Effect of heave plate hydrodynamic force parameterization on a two-body wave energy converter

Abstract

Heave plates are one approach to generating the reaction force necessary to harvest energy from ocean waves. In a Morison equation description of the hydrodynamic force, the components of drag and added mass depend primarily on the heave plate oscillation. These terms may be parameterized in three ways: (1) as a single coefficient invariant across sea state, most accurate at the reference sea state, (2) coefficients dependent on the oscillation amplitude, but invariant in phase, that are most accurate for relatively small amplitude motions, and (3) coefficients dependent on both oscillation amplitude and phase, which are accurate for all oscillation amplitudes. We validate a MATLAB model for a two-body point absorber wave energy converter against field data and a dynamical model constructed in ProteusDS. We then use the MATLAB model to evaluate the effect of these parameterizations on estimates of heave plate motion, tension between the float and heave plate, and wave energy converter electrical power output. We find that power predictions using amplitude-dependent coefficients differ by up to 30% from models using invariant coefficients for regular waves ranging in height from 0.5 to 1.9 m. Amplitude- and phase-dependent coefficients, however, yield less than a 5% change when compared with coefficients dependent on amplitude only. This suggests that amplitude-dependent coefficients can be important for accurate wave energy converter modeling, but the added complexity of phase-dependent coefficients yields little further benefit. We show similar, though less pronounced, trends in maximum tether tension, but note that heave plate motion has only a weak dependence on coefficient fidelity. Finally, we emphasize the importance of using experimentally derived added mass over that calculated from boundary element methods, which can lead to substantial under-prediction of power output and peak tether tension.

Appendices

Appendices

A WEBS specifications

Table 3 System properties

Table 4 System properties for numerical simulation

Fig. 10

Design measurements for the WEBS open hexagonal conic heave plate

Fig. 11

Model of WEBS PTO used to estimate electrical power output from generator shaft speed

Here, we show the specifications for WEBS, as presented in Rusch (2020), which are also used in the Proteus and MATLAB models. Table 3 outlines specifications for all critical WEBS components. Table 4 shows secondary specifications required for numerical simulation. The values listed for center of gravity are for the device in a horizontal position, where the nacelle arm is parallel to the sea surface (i.e., $\phi = 90^{\circ }$).

Additional detail for the heave plate design is given Fig. 10. Elongation properties used to determine tether stiffness properties are based on manufacturer specifications (3/4” Samson Tenex). In addition, the model used to estimate electrical power output for WEBS from PTO (Power Take-Off) velocity (${\dot{\phi }}$) is shown in Fig. 11. This model was constructed using synchronized measurements of electrical power output and PTO velocity from WEBS field testing.

B Proteus model

The Proteus model (ProteusDS) relies on the parameters outlined in Appendix A. Floats and nacelle are constructed from standard cylinder rigid body features. The PTO is represented using a “RigidBodyConnectionABAJoint” with spring and damping coefficients as specified in Table 3. As ProteusDS may experience difficulties with hydrodynamics on concave surfaces, a solid cone is used to represent the heave plate, and buoyancy calculations within Proteus DS are turned off for the heave plate and specified manually.

The wave spectrum is defined using a Pierson–Moskowitz spectrum as described in DSA (2018). This spectrum has a wave heading of $0^{\circ }$, peak period of 9.66 s, significant wave height of 1.33 m, with 270 wave segments and a wave seed of 12,345. The number of wave segments and peak period affect the return period of the wave field, or the length of time that a simulation is run before the wave field repeats. For this case, the maximum run time is 695 s. No wind or currents are included in this simulation.

C MATLAB model

Fig. 12

Free body diagram for WEBS surface body and heave plate. $l_{\text {cb}}$ denotes the distance from nacelle to center of buoyancy of the float, and $l_\mathrm{cm}$ denotes the distance from nacelle to the center of mass of the float

The MATLAB model is constructed from the equations of linear motion governing the components of WEBS in the vertical direction and from the equations of rotational motion about the nacelle. From an initial set of equations describing the floats, nacelle, and heave plate, as well as compatibility conditions, these can be simplified to one equation in the vertical direction of motion and one equation to describe the rotation about the nacelle. A free body diagram describing the direction of force acting on the device in equilibrium is shown in Fig. 12. The displacement between the center of mass and center of buoyancy (Fig. 12a) affects the moments induced on the PTO by the forces of buoyancy and gravity, reflected in the equation of motion describing rotation about the nacelle:

$$\begin{aligned} \begin{aligned}&\ddot{\phi } (J_\mathrm{PTO} - J_1 - J_2) = \sin \phi [l_{\text {cb}}(F_\mathrm{b,1} \\&\quad + F_\mathrm{b,2}) - l_\mathrm{cm} g(m_1+m_2)] \\&\quad + 2k (\phi - \phi _0) - 2c{\dot{\phi }}. \end{aligned} \end{aligned}$$

(3)

In this equation, $J_\mathrm{PTO}$ is the rotational moment of inertia for the PTO, and $J_1$ and $J_2$ are the moments of inertia for the two floats about the nacelle. $F_b$ represents forces of buoyancy on the nacelle (subscript 0), aft float (subscript 1), fore float (subscript 2), and heave plate (subscript hp). m represents mass, g is the gravitational constant, $l_\mathrm{cm}$ is the distance from the center of the nacelle to the center of mass of the float, $l_\mathrm{cb}$ is the distance to the center of buoyancy of the float, and phi is the angle of the arm from vertical. As $\phi$ is half of the angle between floats, $\phi _0 = 130^{\circ }$. Finally, k is the spring constant and c is the damping constant used to approximate the PTO. The spring reference angle shown in Table 3, represents the angle between floats at which the spring from the PTO exerts no force in the positive $\phi$ direction.

The equation of motion in the vertical (z) direction is

$$\begin{aligned} \begin{aligned}&(m_0 + m_1 + m_2 + m_\mathrm{hp} + C_{\text {a}} \rho \pi /6 D_\mathrm{hp}^3 )\ddot{z}_0\\&\quad = g (m_0 + m_1 + m_2 + m_\mathrm{hp})\\&\qquad -[F_\mathrm{B} + C_{\text {d}}(1/8 \rho \pi D_{hp}^2){\dot{z}}_0 |{\dot{z}}_0|]\\&\qquad -(m_0 + m_1)(l_\mathrm{cm} {\dot{\phi }}^2 \tan \phi \sin \phi + l_{cm} {\dot{\phi }}^{2} \cos \phi )\\&\qquad - m_2(l_\mathrm{cm} {\dot{\phi }}^2 \tan \phi \sin \phi + l_\mathrm{cm} {\dot{\phi }}^{2} \cos \phi ). \end{aligned} \end{aligned}$$

(4)

The added mass force of the heave plate is represented as a product of the coefficient of added mass $C_{\text {a}}$, the fluid density ($\rho$), the volume of a sphere with an effective diameter equal to that of the heave plate ($D_\mathrm{hp}$), and the vertical acceleration of the plate ($\ddot{z}$). Buoyancy of all bodies is represented as $F_{\text {B}} = F_\mathrm{b,0} + F_\mathrm{b,1} + F_\mathrm{b,2} +F_\mathrm{b,hp}$. Drag is calculated as a product of the drag coefficient ($C_{\text {d}}$), the area of the heave plate, and the square of vertical velocity (${\dot{z}}$). Because of the rigid coupling, heave plate motion is equal to nacelle motion.

In the model, these equations are solved for $\ddot{\phi }$ and $\ddot{z}$, respectively. We then rewrite the equations as a set of first order linear ordinary differential equations, which we solve using ODE45 in MATLAB. The model updates every time step with new values for $F_\mathrm{b,1}$ and $F_\mathrm{b,2}$. These change based on the position of the floats with respect to mean water level ($z = 0$) at the current time step and the position of the free surface for the next time step. We assume the water surface is uniform and level across the floats.