Boltzmann model for determining conflict probability in ATM

This study is in the framework of the air traffic management and proposes a new definition of the conflict probability for an aeroplanes pair, which is valid in three dimensions. Following this method, the conflict probability is defined as the space correlation probability between the regions which correspond to the Protected Airspace Zone. A distribution function of space which represents the Protected Airspace Zone, is associated to each aircraft, and an error model, based on the Fokker-Planck equation, accounts for the trajectory prediction errors. The conflict probability between two airplanes is calculated in the sense of the Boltzmann theory, i.e. through the space integral of the product of the distribution functions of the two aircraft. An analytical expression of the conflict probability is established which is valid in a general three-dimensional case. This approach is capable to account for the effects of turbulence on the trajectory error and on the conflict probability by means of the Taylor scale concept. To validate the model, present results are compared with those known in the literature and with those obtained through Montecarlo simulations. To evaluate the opportunity of using the proposed conflict probability definition, several simulations are made, where an optimization code based on the Davidon-Fletcher and Powell method which minimizes the conflict probability of the diverse airplanes pairs is adopted.