Robust aircraft conflict resolution under trajectory prediction uncertainty

Abstract: We address the aircraft conflict resolution problem under trajectory prediction uncertainty. We consider that aircraft velocity vectors may be perturbed due to weather effects, such as wind, or measurement errors. Such perturbations may affect aircraft trajectory prediction which plays a key role in ensuring flight safety in air traffic control. Our goal is to solve the aircraft conflict resolution problem in the presence of such perturbations and guarantee that aircraft are separated for any realization of the uncertain data. We propose an uncertainty model wherein aircraft velocities are represented as random variables and the uncertainty set is assumed to be polyhedral. We consider a robust optimization approach and embed the proposed uncertainty model within state-of-the-art mathematical programming formulations for aircraft conflict resolution. We then adopt the approach of Bertsimas and Sim (2004) to formulate the robust counterpart. We use the complex number reformulation and the constraint generation algorithm proposed by Dias et al. (2020a) to solve the resulting nonconvex optimization problem on benchmarking instances of the literature. Our numerical experiments reveal that perturbations of the order of $\pm 5\%$ on aircraft velocities can be accounted for without significantly impacting the objective function compared to the deterministic case. Our tests also show that for greater levels of uncertainty, several instances fail to admit conflict-free solutions, thus highlighting existing risk factors in aircraft conflict resolution. We attempt to explain this behavior by further analyzing pre- and post-optimization aircraft trajectories. Our findings show that most infeasible instances have both a relatively low total aircraft pairwise minimal distance and a high number of conflicts.