Wave Energy Converter Buoy with Variable Geometry

Abstract

A nonlinear control design technique capitalizes on a wave energy converter comprising a shaped buoy having a variable geometry wave energy. For example, the shaped buoy can have an hourglass (HG) geometry having a variable cone or steepness angle. The HG buoy is assumed to operate in the heave motion of the wave. The unique interaction between the HG buoy and the wave creates a nonlinear cubic storage effect that produces actual energy storage or reactive power during operation. A multi-frequency Bretschneider spectrum wave excitation input was simulated for the HG design both with constant and varying steepness angle profiles which demonstrated further increased power generation with changing sea states for the variable design.

Claims

1. A wave energy converter, comprising: a shaped buoy in a body of water having a wave motion, wherein the waves impacting the buoy exert an excitation force with a plurality of excitation frequencies on the buoy that causes a buoy motion in a heave direction relative to a reference and wherein the buoy has a geometry such that a water plane area of the buoy increases with distance away from the water line in the heave direction both above and below the water line, thereby producing reactive power from the wave motion; and a controller configured to vary the geometry of the buoy in response to the wave motion.

2. The wave energy converter of claim 1, wherein the shaped buoy comprises an hourglass geometry.

3. The wave energy converter of claim 2, wherein the hourglass geometry comprises mirrored right circular cones having a steepness angle.

4. The wave energy converter of claim 3, wherein the steepness angle is variable from 35 to 75 degrees.

5. The wave energy converter of claim 3, wherein the steepness angle is varied to harvest maximum energy from the wave motion.

6. The wave energy converter of claim 1, wherein the shaped buoy comprises opposing geometries that are mirrored about the water line.

7. The wave energy converter of claim 6, wherein the shaped buoy comprises mirrored hemispheres, pyramids, ellipsoids, paraboloids, or hyperboloids.

8. The wave energy converter of claim 1, wherein the shaped buoy comprises opposing geometries that are not mirrored about the water line.

9. The wave energy converter of claim 1, wherein the shaped buoy comprises a geometry of a polynomial spline expansion.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] The detailed description will refer to the following drawings, wherein like elements are referred to by like numbers.

[0009] FIG. 1 is a schematic illustration of an hourglass variable geometric buoy design with two configurations shown: maximum and minimum steepness angles which are superimposed.

[0010] FIG. 2 is a block diagram of a control apparatus for variable geometry WEC system.

[0011] FIG. 3 is a graph of a Bretschneider wave input profile.

[0012] FIG. 4 is a graph of buoy steepness and draft limit specifications as a function of time for the variable geometry hourglass buoy.

[0013] FIG. 5 is a graph of external force responses for the variable and constant steepness angle hourglass buoys.

[0014] FIG. 6 is a graph of control force responses.

[0015] FIG. 7 is a graph of real power responses.

[0016] FIG. 8 is a graph of reactive power responses.

[0017] FIG. 9 is a graph of buoy position responses.

[0018] FIG. 10 is a graph of buoy velocity responses.

DETAILED DESCRIPTION OF THE INVENTION

[0019] In their simplest form, linear WEC point absorbers can be defined for a regular wave, where the excitation force has only one frequency, co, and it can be shown that the radiation term can be quantified using an added mass and a radiation damping term, each considered at a constant frequency only. See J. N. Newman, Marine Hydrodynamics, The MIT Press, USA (1977); and J. Falnes, Ocean Waves and Oscillating Systems, Cambridge: NY: Cambridge University Press (2002). The equation-of-motion for this simple case is expressed as

m{umlaut over (z)}+c{umlaut over (z)}+kz=F.sub.e+F.sub.u  (1)

where m and c are constant mass and damping terms for a given excitation frequency, and k is the linear stiffness term. F.sub.e is the input excitation force and F.sub.u is the control force. Further details for a heave motion linear WEC system can be found in Song et al. See J. Song et al., Ocean Eng. 27, 269 (2016).

[0020] The heave oscillations for a 1-DOF (degree-of-freedom) buoy relative to a reaction mass can be modeled simply with a power-take-off (PTO) system consisting of a linear actuator as part of the power conversion from mechanical to electrical power. The hourglass (HG) buoy nonlinear variable geometry is shown in FIG. 1. This exemplary HG buoy comprises mirrored right circular cones having a draft, 2h, radius, r, and cone or steepness angle, α. The buoy is mechanically coupled to a reference, in this example a deeply submerged reaction mass. The reactive mass is submerged deep enough for its oscillations to be negligible in wave conditions of interest for power conversion. Therefore, the reaction mass can remain essentially stationary as the buoy moves. The mirrored cone buoy can move up and down along a vertical Z axis in a heaving motion (a real buoy would generally move with three degrees-of-freedom, further including an up/down rotation about a center-of-gravity in a pitching motion, and back-and-forth, side-to-side displacement in a surging motion). The PTO can comprise an actuator that couples the buoy with the reaction mass and can be used to apply a control force or reactive power to the buoy. The PTO actuator assembly can further be configured to convert the buoy mechanical motion to electrical energy, which can then be output by way of a transmission line (not shown). For the variable design, the steepness angle variation with time can be achieved using an angle servo.

[0021] The corresponding range of parameters investigated herein is shown in Table I.

[0022] Table I. WEC Hourglass Variable Geometry Parameters.

TABLE-US-00001 TABLE I WEC Hourglass Variable Geometry Parameters. Buoy r (m) h (m) α (degrees) HG 5.72-10.0 8.18-2.68 35-75

[0023] Nonlinear Control Driven Buoy Design

[0024] At resonance, a WEC device operates at maximum energy absorption. See D. G. Wilson et al., J. Mar. Sci. Eng. 8(2), 84 (2020); and U.S. application Ser. No. 16/792,749. In off-resonance the WEC absorbs less real power and will require reactive power to increase energy capture by enabling resonance. Practically, this can be achieved with model predictive control (MPC) or PDC3 (Proportional-Derivative C3). See J. Hals et al., J. Offshore Mech. Arct. Eng. 133(3), 031101 (2011); G. Li et al., Renew. Energy 48, 392 (2012); J. A. Cretel et al., Maximization of energy capture by a wave-energy point absorber using model predictive control, in 18th IFAC World Congress, Milano (Italy) Aug. 28-Sep. 2, 2011; J. Song et al., Ocean Eng. 27, 269 (2016); and D. G. Wilson et al., Order of Magnitude Power Increase from Multi-Resonance Wave Energy Converters, OCEANS' 17 MTS/IEEE, Anchorage, Ak., Sep. 20-22, 2017. Both techniques require energy storage and power electronic elements. MPC will also need additional wave prediction as a priori input. The present invention utilizes a nonlinear (NL) control design to realize a nonlinear buoy with variable geometry to produce the energy storage and reactive power through the nonlinear coupling between the buoy and wave interaction, thus eliminating the need for energy storage and power electronic elements. See R. D. Robinett III and D. G. Wilson, Nonlinear Power Flow Control Design: Utilizing Exergy, Entropy, Static and Dynamic Stability, and Lyapunov Analysis, Springer-Verlag London Limited 2011; D. G. Wilson et al., J. Mar. Sci. Eng. 8(2), 84 (2020); and U.S. application Ser. No. 16/792,749.

[0025] NL Control Design for NL HG Buoy Geometry WEC

[0026] A cubic hardening spring can be created by defining the buoy shape as an HG geometry as shown in FIG. 1. See D. G. Wilson et al., J. Mar. Sci. Eng. 8(2), 84 (2020); U.S. application Ser. No. 16/792,749; D. G. Wilson et al., Nonlinear Control Design for Nonlinear Wave Energy Converters, John L. Junkins Dynamical Systems Symposium, College Station, Tex., May 20-21, 2018; and D. G. Wilson et al., Nonlinear WEC Optimized Geometric Buoy Design for Efficient Reactive Power Requirements, OCEANS' 19 MTS/IEEE, Seattle, Wash., Oct. 27-31, 2019. The model uses a small body approximation. The non-uniform water plane area, S.sub.w, for the hourglass cone is

S.sub.W=πr(z).sup.2=πα.sup.2h.sup.2, where r=h tan α=αh  (2)

The hydrostatic force is proportional to the submerged volume of the body. For very long waves, the wave profile can be considered as having the same value as the vertical coordinate across the cone. That is, z˜η where η is the wave elevation. Assuming the neutral buoyancy or water line (i.e., the equilibrium position) is located at the apex of the mirrored cones, the volume as a function of position of the center-of-volume is

V(z)=⅓πα.sup.2h.sup.3−⅓πα.sup.2z.sup.3=⅓πα.sup.2[h.sup.3−z.sup.3]  (3)

The hydrostatic force for the buoy staying in the water is

F.sub.h=F.sub.g+F.sub.buoy=−mg+pgV(z)=−⅓πρgα.sup.2z.sup.3(4)

A nonlinear WEC model for the HG can be developed from Falnes and Wilson, where the excitation force in heave is dominated by the hydrostatic force. See J. Falnes, Ocean Waves and Oscillating Systems, 1st ed., Cambridge University Press, Cambridge, N Y, 2002; and D. Wilson et al., “10x Power Capture Increased from Multi-Frequency Nonlinear Dynamics,” Sandia National Laboratories, SAND2015-10446R (2015). The summarized equation-of-motion is

m{umlaut over (z)}+cż+K.sub.HG(α)[⅓z.sup.3−ηz.sup.2+η.sup.2z]=⅓K.sub.HG(α)n.sup.3+F.sub.u    (5)

which contains the cubic spring term given by ⅓ K.sub.HG(α)z.sup.3. The parameter K.sub.HG(α) is a function of the steepness angle α, buoy mass and geometric properties, as shown in Equation (4) for the case of the HG buoy.

[0027] FIG. 2 shows a block diagram of a control apparatus for variable geometry WEC system. A Wave Prediction Buoy can be used to measure the incoming wave height and direction. A Wave Prediction algorithm can be used to predict a statistical estimate of the incoming wave height and duration at the buoy from the measured wave height and direction at the Wave Prediction Buoy as controller inputs. The Wave Prediction Buoy and Wave Prediction are a priori inputs to the variable geometry WEC system, as indicated by the dashed line boxes in the block diagram. A Controller provides control for rate feedback control to the PTO and steepness angle control to the angle servo. control to adjust the steepness angle of the variable design, subject to the relative wave height. The PTO control is defined simply as rate feedback or F.sub.u=−K.sub.Dż. The servo control adjusts the steepness angle of the variable design, subject to the relative wave height, according to ⅓K.sub.HG (α)z.sup.3. Feedback Sensors include a PTO rate or velocity feedback sensor and a steepness angle feedback sensor.

[0028] Multi-frequency Numerical Simulations

[0029] Numerical simulations were performed for a Bretschneider spectrum for both a constant and variable steepness angle for the HG WEC design. In U.S. application Ser. No. 16/792,749, four varying Bretschneider Sea States (SS) were investigated with a constant steepness angle, α. Five minute Bretschneider profiles were generated from the MATLAB toolbox. See T. Perez and T. Fossen, Model. Identif. Control 30(1), 1 (2009). The steepness angle was increased in five degree increments until the HG buoy draft constraint was violated. The maximum safe angle was set to the previous value such that the HG buoy would not overtop or exit the water. The results are summarized in Table II for a fixed or constant steepness angle.

TABLE-US-00002 TABLE II Bretschneider Spectrum Sea State Results Sea Steepness angle Energy state (degrees) (MJ) 1 65 67.170 2 70 92.752 3 55 174.63 4 65 69.790
In Wilson et al., the steepness angle was relaxed for one of the larger waves in Sea State 4 and increased power and energy capture was observed. See D. G. Wilson et al., Nonlinear WEC Optimized Geometric Buoy Design for Efficient Reactive Power Requirements, OCEANS' 19 MTS/IEEE, Seattle, Wash., Oct. 27-31, 2019. For this study Sea State 4 was further reviewed and a new scenario was defined that investigates the benefit of utilizing wave estimations with a slower update on α . The following numerical simulation results were produced.

[0030] A Bretschneider spectrum with T.sub.p=11 seconds and H.sub.s=6.9 meters was employed with the corresponding wave input shown in FIG. 3. The steepness angle was varied as a function of time for the variable HG design, as shown in FIG. 4. FIG. 4 also includes the constrained draft heights for SS4. In comparing FIGS. 3 and 4, it is apparent that the steepness angle decreases in response to an increase in the wave height. The variable HG buoy was compared to an HG buoy with a constant steepness angle, set at α=65°. The external forces F.sub.e and control forces F.sub.u for both constant and variable steepness angles are shown in FIGS. 5 and 6. The real and reactive powers are shown in FIGS. 7 and 8. Note that the nonlinear HG buoy design inherently produces the required reactive power shown in FIG. 8. The buoy positions and velocities are given in FIGS. 9 and 10. The energy captured for both the constant and variable HG buoys are given in Table III. By the varying steepness angle, further increases in power and energy capture were achieved.

TABLE-US-00003 TABLE III Energy Captured Bretschneider Comparison. Steepness angle Energy (MJ) Constant (65°) 69.80 Variable 104.6

[0031] The example described above assumed a mirrored right circular cone HG buoy with variable cone or steepness angle. However, other shapes and variations thereof can provide a cubic hardening spring equivalent. Indeed, whenever the buoy has a geometry such that a water plane area increases with distance away from the water line in the heave direction, then the hydrostatic force will be nonlinear. Typically, the buoy shape can be axisymmetric about the buoy axis but is not required to be so. Typically, the buoy can comprise opposing shapes that are mirrored about the water line. For example, the buoy shape can comprise a polynomial spline expansion of the form, z=a+bx+cx.sup.2+dx.sup.3+ex.sup.4+ . . . where a, b, c, d, and e are arbitrary coefficients, rotated about the vertical axis. For example, the shaped buoy can comprise a hyperboloid of revolution about the buoy axis. For example, the shaped buoy can comprise opposing hemispheres, pyramids, ellipsoids, or paraboloids. However, the opposing surfaces need not be mirrored geometries, symmetric about the water line, or of the same shape. In such cases, the shape of the buoy—i.e., the water plane area as a function of distance away from the water line —can be varied in time in response to the wave motion.

[0032] The present invention has been described as a wave energy converter buoy with variable geometry. It will be understood that the above description is merely illustrative of the applications of the principles of the present invention, the scope of which is to be determined by the claims viewed in light of the specification. Other variants and modifications of the invention will be apparent to those of skill in the art.