Hydrodynamic Considerations in Near-Optimal Control of a Small Wave Energy Converter for Ocean Measurement Applications

Abstract

AbstractThis paper investigates the use of wave energy to power long-term ocean sensing systems. The device examined here consists of an oceanographic buoy and a shallow-submerged reaction frame that may carry a science instrument. Power conversion is from the relative heave oscillation between the two bodies. The oscillation is controlled on a wave-by-wave basis using near-optimal feedforward control, which requires up-wave surface elevation measurement and deterministic prediction at the device location. This paper presents the dynamic formulation used to evaluate the near-optimal, wave-by-wave control forces in the time domain. Also examined are reaction-frame geometries for their impact on overall power capture through favorable hydrodynamic interactions. Performance is evaluated in a range of wave conditions (from most to least favorable for conversion) sampled over a year at a chosen site of deployment. It is found that control may be able to provide the required amounts of power to sustain instrument operation at the chosen site but also that energy storage options may be worth pursuing.<def-list>Nomenclature<def-item><term>αr</term><def>Maximum displacement allowed by the swept volume constraint</def></def-item><def-item><term>βr (ω)</term><def>Velocity constraint</def></def-item><def-item><term>η(x; iω)</term><def>Frequency-domain expression for wave surface elevation</def></def-item><def-item><term>η(x, t)</term><def>Time-domain wave surface elevation at point x and time t</def></def-item><def-item><term>ω</term><def>Angular frequency of wave/oscillation</def></def-item><def-item><term>āt(ω), āb(ω)</term><def>Added mass variations for the top and bottom bodies, respectively, inclusive of infinite-frequency parts</def></def-item><def-item><term>A</term><def>Incident wave amplitude</def></def-item><def-item><term>Ac(ω), bc(ω)</term><def>Added mass and radiation damping coefficients representing the frequency-dependent radiation coupling between the top and bottom bodies</def></def-item><def-item><term>bt(ω), bb(ω)</term><def>Radiation damping variations for the top and bottom bodies, respectively</def></def-item><def-item><term>cdt, cdb</term><def>Linearized, constant viscous damping coefficients for the top and bottom bodies, respectively</def></def-item><def-item><term>D</term><def>Constant damping load applied on the relative heave oscillation</def></def-item><def-item><term>D</term><def>Distance between the up-wave measurement point and the device centroid; xB ‐ xA</def></def-item><def-item><term>Fa(t)</term><def>Reactive control force applied by the power takeoff</def></def-item><def-item><term>Fe(iω)</term><def>Effective heave force</def></def-item><def-item><term>Fl(t)</term><def>Resistive control force applied by the power takeoff</def></def-item><def-item><term>Ffb(iω)</term><def>Exciting force coefficient of reaction frame</def></def-item><def-item><term>Fft, Ffb</term><def>Exciting forces on the top and bottom bodies, respectively</def></def-item><def-item><term>Fft (iω)</term><def>Exciting force coefficient of standard buoy</def></def-item><def-item><term>Frelative (iω)</term><def>Relative exciting force coefficient</def></def-item><def-item><term>Ftotal (iω)</term><def>Total exciting force coefficient</def></def-item><def-item><term>g</term><def>Acceleration of gravity</def></def-item><def-item><term>hl (t; d)</term><def>Impulse response function defining the deterministic propagation model for distance D</def></def-item><def-item><term>hs1(2)</term><def>Significant wave height for swell (wind) seas</def></def-item><def-item><term>k(ω)</term><def>Wave number; related to angular frequency ω through the dispersion relation</def></def-item><def-item><term>kt, kb</term><def>Stiffness constants determining the restoring forces on the top and bottom bodies, respectively</def></def-item><def-item><term>mt, mb</term><def>In-air masses of the top and bottom bodies, respectively</def></def-item><def-item><term>Pω</term><def>Average power absorbed over time t</def></def-item><def-item><term>Ri</term><def>Relative radiation damping coefficient</def></def-item><def-item><term>Ri(ω), ci(ω)</term><def>Equivalent hydrodynamic damping and reactance components “acting on” the relative oscillation between the two bodies</def></def-item><def-item><term>s</term><def>Geometric scale factor, defined as ratio of full-scale length dimension and model-scale length dimension</def></def-item><def-item><term>te1(2)</term><def>Energy period for swell (wind) seas</def></def-item><def-item><term>vr, xr</term><def>Relative heave velocity and displacement between the top and bottom bodies</def></def-item><def-item><term>vt, vb</term><def>Heave oscillation velocities of the top and bottom bodies, respectively</def></def-item><def-item><term>vro (iω)</term><def>Hydrodynamically optimum velocity</def></def-item><def-item><term>Zb</term><def>Complex impedance of the bottom body</def></def-item><def-item><term>Zc</term><def>Complex impedance representing radiation coupling between the top and bottom bodies</def></def-item><def-item><term>ZL</term><def>Complex impedance representing resistive and reactive loads</def></def-item><def-item><term>Zt</term><def>Complex impedance of the top body</def></def-item></def-list>