Variational control approach to energy extraction from a fluid flow

Energy harvesting from the environment is an important aspect of many technologies. The scale of energy capturing and storage can involve the power range from mWatt up to MWatt, depending on the used devices and the considered environments (from ambient acoustic and vibration to ocean wave motion, or wind). In this paper, the wind turbine energy harvesting problem is approached as an optimal control problem, where the objective function is the absorption of an amount of energy in a given time interval by a fluid-flow environment, that should be maximized. The interest relies on outlining general control models of fluid-flow-based extraction plants and identifying an optimum strategy for the regulation of an electrical machine to obtain a maximum-efficiency process for the related energy storage. The mathematical tools are found in the light of optimal control theory, where solutions to the fundamental equations are in the frame of Variational Control (the basis of the Pontryagin optimal control theory). A special problem, named Optimally Controlled Betz’s Machine OCBM-optimal control steady wind turbine, is solved in closed form, and it is shown that, in the simpler steady case, it reproduces the maximum efficiency machine developed in Betz’s theory.