An Approach to Determining the Number of Time Intervals for Solving Dynamic Optimization Problems

To numerically solve a dynamic optimization problem, the model equations need to be discretized over a time horizon. The very first step therefore is to decide the number of time intervals. In principle, the decision is made to achieve a compromise between the numerical accuracy of the discretization and the computation load for solving the discretized optimization problem. However, there have been no comprehensive rules for this purpose. In the context of collocation on finite elements, we propose a novel bilevel approach to evaluate an upper limit of the approximation error by formulating and solving an error maximization problem. In this way, a minimum number of time intervals can be determined a priori, which guarantees a user-defined error tolerance. In addition, the impact of the initial conditions on the maximum approximation error is taken into account so that the determined number of intervals is valid for varying initial conditions and thus can be applied to nonlinear model predictive control. Several case studies are used to demonstrate the efficacy of the proposed approach.