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This paper introduces a novel reformulation and numerical methods for optimal control of complementarity Lagrangian systems with state jumps. The solutions of the reformulated system have jump discontinuities in the first time derivative instead of the trajectory itself, which is easier to handle theoretically and numerically. We cover not only the easier case of elastic impacts, but also the difficult case, when after the state jump the system evolves on the boundary of the dynamic's feasible set. In nonsmooth mechanics this corresponds to inelastic impacts. The main idea of the time-freezing reformulation is to introduce a clock state and an auxiliary dynamic system whose trajectory endpoints satisfy the state jump law. When the auxiliary system is active, the clock state is not evolving, hence by taking only the parts of the trajectory when the clock state was active, we can recover the original solution. We detail how to recover the solution of the original system, show how to select appropriate auxiliary dynamics and give practical numerical methods to handle discontinuous ODEs with nonunique sliding motions. Moreover, we introduce a novel auxiliary ODE for time-freezing for elastic impacts and overcome some drawbacks of [22]. The theoretical findings are illustrated on the nontrivial numerical optimal control example of a hopping one-legged robot.

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This article discusses how to use optimization-based methods to efficiently operate microgrids with a large share of renewables. We discuss how to apply a frequency-based method to tune the droop parameters in order to stabilize the grid and improve oscillation damping after disturbances. Moreover, we propose a centralized real-time feasible nonlinear model predictive control (NMPC) scheme to achieve efficient frequency and voltage control while considering economic dispatch results. Centralized NMPC for secondary control is a computationaly challenging task. We demonstrate how to reduce the computational burden using the Advanced Step Real-Time Iteration with nonuniform discretization grids. This reduces the computational burden up to 60 % compared to a standard uniform approach, while having only a minor performance loss. All methods are validated on the example of a 9-bus microgrid, which is modeled with a complex differential algebraic equation.

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