5.1. Computational Efficiency
The computational efficiency in relation to the amount of handled data is an important factor and often a trade-off between accuracy and efficiency. To increase efficiency, we reduce the amount of common point data and the search space of the DBSCAN algorithm. Common points with a distance between them of less than 0.1 NM were defined as equal (
Section 3.1). As a result, the number of possible cluster elements decreases considerably in comparison to the original number of calculated common points.
To create the ε-neighborhood for the DBSCAN algorithm, the search space is restricted to the surrounding airspace in dependence on the position of the observed cluster element. For this, the observation area is overlaid with a grid with cell size
and the search space is then restricted to the cluster element’s and the adjacent cells. With these adaptions, the computation time for the DBSCAN itself (step B,
Section 3.3.2) decreased considerably while the time needed for pre-processing the common points (step A) increased for higher flight numbers (
Table 2).
The computation time for step C depends strongly on the number of cluster centers and the connecting links, because the algorithm uses the existing links between cluster centers as possible parts of the new trajectory. Furthermore, the number of optimized trajectories influences the computation time. For step C, where an algorithm calculates the shortest route, each trajectory part between a selected pair of origin and destination point is determined only once. This optimized trajectory part is assigned to all trajectories with the same pair. This is not possible when the advanced algorithm is applied since the evaluation value of a route depends on the already optimized trajectories. This leads to a considerably higher simulation time for step Cadv. Step D includes the point-balancing functionality which as well works on each identified cluster element.
As hardware a workstation with an Intel® Xeon® E5-2186G 3.8 GHz processor (6 cores), 64 GB Ram, and Windows 10 as operating system was used. The algorithms were implemented in Java.
5.2. Main-Flow Network
Here, the goal is to emulate the real flight route system as close as possible, independently of the number of necessary cluster centers.
Table 3 shows the general results for both scenarios where the cost function of the
algorithm is solely based on the length of the trajectory. The numbers of common points are very high due to the high number of flights and the dissection of trajectories into segments, which are all tested for common points separately. Furthermore, several different flights use identical trajectory segments and thus intersections (start and end point) of these segments are identified as common points. As a side effect, the number of different common points is very low compared to their total number. For Scenario 2, the number of common points is considerably lower than for Scenario 1 due to the lower number of flights, but the percentage of different common points is higher for Scenario 2 (7.7%) than for Scenario 1 (1.3%). This may be caused by the less dense traffic on different routes particularly in the northern sea leading to fewer intersections at the same or a similar position.
The number of cluster centers is reduced by 50%, resp. 58% from adapted (step B) to reduced main-flow network (step D) for Scenario 1, resp. 2. The percentage of noise is very low for Scenario 1.
Figure 10 shows the cluster elements for each cluster in an assigned color for both scenarios. The main traffic streams of
Figure 9 are easily recognized. Taking the high number of common points into account it can be concluded that most flights used the main routes. This would reduce the number of common points in areas with less dense traffic and therefore the chance for the creation of a cluster center.
Figure 10 shows the cluster elements, i.e., common points assigned to cluster centers in step B and not identified as noise. Most clusters are small with a high number of (nearly identical) elements. Comparing
Figure 9b to
Figure 10b shows that the north-west region has a very low number of clusters. In turn, many intersections in this area are marked as noise. This is not the case for Scenario 1 due to the dense traffic and the extensively used route system. The reduced main-flow networks presented in
Figure 11 follow pre-defined flight routes and this leads to hot spots for common points, which can be easily identified. It is clearly visible that the main flows are very close to the underlying traffic for both scenarios (cp.
Figure 9).
Entry and exit points were clustered separately (
Section 3.3.1). For both scenarios, the minority of entry/exit points are combined points belonging to both groups (
Table 4). This underpins the advantage of handling these points separately, especially with the creation of artificial traffic samples based on the main-flow network in mind. Furthermore, dividing the entry/exit points into separate groups ensures a more homogenous flow of traffic, which can be easier supervised by airspace controllers.
Table 5 shows results for both scenarios for the reduced main-flow network. For both scenarios, the SSPD values are very low and the interquartile range between quartile 1 and 3 is only 2.4 in both cases. The higher median value for Scenario 2 is caused by the lower number of cluster centers and corresponding low number of route segments, particularly in the north-west part of the observed airspace.
Thus, several flights were unable to maintain their short as-flown routes. The results for the SSPD metric are shown in more detail in
Figure 12 and confirm that most flights have very similar routes. For these routes, the SSPD is within the interquartile range while some outliers have SSPDs outside. Especially the whiskers, which mark a distance of 1.5 times the interquartile range to the median, are very close to the first and third quartile, around a value of 7 for both scenarios.
Figure 13 shows two examples for higher SSPD values for Scenario 1 and 2. In both cases, the as-flown trajectory (green) is nearly a direct connection between entry and exit points. Since no direct connection between these points exists within the reduced main-flow network, the associated reduced route (in blue) has a significant difference in comparison to the original one.
A comparison of the main-flow system’s median route length to the length of the original trajectories (100%) is also a metric denoting the quality of the solution. The interquartile ranges for the relative route length (
Table 5) for both scenarios are small, indicating that the reduced trajectories of most flights have a comparable length.
The structural complexity SC of the main-flow networks is very low in comparison to the intersection-based SC and nearly the same for both scenarios (
Figure 14 and
Figure 15). The cell size is 10 NM × 10 NM and the grid color ranges from 1 (dark green) to 10 (dark red). The low values indicate a highly structured traffic network with clearly defined main flows, which are separated by different flight levels for opposing traffic streams. Only a low number of flights intersect these traffic streams from other directions. Therefore, their influence on the structural complexity is small. The high values for the intersections, especially for Scenario 2, may be caused by the individual handling of traffic by controllers, e.g., allocating a direct route in times or areas of low traffic. Since a main-flow network is based on general route structures (
Section 3.3), points common for a few individual trajectories are not considered by the selected clustering algorithm. Instead, these points are marked as noise.
Nevertheless, the reduced individuality of flight routes may lead to overloaded cluster centers when all flights use the prescribed flight routes. Therefore, the number of trajectories moving through a cluster center is another hint for a main-flow network’s suitability. Together with the structural complexity it can be used as an indicator for the controller workload expected when supervising these nodes. Together, they can be used as measures of safety because the number of trajectories through a cluster center and the structure of the traffic pattern influence the probability of conflicts for this point. For Scenario 1, the high number of flights compared to Scenario 2 has caused a considerably higher number of trajectories per cluster center than for Scenario 2. The interquartile range of 59 for Scenario 1 in comparison to 44 for Scenario 2 suggests a higher traffic variation among cluster centers. Therefore, it can be expected that the traffic in Scenario 1 is distributed to more cluster centers than for Scenario 2, but with a varying number of flights. This leads to an imbalance in controller workload. Nevertheless, the higher the number of cluster centers the better are the possibilities to distribute the traffic. Unfortunately, a higher amount of cluster centers tends to increase noise values, so a compromise must be found. The low number of cluster centers for Scenario 2 has led to a higher number of trajectories per cluster center as it could be expected in comparison to Scenario 1, which has more than twice the number of flights.
5.3. Flight Parameters
In addition to a general main-flow system, we have shown in
Section 3.3.3 how the flight parameters altitude and speed can be reflected in such a system. Within this paper we chose the wake vortex classes Heavy, Medium and Light as an example for the distribution of flights to classes for each cluster center (
Section 3.3.3). The results are displayed in
Table 6 for the average flight levels and
Table 7 for the average flight speeds. In addition, each identified common point was assigned to the appropriate semicircular cruising level.
Table 6 shows that average flight levels of the flight classes are similar within a semicircular cruising level for the three classes, but as expected vary for different semicircular cruising levels. Even the small differences between the wake vortex categories imply that different flight levels are used for different categories in several cases. Aircraft with wake vortex class Light are less common in the considered scenarios due to the selected altitude range. Of the westbound flights in Scenario 2, only 63 flights belong to category Light compared to 540 in Heavy. So Light results are less reliable and presented for completeness only.
Speeds are quite different for the flight classes (
Table 7). As expected, flights of wake vortex category Heavy have a higher speed than Medium flights, which are themselves on average faster than flights with wake vortex category Light. As for the flight levels, there is again a difference between east- and westbound flights, which is much more evident and may be caused by the prevailing wind direction.
5.4. Adapted Cost Function
The influence of the advanced cost function
(
Section 4.2.3) is assessed in this section. The idea of an additional structural complexity factor as weighted influence factor
within the cost function was to reduce the number of link combinations crossing through each identified cluster center and thereby the complexity to supervise such a center. Furthermore, bypasses for nodes used by many flights are induced resulting in an increased flexibility of the resulting route network. Nevertheless, the main cost factor for the
algorithm is still the trajectory length.
With (8),
and therefore
adopt values between 0 and 9. Thus,
in Equation (9) can increase the route length of
at most by nine percent. The average complexity values over all cluster centers and the proportion of average and original trajectory lengths in percent to the original trajectory have been compared for several denominator values (
Figure 16). A denominator of 100 leads to a good compromise between the length of the trajectory and the reduction of the structural complexity. The “Percentage Route Length” relates the lengths of optimized trajectories created with
(step C) to their original length. The average complexity value is determined in step D and therefore corresponds to the reduced trajectories. Although the results for the scenarios are on different levels in
Figure 9, the structures are the same. A denominator of 50 leads in both cases to inferior results for the complexity value and the route length. A value of 10 prefers structural cluster complexity to trajectory length.
Additionally, some tests were carried out were all flights between the same origin and destination got the same trajectory as the first optimized flight of this entry/exit combination (
Section 5.1). In the case of
, structural cluster complexity and percentage of route length increased slightly by 0.1, resp. 0.15. This indicates clearly that different trajectories are allocated to flights with the same origin and destination depending on the set of already used links between cluster centers at the time of optimization. Moreover, this leads to a higher number of flights per cluster center. Even the SSPD value increases slightly indicating that the controllers have not always assigned the same shortest route but tried to avoid e.g., crowded areas.
Table 8 compares results for
and
(c100) after step C for both scenarios (
Section 3.3.2). The results for the structural complexity belong to step D to allow a comparison to the results of
Section 5.2. For both scenarios, the route lengths with the
algorithm are slightly higher and the structural complexities lower than for
. Despite this, the number of links in the OCA is considerably higher for the advanced version.
With more links, necessary crossing points were able to move to a different position between two routes if many crossing links or an inappropriate crossing angle increased the cost function of . Thus, the creation of trajectories is more flexible as intended and this helps to avoid complex structures. In turn, the length of the main-flow system increases. However, SSPD and node number per optimized trajectory stay the same for Scenario 1. For Scenario 2, the trajectories for the advanced version are unable to stay as closely to the original trajectory as before. This may be caused by the considerably lower number of cluster centers and the resulting limited number of choices in case of Scenario 2.