*2.3. Modeling the Spatial Allocation of ITUs and PSUs*

In Figure 2, we show in red a part of a fictitious allocation of ITUs and PSUs on some of the roads (in black) used by passenger buses for the Vienna Airport. Each ITU has to be part of a contiguous structure which is in turn connected to a PSU. Some locations, in particular those close to buildings, might be natural candidates to establish a PSU. On the other hand, there may be parts of the apron road network where installing ITUs could be impossible or unattractive due to interference with aircraft or the nature of road surfaces.

To represent the airport apron in the mathematical optimization model, we model the road network as a directed graph, as shown in Figure 3. Gates, parking positions and road intersections are modeled as nodes. The lanes of the airport apron roads are represented as links (directed arcs) in the graph. We assume that, on these links, the ITUs can be installed. If there are multiple adjacent edges, each with an installed ITU in the graph, they represent one contiguous ITU structure on the real airport apron. An example is given in Figure 3, where the ITU is installed between the nodes *g*2, *i*6, *i*5, *i*14 and *i*15. This contiguous ITU structure can then be powered by one PSU, which is installed at node *g*2 in the example. A consistent connection from each ITU segment to a PSU is required. This connection can be set up directly if the ITU is directly adjacent to a PSU node or indirectly via other ITU segments (e.g., the ITU segment between *i*5 and *i*6 is connected via the ITU segment between *g*2 and *i*6 to the PSU at node *g*2).

We are now in the position to state in a non-technical manner the infrastructure allocation problem. The overall objective is to minimize the investment in ITUs and PSUs. The selection of links to be equipped with ITUs as well as the installation of PSUs must be such that:


This modeling approach relieves us from the need to track the SOC of the vehicles' batteries. Figure 4 illustrates this. It shows an example SOC curve for a service request. The service request consists of the links *l*1 to *l*6. At links *l*2 and *l*5, an ITU is installed. When the vehicle passes over an ITU, it absorbs energy and the curve increases. In all other cases, the SOC decreases. According to our assumption, the vehicle must absorb at least as much energy as it consumes for the service request. For this reason, the SOC at the end might be greater than at the beginning. Of course, it may happen that the vehicle cannot absorb the energy because the battery is already full. In that case, the SOC can be lower at the end. We assume the battery is large enough to survive a longer part of the service request without energy intake. This results in the SOC never falling below zero.

**Figure 3.** Representation of an airport as a directed graph. (Source: Adapted from Broihan et al. [8]).

**Figure 4.** Exemplary SOC. (Source: Adapted from Broihan [21]).
