This study proposes a flow model using a modified Lighthill-Whitham-Richards highway model. The proposed model treats each aircraft on an airway as a continuous distribution of air collision probability, which is called the danger value distribution. With the proposed flow model, collision can be easily predicted by the peak value of the overlap of the danger value distribution of each aircraft. The study further proposes a velocity adjustment method that can be used to resolve the conflict. The proposed method can be applied for aircraft separation during the landing process, in which the separation time is different for different combinations of aircraft types.
Currently, air travel is one of the major methods of transportation for people around the world. The rapid growth of the aviation industry has resulted in heavier air traffic. Consequently, airways and airports are both busier. Because of heavy air traffic, flight safety and air traffic management have increased in importance. To this point, air traffic management has relied on air traffic controllers (ATCs). Radar control allows ATCs to coordinate aircraft spacing directly by using visible information through the radar display. Aircraft spacing is mainly used to avoid the danger of possible collision and prevent aircraft from being affected by wake vortices induced by the preceding aircraft. The spacing can be based on time or distance. Distance-based separation is easily managed by ATCs through radar displays. However, when compared with time-based separation, distance-based separation is known to be less efficient when considering different weather conditions [
Air traffic flow must be modeled before it can be controlled. Along with the development of civil aviation, several evolutions in modeling air traffic flow have been proposed. In addition, concepts of modeling traffic have also evolved with the growth of air traffic flow. The Eulerian network model of air traffic is inspired by the Lighthill-Whitham-Richards (LWR) model [
This study presents a danger value distribution flow model that captures the time-based separation characteristics, permitting its use to manage time-based separations. Using the proposed model, we can determine how crowded the airspace is and detect whether the separation between the aircraft is sufficient. Thus, the proposed model can provide more detailed separation information to ATCs. In the proposed model, different separation constraints are considered for different combinations of aircraft types, which make our method applicable for complicated separation situations during the aircraft landing process. Moreover, we have also provided an automatic speed adjustment procedure that maintains minimum separation time between aircraft and thus maximizes runway capacity.
The organization of this paper is as follows. In Section
Wake vortices are turbulent airflows generated during all phases of flight as a byproduct of the wing that is generating the lift. This turbulent flow of air may cause an aircraft to undergo an unstable flight period, resulting in injuries as well as loss-of-control accidents. As aircraft get closer to each other during the landing procedure, the wake turbulence problem is especially prominent during landing, particularly for a smaller aircraft following a larger aircraft. This is because a larger aircraft may generate stronger turbulent flow than a smaller aircraft, whereas a smaller aircraft may have fewer methods of resisting the strong turbulent flow than a larger aircraft. Therefore, the ability of different types of aircraft to resist and induce vortices should be considered when developing the danger value profiles for wake avoidance separation procedure. The width and length of the area, which are influenced by the generated vortex, are related to the speed and type of the aircraft [
For the best utilization of runway capacity, the optimum situation is that the trailing aircraft should follow the preceding aircraft by the distance specified by ICAO's separation criteria during the landing process. As there is no time-based separation rule for landing, we have used Freville's method to convert ICAO's rule to a time-based rule by assuming an average landing speed of
Time-based separation criteria derived with average landing speed.
Following aircraft (second) | Leading aircraft | |||
---|---|---|---|---|
Small | Large | B757 | Heavy | |
Small | 79 | 106 | 132 | 159 |
Large | 79 | 79 | 106 | 132 |
B757 | 79 | 79 | 106 | 132 |
Heavy | 79 | 79 | 106 | 106 |
We used a single critical value as an indicator of insufficient separation for all types of aircraft. This critical value should be higher than the peak danger value in the presence of an aircraft, and be lower than the peak danger value when two aircraft collide with each other. In our design, the danger value profile of an aircraft has a peak value of
The wake vortex separation scheme.
Let the time at which the danger value is
Variables of different types of aircraft.
Aircraft type |
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Small |
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Large |
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B757 |
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Heavy |
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The separation criteria determined using linear programming are shown in Table
The separation time using the danger value distribution.
Following aircraft (second) | Leading aircraft | |||
---|---|---|---|---|
Small | Large | B757 | Heavy | |
Small |
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106 | 132 | 159 |
Large | 79 | 79 | 106 | 132 |
B757 | 79 | 79 | 106 | 132 |
Heavy | 79 | 79 | 106 |
|
Figure
Danger value profiles for landing.
In this section, we introduce the modified LWR model used in air traffic control. As the arriving aircraft flows are not as dense as those on the en-route airway, one may suspect whether the LWR model, which was originally designed for modeling high-way traffic in a continuous way, is appropriate for modeling the arriving aircraft flow, which seems more discretized than high-way traffic. Our selection of the LWR model is based on various successful examples and applications in the literatures [
The LWR model can be numerically solved in either the classical Eulerian scheme or a Lagrangian scheme. As we had added the danger values associated with each aircraft at fixed locations, the Eulerian scheme was opted in this study. Moreover, the Eulerian scheme can be easily used for control purposes than the Lagrangian scheme [
Let
If the danger value is treated as a density distribution on the airway, its peak value decreases when the density distribution becomes wider, which is a consequence of increased velocity. This phenomenon is undesirable for time-based separation because we use the summed danger value for insufficient separation detection. To solve this problem, we designated danger value distribution flows in a velocity-related coordinate
Subsequently, the time derivation of
We used finite difference methods to approximate the solutions of (
Several numerical schemes can be used to compute the danger value distribution flow model. The Lax-Wendroff scheme [
When insufficient time-based separation between aircraft is detected, ATCs should make necessary arrangements for the involved aircraft to avoid danger of possible collision. In this section, we present a method of velocity adjustment that maintains the minimally required separation time between aircraft so that the capacity of an airway can be optimally utilized. The proposed approach considers the case where two adjacent aircraft are flying along the same airway in the same direction at the same altitude. That is, one aircraft is chasing the other aircraft at the same flight level. If the two aircraft are traveling with the same velocity profile, the time-based separation between these two aircraft remains fixed. This idea inspired the proposed velocity adjustment procedure.
Let us assume that the preceding and trailing aircraft have danger value profiles
Minimum time-based separation.
Consider the case shown in Figure
Velocity profiles of two aircraft.
Let us suppose that the trailing aircraft can decelerate at a deceleration speed
Then, we used bisection searching by moving the curve up or down to satisfy (
Let
Let
Update
Repeat Step
If the speed of the preceding aircraft is so slow that the trailing aircraft cannot travel using the velocity profile of the preceding aircraft, then the only allowable location for minimum time-based separation to occur is at the end of the airway. In this situation, the velocity change curve will be used as the transition from the original velocity profile to the minimum speed,
This section presents an example of the time-based separation using the danger value distribution flow model with the proposed danger value profiles shown in Figure
Velocity profiles of the trailing and preceding aircraft.
Here, we present a scenario in which a small aircraft is following a B757 aircraft during the landing process. Figure
Danger value flow without velocity adjustment.
Danger value flow with velocity adjustment.
To better illustrate the actual separation condition between these aircraft, the separation distance and time are shown in Figure
Separation in seconds and nautical miles.
The total computation time for the separation detection process as well as the speed adjustment process, determined using a laptop with Intel i5 M540 CPU and
This study presented a flow model that uses the danger value distribution to represent possible dangers of collision surrounding each aircraft on an airway. Overlapping of the danger value distributions of each aircraft on the route can indicate the level of the time-based separation. This method can be used as an indicator of insufficient time-based separation. With the proposed flow model, a velocity adjustment algorithm using the deceleration of the trailing aircraft was presented in this paper. Using the proposed algorithm, the trailing aircraft can decelerate according to its capabilities. The modified velocity profile of the trailing aircraft can be used to maintain the separation time at its minimum required value and thus can better utilize the capacity of an airway.
Through the development of danger value profiles, we demonstrated that both the forward and backward danger value profiles have practical importance. Different danger value profiles can be constructed for various types of aircraft, and a more sophisticated time-based separation can be applied to further optimize the usage of limited runway/airport capacity. Apart from the influence of wake vortices, the estimation error of aircraft positions can also be included into our method. The danger value distribution can be modified or expanded to incorporate the estimation error of aircraft position. In the future, the study of how the estimation error affects the danger value profile will be pursued.
This work was supported by the National Science Council under Grant NSC-96-2218-E-006-283-MY3, which is greatly appreciated.