Geometrical Framework for Hydrodynamics and Control of Wave Energy Converters

Abstract

This article presents a simple geometrical approach to visualize the hydrodynamic properties of wave energy converters (WECs), in terms of wave reflection, transmission, and absorption, and how those properties are governed by the WEC control parameters. The problem is modeled as an array of periodic rows of WECs parallel to the wave front, which is representative of WEC farms located along the shoreline, and allows for a straightforward two-dimensional analysis of energy fluxes. The WECs are assumed to be symmetric with respect to the plane perpendicular to the wave propagation direction, and they operate in a single degree of freedom. Under those assumptions, fundamental hydrodynamic relationships allow the WEC operation at a given frequency to be mapped to a single complex number, T[over ^], which represents the WEC (complex) transmission coefficient, located in a circle with center 1/2 and radius 1/2. The WEC hydrodynamic and control parameters (added mass, stiffness, damping) govern the precise location of T[over ^] within the circle. The distance of T[over ^] to the center of the circle determines the hydrodynamic efficiency, between 0 (when T[over ^] is on the circle border) and 1/2 (which is achieved when T[over ^] is at the center of the circle). Therefore, the representation of T[over ^] provides an immediate insight into the balance between reflection, transmission, and absorption. The proposed unified representation reflects, in a didactic way, some fundamental wave energy concepts, common to all WECs, such as the Haskind relationship, or the impedance matching condition for optimal wave power absorption. Two numerical examples illustrate how the locus of T[over ^], across a prescribed frequency range, provides a distinctive “signature” specific to the WEC geometry, mode of operation, and control strategy. Finally, the proposed representation shows many similarities to the Smith chart and, as such, is but one additional analogy between wave energy conversion and electrical engineering.