Abstract
We present two bilevel programming formulations for the aircraft deconfliction problem: one based on speed regulation in k dimensions, the other on heading angle changes in 2 dimensions. We propose three reformulations of each problem based on KKT conditions and on two different duals of the lower-level subproblems. We also propose a cut generation algorithm to solve the bilevel formulations. Finally, we present computational results on a variety of instances.
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Notes
1 NM = 1852 m.
References
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This research was partly funded by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 764759 ETN “MINOA”.
This publication was supported by the Chair “Integrated Urban Mobility”, backed by L’X—École Polytechnique and La Fondation de l’École polytechnique and sponsored by Uber. The Partners of the Chair shall not under any circumstances accept any liability for the content of this publication, for which the authors shall be solely liable.
Appendices
Appendix
Returning aircraft to original trajectories
The optimal heading angle change \(\theta _i^\star \) for each aircraft i is obtained by solving Eqs. (14a)–(14c). The trajectory deviation is followed until necessary to guarantee the safety distance, then the aircraft must return to their initial trajectories. Following what is done in [17, 35], for each pair of aircraft the convex unconstrained QP (22) is solved as a post-processing step to return each aircraft to its original flight plan as soon as possible after conflict resolution:
The objective function of the problem (22) is the relative squared Euclidean distance between aircraft, which is computed using the optimal heading angles of the proposed bilevel problem (14a)–(14c).
Once the optimal solution \(\tau _{ij}^\star \) for problem (22) is found, we compute
as the optimal time for which aircraft i can return to its initial trajectory after the deconfliction (there will be, for each i, a different \(\tau _{ij}\) for every pair of aircraft (i, j)).
Knowing \((x_i(T_i^\star ),y_i(T_i^\star ))\) and the exit point from the air sector, it is easy to determine the new trajectory each aircraft has to follow in order to go back to its initial trajectory, as shown in Fig. 7.
As clarified in [35], when the aircraft are returning to the initial trajectories, new conflicts may occur. In order to ensure a conflict-free situation, the HACADP must be solved again. Sometimes, the maneuver to return to the initial trajectory must start when the aircraft is already close to the boundary of the air sector. This could lead to an angle variation exceeding the bounds. In this case, Alonso-Ayuso et al. [35] proposes turning at the maximum bound and sending a warning message to the air traffic controllers of the following sector notifying that the aircraft is arriving in that sector at a different entry point w.r.t. the scheduled one.
Cut dominance
The time per iteration taken by Algorithm 1 and Algorithm 2 increases with the number of cuts added to the formulation. In fact, while solutions of the lower-level subproblems are easily computed in closed form in Step 4 of both algorithms respectively, increasing the number of quadratic constraints (20) and (21) yields a time increase when solving \(R_h\). It would therefore be desirable to remove as many unnecessary cuts as possible. Consequently, we study the existence of dominance relationships between cuts in the proposed cut generation algorithms.
Proposition 5
Let (i, j) be a pair of aircraft. There is no dominance relationship between any pair of the cuts added for (i, j) by either Algorithms 1 or 2.
Proof
Let \(h, h'\) be two different iterations of the cut generation algorithm in which cuts were added for the pair (i, j). We consider the time instants at which i and j are closest in each iteration, \(\tau _{ij}^h\) and \(\tau _{ij}^{h'}\), with \(\tau _{ij}^h\ne \tau _{ij}^{h'}\). Taking the case of Algorithm 1 (the proof for Algorithm 2 is analogous), we proceed by contradiction and suppose that the cut added in iteration h dominates that of iteration \(h'\). That is, for all feasible \(q_i, q_j\):
implies
or, equivalently,
In particular, for \(q_i, q_j\) equal to the solution obtained at iteration \(h'\), \(q_i^{h'}, q_j^{h'}\), the inequality
must hold. Being \(\tau _{ij}^{h'}\) a minimizer of the function
it has to be:
Since \(\tau _{ij}^h\ne \tau _{ij}^{h'}\), it yields that f(t) attains its minimum at two different points. This is only possible if f(t) is constant for all t, i.e., if \(q_i^{h'} v_i u_{ik} = q_j^{h'} v_j u_{jk}\). But, if a cut is added for (i, j) at iteration \(h'\) is because the separation distance was violated also at \(t=0\) (since f is constant), something that we discarded by assumption. \(\square \)
Proposition 5 ensures that a cut added in a certain iteration for a pair of aircraft will not be dominated by any cut added in future iterations for the same pair. However, whether several cuts together dominate a single cut is a question that remains open. Similarly, the study of potential dominance between cuts involving conflicting triplets of aircraft in special cases would be worth-studying for future work.
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Cerulli, M., D’Ambrosio, C., Liberti, L. et al. Detecting and solving aircraft conflicts using bilevel programming. J Glob Optim 81, 529–557 (2021). https://doi.org/10.1007/s10898-021-00997-1
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DOI: https://doi.org/10.1007/s10898-021-00997-1