Abstract
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.
Highlights
The problem of energy extraction from an environmental source is certainly one of the key problems in the history of technology
optimally controlled Betz’s machine (OCBM)-optimally controlled steady wind turbine problem consists in determining how the regulation law of the electrical machine c(t) should be designed to maximize the energy E extracted from the flow
The following analysis focuses on the fluctuating component V1 (ω) of the wind speed, in order to provide the conditions in which it is better to apply the OCBM controller rather than the steady wind turbine, and an estimation of the amount of increased energy storage
Summary
The problem of energy extraction from an environmental source is certainly one of the key problems in the history of technology This issue has set the basis of thermal engines and the related thermodynamic theory since the 17th century and recalls the notion of engine efficiency that reaches its highest point in the statement of the Carnot’s theorem. In the theory of wind turbines, developed by Betz [2] in the 1930s, the maximum efficiency of a windmill that extracts energy from a steady wind flow has been investigated, unveiling the existence of a limit to the maximum extractable power, analogously with the thermodynamic efficiency approached by Carnot. The paper shows numerical simulations to validate the analytical results for different wind conditions, such as harmonic oscillations, wind gusts, and random fluctuations
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