Bi-level homotopic aircraft sequencing using gradient-based arrival time assignment and direct optimal control

A bi-level homotopy approach is investigated for sequencing arriving aircraft under the assumption, that each aircraft is operated optimally. The lower level of this bi-level problem is represented by an optimal control problem for each individual aircraft, which have to respect separation constraints with regard to each other. This lower level algorithm is embedded into the upper level, which finds the optimal arrival sequence using a gradient based approach. In order to avoid a discrete combinatorial problem in the upper level, the sequence is represented by the continuous arrival time of each aircraft. The Lagrangian multipliers of the discretized lower level problems are used as an approximation of the gradient of the upper level Lagrange function. Furthermore, a homotopy approach is introduced to facilitate the solution process by incrementally increasing time and distance separation. The algorithm is validated by implementing a scenario comprising five aircraft, which is solved within 17min on a desktop PC.