrastructure's components. The model presented here hence provides substantial opportunities for such extensions.

**Author Contributions:** Conceptualization, N.P., I.N., J.B. and S.H.; methodology, N.P.; software, N.P. and J.B.; validation, N.P., I.N., J.B. and S.H.; formal analysis, N.P., I.N. and J.B.; investigation, N.P., I.N. and J.B.; data curation, N.P., I.N. and J.B. ; writing—original draft preparation, N.P., I.N. and J.B.; writing—review and editing, S.H.; visualization, N.P., I.N. and J.B.; supervision, S.H.; project administration, S.H.; funding acquisition, S.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** We would like to acknowledge the funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—EXC 2163/1—Sustainable and Energy Efficient Aviation—Project-ID 390881007. The publication of this article was funded by the Open Access Fund of the Leibniz University Hannover.

**Data Availability Statement:** The data presented in this study are openly available in the Research Data Repository of the Leibniz University Hannover at https://doi.org/10.25835/itr694hg (accessed on 8 August 2022).

**Acknowledgments:** We acknowledge the support of the cluster system team at the Leibniz University Hannover, Germany, in the production of this work.

**Conflicts of Interest:** The authors declare no conflict of interest.
