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

This 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. Nomenclature α r Maximum displacement allowed by the swept volume constraintβ r ( ω ) Velocity constraintη ( x; iω ) Frequency-domain expression for wave surface elevationη ( x , t ) Time-domain wave surface elevation at point x and time tω Angular frequency of wave/oscillationā t ( ω ),  ā b ( ω ) Added mass variations for the top and bottom bodies, respectively, inclusive of infinite-frequency partsA Incident wave amplitudeA c ( ω ), b c ( ω ) Added mass and radiation damping coefficients representing the frequency-dependent radiation coupling between the top and bottom bodiesb t ( ω ), b b ( ω ) Radiation damping variations for the top and bottom bodies, respectivelyc dt , c db Linearized, constant viscous damping coefficients for the top and bottom bodies, respectivelyD Constant damping load applied on the relative heave oscillationD Distance between the up-wave measurement point and the device centroid; x B ‐ x AF a ( t ) Reactive control force applied by the power takeoffF e ( iω ) Effective heave force F l ( t ) Resistive control force applied by the power takeoffF fb ( iω ) Exciting force coefficient of reaction frameF ft , F fb Exciting forces on the top and bottom bodies, respectivelyF ft ( iω ) Exciting force coefficient of standard buoyF relative ( iω ) Relative exciting force coefficientF total ( iω ) Total exciting force coefficientg Acceleration of gravityh l ( t; d ) Impulse response function defining the deterministic propagation model for distance Dh s1(2) Significant wave height for swell (wind) seask ( ω ) Wave number; related to angular frequency ω through the dispersion relationk t , k b Stiffness constants determining the restoring forces on the top and bottom bodies, respectivelym t , m b In-air masses of the top and bottom bodies, respectivelyP ω Average power absorbed over time tR i Relative radiation damping coefficientR i ( ω ), c i ( ω ) Equivalent hydrodynamic damping and reactance components “acting on” the relative oscillation between the two bodiess Geometric scale factor, defined as ratio of full-scale length dimension and model-scale length dimensiont e1(2) Energy period for swell (wind) seasv r , x r Relative heave velocity and displacement between the top and bottom bodiesv t , v b Heave oscillation velocities of the top and bottom bodies, respectivelyv ro ( iω ) Hydrodynamically optimum velocityZ b Complex impedance of the bottom bodyZ c Complex impedance representing radiation coupling between the top and bottom bodiesZ L Complex impedance representing resistive and reactive loadsZ t Complex impedance of the top body