Overshoots in the water-level control of hydropower plants

In the field of renewable-energies, a number of engineering problems are modeled as dynamical systems, and a key issue is the assessment of their stability to external disturbances. Stability analyses typically focus on the asymptotic stability, i.e., the fate of perturbations after long times from their onset. The system behavior at finite-times has attracted much less attention, although it plays a crucial role in determining the system dynamics. In this work, we focus on the response at finite-times to perturbations in run-of-river hydropower plants. These are widespread systems in the hydropower industry. We show that their response at finite-times (i) can be analytically studied by the non-modal analysis, and (ii) can be very different from the asymptotic-times response. In particular, perturbations can exhibit very relevant transient amplifications (with important technical consequences), although the system is asymptotically stable. The proposed analytical approach is general and can be applied to investigate the finite-time response of any dynamical system.