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An Eulerian network model for air traffic flow in the National Airspace System is developed and used to design flow control schemes which could be used by Air Traffic Controllers to optimize traffic flow. The model relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE), which contains a velocity control term inside the divergence operator. This PDE can be related to aircraft count, which is a key metric in air traffic control. An analytical solution to the LWR PDE is constructed for a benchmark problem, to assess the gridsize required to compute a numerical solution at a prescribed accuracy. The Jameson-Schmidt-Turkel (JST) scheme is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against this analytical solution. Linear numerical schemes are discarded because of their poor performance. The model is validated against actual air traffic data (ETMS data), by showing that the Eulerian description enables good aircraft count predictions, provided a good choice of numerical parameters is made. This model is then embedded as the key constraint in an optimization problem, that of maximizing the throughput at a destination airport while maintaining aircraft density below a legal threshold in a set of sectors of the airspace. The optimization problem is solved by constructing the adjoint problem of the linearized network control problem, which provides an explicit formula for the gradient. Constraints are enforced using a logarithmic barrier. Simulations of actual air traffic data and control scenarios involving several airports between Chicago and the U.S. East Coast demonstrate the feasibility of the method.
Bayen et al. (Tue,) studied this question.