Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation

Objective: comparative assessment of variants of numerical integration of the longitudinal equation of motion of an electric train and identification of rational combinations of method and step size for the computational core of an automatic train operation system according to the criteria of accuracy and computation time. Methods: integration of the longitudinal equation of motion with respect to time, speed and distance was considered using separate sets of numerical methods. For integration with respect to distance, an energy transformation of the equation was additionally investigated. For the methods considered, comparative simulation of the electric train in acceleration mode was performed for four variants of the longitudinal profile. Accuracy was evaluated by the absolute deviations of the final values of distance and time from the reference calculation results. The Derringer–Suich method was used to form a composite indicator accounting for accuracy and computation time. Results: the choice of integration method for the computational core of an automatic train operation system should take into account the independent variable and step size. Integration with respect to speed provides high computational performance over intervals of monotonic speed change, but the calculation becomes more complicated when speed is maintained. Integration with respect to time is convenient when the duration of individual driving modes has to be maintained. Since in both cases the path is a dependent variable, changes in slope, speed limits and control commands associated with certain coordinates may occur within a step and must be taken into account separately. When integration is performed with respect to distance, the coordinates of these changes are used directly as step boundaries. For explicit second-order methods used for integration with respect to distance, a broad range of step sizes with a high composite indicator was identified across the profiles considered. Practical significance: the results make it possible to select the mathematical basis for the computational core of an automatic train operation system, taking into account the required balance between accuracy and computational performance, the features of the input dependencies and the applicability range of each method.