*5.1. Main Features of the Aerial Network Model*

The aerial network model integrates the concept of dynamic air corridors (e.g., DDCs) [56] in a multilayer structure [57] in order to define a three-dimensional graph that involves the vertical dimension for ensuring trips between origin/destination points. Here a 3D-UAN model is proposed, given by the union of bi-dimensional graphs, *GL*, in multiple layers, *L*, which includes the set of Fixed Nodes (*NF*,*L*), a set of Transition Nodes (*NT*,*L*) and a set of Dynamic Links (*DL*) connecting the nodes and the layers.

The 3D Graph (Θ) model [23] is summarized by the following formulation:

$$
\Theta \, = \cup\_{L=\{1,\dots,n\}} G\_L \cup D\_{v,L} \tag{2}
$$

where *GL* = (*NF*,*L*, *NT*,*L*, *Dh*,*L*).

In more detail, Fixed Nodes correspond to vertiports, which are the access to egress from UAM services, and are located based on the digital twin of the urban system, as described in Section 3. Transition Nodes are set to allow horizontal crossings and shifting to an upper or lower layer; some of them may be located at the same coordinates as Fixed Nodes, except for the vertical coordinate. Again, the location of Transition Nodes—or just their vertical position if they correspond to Fixed Nodes—is obtained by the digital twin model, particularly by using the information related to height of buildings, potential obstacles, and no-fly zones, among others. Pairs of nodes (both fixed and transition) are connected by Dynamic Links, *dm*,*L*, which belong to the set *DL* = {*dm*,*L*} | m = {1, 2 ... *PL*}, where *PL* is the total number of links for layer *L*. More specifically, the dynamic link set consists of horizontal and vertical link subsets, respectively, *Dh*,*<sup>L</sup>* <sup>=</sup> { *hmL*} <sup>⊂</sup> *DL* <sup>|</sup> m = {1, 2 ... } and *Dv*,*<sup>L</sup>* <sup>=</sup> { *vmL*} <sup>⊂</sup> *DL* | m = {1, 2 ... }. Such links are identified by ensuring safe flight conditions with respect to the external features, i.e., urban or extraurban environment.

The proposed 3D-UAN model also includes a cost function, defined on each link, with the aim of providing minimum cost origin/destination connections, suitable ACV separation and in-flight safety. The following link cost function *c Tt* , *Tg* has been defined for each link belonging to *DL*:

$$\mathcal{L}\left(T\_{l\prime}, T\_{\mathcal{S}}\right) = \left\{ \begin{array}{c} T\_{l\_{j}} \text{ for } j = 1\\ T\_{l\_{j}} + T\_{\mathcal{S}\_{(j, j-1)}} \forall \, j > 1 \end{array} \right\} \tag{3}$$

where *j* is the *j*-th ACV using the dynamic link *dm*,*<sup>L</sup>* at a given time period; *Ttj* is the travel time of j on *dm*,*L*; *Tg*(*j*, *<sup>j</sup>*−1) is the time gap between *<sup>j</sup>* and *<sup>j</sup>* <sup>−</sup> 1.

Depending on *dm*,*L*—horizontal or vertical link—the travel time, *Ttj* , will change. By considering vertical links, if *j* = 1 the *Ttij* may be climbing (*Taj* ) or descent (*Tfj* ) time depending on the link direction—i.e., to upper layers or to lower layers. If there are several ACVs on the same link, i.e., *<sup>j</sup>* > 1, the time gap *Tg*(*j*,*j*−1) guarantees suitable separation between two following ACVs along vertical links. For horizontal links, *Ttj* is the running time, *Trj* , if *<sup>j</sup>* = 1, while if *<sup>j</sup>* > 1, the time gap *Tg*(*j*,*j*−1) guarantees appropriate separation between two following ACVs. To assess the waiting time component, time gap *Tg*(*j*,*j*−1) is assigned at a fixed node before ACV departure, to keep safe travel conditions among them.

Equation (3) may be re-written for different types of links as:

$$c\_{h,L}\left(T\_{r\prime}\ T\_{\mathcal{S}}\right) = \left\{ \begin{array}{c} T\_{r\_j} \text{ for } j=1\\ T\_{r\_j} + T\_{\mathcal{S}\_{(j,\,j-1)}} \forall \, j>1 \end{array} \right\} \tag{4}$$

$$\mathfrak{c}\_{\mathbf{c},\mathbf{L}}\left(T\_{a\_{j}f\_{\mathbf{c}}},T\_{\mathcal{S}}\right) = \begin{cases} T\_{a\_{j}} \text{ for upper layer transitions, } j=1\\T\_{a\_{j}} + T\_{\mathcal{S}\_{\{i,j-1\}}} \text{ for upper layer transitions, } j>1\\ \begin{array}{c} T\_{f\_{j}} \text{ for lower layer transitions, } j=1\\T\_{f\_{j}} + T\_{\mathcal{S}\_{\{i,j-1\}}} \text{ for lower layer transitions, } j>1 \end{array} \end{cases} \tag{5}$$

Data regarding travel, climbing and descendent times; physical obstacles; day time (related to social economics habits); overflight zones; and environmental conditions—such as wind phenomena or meteorological conditions, which affect flight conditions—should be included in the digital twin model, together with real time information on aerial traffic flows in more advanced versions.

The cost function (4) is used to compute the minimum cost paths that each ACV will use to realize the trip between the origin and destination vertiports, and to ensure appropriate ACV safety separation (defined by *Tg*(*j*,*j*−1) ). Shortest paths based on such link cost function may be found by iterative search algorithms—such as Dijkstra [58] or A\* [59].

Flight operations within the same layer, *L*, occur along horizontal dynamic links (connecting fixed and transition nodes) belonging to *Dh*,*L*. The vertical links belonging to *Dv*,*<sup>L</sup>* guarantee some specific procedures—i.e., landing and take-off operations, as well as layer transitions to/from the upper or lower layers. As for the dynamic nature of both horizontal and vertical links, it consists of enabling/disabling them in compliance with data on environmental conditions and traffic capacity (e.g., operational delays, unfavourable weather conditions). In addition, the features of the dynamic links may change according to ACV size, which requires different features of *dm*,*<sup>L</sup>* cross sections. For example, large ACVs require a greater distance between the layers and vertical link length increase and changes in transition node positions in order to ensure suitable protection volumes around them [60].
