**4. Generating a Set of Instances for Dynamic Charging Infrastructures Problems**

*4.1. Purpose and Objective of the Instance Generation Process*

Our previous and preliminary numerical results (see Helber et al. [7] and Broihan et al. [8]) showed that any attempt to use a high-end commercial mixed-integer programming solver like Gurobi or CPLEX to solve the DICP shows very mixed results with respect to computation times and solution quality. In particular, we observed that when the instances


is then those commercial solvers could often solve the resulting instances of the DICP to proven optimality within a few seconds or minutes.

However, real-world airports often have large and complex apron road networks, many passenger gates and many aircraft parking positions. As one consequence, the operational variance of the possible routings of the vehicles (passenger buses in our example) can be substantial. As we aim at obtaining a charging infrastructure allocation that is robust over a wide variety of such service requests, many of them have to be considered simultaneously in the infrastructure design decision, which is one factor leading to large model instances that tend to be hard to solve, i.e., having intolerably long computation times as well as potentially large optimality gaps.

A further problem of dealing with larger real-world airports is that the length of road segments, say, between terminals and aircraft parking positions, can be substantial. It could be desirable to equip only small fractions of those long road segments with the ITUs. However, to represent those fractions of the road segments in our model, we have to subdivide those road segments into sub-segments by adding additional vertices to the graph. An example of this problem aspect is depicted in Figure 5. Here, between the two nodes denoted as *i*44 and *i*51, two long road segments exist, one for each direction and each having a length of 300 m. By introducing two or even five further vertices, link lengths of 100 m or even 50 m are created, respectively.

**Figure 5.** Extending an initial graph with link length maximum (**top**), 100 m (**mid**) and 50 m (**bottom**).

Having such finer granularity of the road system network in our model makes more cost-efficient infrastructure solutions possible, which is attractive. However, this comes at the price of operating again with larger models, i.e., models with a larger number of links and vertices that tend to be numerically more difficult to solve.

Another element that affects the difficulty of solving the problem by using standard commercial solvers to proven optimality is the cost ratio of the elements of the charging infrastructure, i.e., the necessary investment per PSU relative to the investment per ITU unit. If the PSUs tend to be relatively expensive, the resulting structures tend to operate with smaller numbers of PSUs to which then relatively large ITU structures are connected. In the opposite case of relatively inexpensive PSUs, larger numbers of ITU structures are placed over the road network.

Finally, the ITUs' power transfer also significantly impacts both the structure of the solutions and the difficulties of finding them. Suppose the ratio of the energy that can be picked up by a vehicle passing along a link is very large relative to the energy required to pass along that link. In that case, it may be possible to equip only a relatively small but well-chosen fraction of the apron road network with ITUs.

We know that all these factors affect:


We also conjecture that there might be cross-effects between the influencing factors. In order to be able to identify those effects and to explore the limitations of using a commercial solver to solve the DICP, we systematically designed a full-factorial test bed consisting of hundreds of instances. We then solved those instances numerically to obtain experimental results shedding some light on the questions outlined above. Before we present the results of those computations, we first describe the underlying system of creating the test instances as well as their characteristic features.
