Derivation and Experimental Validation of a Parameterized Nonlinear Froude–Krylov Force Model for Heaving-Point-Absorber Wave Energy Converters

Abstract

Wave energy converters (WECs) have gained significant attention as a promising renewable energy source. Optimal control strategies, crucial for maximizing energy extraction, have traditionally relied on linear models based on small motion assumptions. However, recent studies indicate that these models do not adequately capture the complex dynamics of WECs, especially when large motions are introduced to enhance power absorption. The nonlinear Froude–Krylov (FK) forces, particularly in heaving-point-absorbers with varying cross-sectional areas, are acknowledged as key contributors to this discrepancy. While high-fidelity computational models are accurate, they are impractical for real-time control applications due to their complexity. This paper presents a parameterized approach for expressing nonlinear FK forces across a wide range of point-absorber buoy shapes inspired by implementing real-time, model-based control laws. The model was validated using measured force data for a stationary spherical buoy subjected to regular waves. The FK model was also compared to a closed-form buoyancy model, demonstrating a significant improvement, particularly with high-frequency waves. Incorporating a scattering model further enhanced force prediction, reducing error across the tested conditions. The outcomes of this work contribute to a more comprehensive understanding of FK forces across a broader range of buoy configurations, simplifying the calculation of the excitation force by adopting a parameterized algebraic model and extending this model to accommodate irregular wave conditions.

1. Introduction

The escalating global demand for clean and sustainable energy sources has fueled extensive research to harness the vast potential of ocean waves. With its consistent availability, wave energy is an attractive renewable resource. Site evaluations have confirmed the potential for ocean wave energy resources [1]. Among the various technologies developed, point-absorber wave energy converters (PAWECs) have emerged as promising devices capable of efficiently capturing and converting wave energy into electrical power [2]. PAWECs, characterized by buoyant structures tethered to the seabed, offer adaptability to varying wave conditions. The wave-induced vertical motion of its buoyant platform is translated into useful work through a power take-off that can act as either a generator or an actuator.

The successful implementation of PAWECs depends on carefully considering design aspects, including buoy size, shape, and mass distribution. Material selection must account for the harsh marine environment, and placement is critical, influenced by factors such as water depth and distance from the shore. Advancements in control systems enable adaptive strategies that maximize energy extraction under varying wave conditions. Improving PAWEC control mechanisms stands out as a crucial avenue with immense potential to improve the feasibility of wave energy utilization [3]. Recent advances highlight the importance of refined control methods, which demonstrate their ability to increase power absorption by up to 20% while simultaneously reducing structural loads [1]. Recent developments in experimental and simulation platforms have also enabled performance validation under irregular wave conditions [4] and improved wave generation algorithms for numerical tests [5].

Extensive research on energy-optimal control, particularly for buoy geometries, has been explored. Topics include establishing upper bounds for energy extraction [6,7,8], impedance matching control [8,9,10,11,12], and exploring both closed-form [13,14] and numerical optimal control solutions [15] such as model predictive control (MPC) [16,17,18,19]. The combination of MPC with data-driven methods, such as LSTM neural networks, has recently shown promising results in improving real-time performance under uncertainty [20]. A common theme is the importance of using accurate WEC math models for analysis and control design. Initially, research focused on buoy response regimes that could be approximated using linear differential equations, valid under the assumption of small motion amplitudes. However, a recent shift toward addressing the nonlinear response of buoys [21,22,23] is underway to improve energy extraction, particularly when the objective is to increase power production. The importance of nonlinearities is that, if the model inaccurately omits or misrepresents them, the resulting optimal control strategy will be fundamentally flawed, leading to incorrect predictions of power extraction and suboptimal energy performance.

Computational fluid dynamics (CFD) models can accurately capture these nonlinearities, but their application in model-based, real-time control laws is impractical [24]. This challenge is a well-known aspect of the “modeling paradox” in wave energy control, where models rich in physics are too complex for real-time use [25]. A previous investigation [26] comparing various modeling approaches showed that an important nonlinearity comes from the Froude–Krylov (FK) force, particularly for axisymmetric buoys with varying cross-sectional areas. This insight underscores the importance of understanding and incorporating the nonlinear effects of FK forces in the modeling process to develop effective control strategies.

The FK forces arise from the pressure field around a submerged or partially submerged body as it moves through waves. Specifically, they are associated with the nonlinear effects induced by the varying shape and submersion of the structure. Together with the scattering forces, they constitute the entire non-viscous force exerted on a floating body subjected to regular waves [27]. The impulse response theory originally proposed by Cummins remains a foundational framework in modeling these hydrodynamic interactions [28]. Developing closed-form, computationally efficient FK force expressions is advantageous for implementing real-time, model-based control strategies. This is especially true for buoys whose shapes result in nonlinear FK forces.

Closed-form nonlinear FK force models have been derived in previous studies [29,30,31], aimed explicitly at using buoy shape effects in model-based control laws for large-motion, nonlinear operating regimes. Giorgi et al. [29] developed a method for generating closed-form FK force expressions using Airy’s wave theory to approximate the pressure on a buoy in regular waves. It has recently been used for model-based control solutions, including sliding model control [32], feedback linearization [33], and latching control [24,34,35]. Shape optimization studies have further explored the use of neural networks and evolutionary algorithms to maximize power capture by optimizing buoy geometry [36,37].

These studies have taken advantage of various buoy shapes, ranging from conventional spheres and cylinders to more unconventional geometries such as the double cone-shaped buoy [21,33]. The FK force approach by Giorgi et al. has been extended in this paper [29] in several ways, including considering flat-bottomed buoys, irregular waves, and a parametrization of a large class of buoy shapes using three parameters. Parameterization allows for rapid evaluation of different geometries, making shape optimization efficient. By generalizing the FK force form, this work provides a powerful tool for improving the design and control of wave energy converters. Furthermore, this paper compares the nonlinear FK model with the buoyancy force approach often used in model-based control [21]. Experiments are used to assess model performance using a spherical buoy subjected to regular waves at several frequencies and amplitudes.

Furthermore, this study focuses specifically on heaving-point-absorbers and introduces three novel contributions to extend the FK force formulation: the incorporation of irregular wave conditions, the inclusion of flat-bottomed buoy geometries, and the development of a three-parameter shape definition that facilitates efficient exploration of buoy designs. These contributions enable a closed-form expression for the FK force and substantially improve the modeling framework used for the optimal design and control of point-absorber wave energy converters.

The paper is organized as follows. The model form is introduced in Section 2, including the distinction between the buoyancy and Froude–Krylov terms that are derived in Section 3 and Section 4. Examples of these terms are compared in Section 5, followed by the experimental validation study in Section 6. Some concluding remarks are provided in Section 7.

2. System Description

Consider the axisymmetric buoy constrained to heave attached by a PTO to the sea floor in Figure 1. In the configuration shown, the buoy has a positive displacement, $\zeta \left(t\right)$, of its draft line relative to the still-water line. The wave elevation, $\eta \left(t\right)$, is also positive in the figure and is assumed to have the general form of Equation (1) that allows the analysis of both regular and irregular waves.

$$\eta (x,t)=\sum _{i=1}^n{A}_i\sin ({\varphi }_i+{\chi }_{i}x-{\omega }_{i}t)=\sum _{i=1}^n{\eta }_i$$

(1)

where $\eta$’s ith component has an amplitude, wave number, angular frequency, and phase shift denoted as $A_i$, ${\chi }_i$, ${\omega }_i$, and ${\varphi }_i$. The wavelengths, ${\lambda }_i=2\pi /{\chi }_i$, are assumed to be large compared to the maximum buoy radius, and therefore the free elevation is considered to be locally horizontal, as shown in Figure 1. A body-fixed coordinate system $(x,y,z)$ is referenced at the bottom of the buoy. The constant water depth, $h_w$, is the distance from the sea floor to the still-water line and is assumed to be large, though it is shown as small in the figure.

Two model variants will be compared, both having the general form of Equation (2)

$$M\ddot{\zeta }=F_P+F_g+F_S+F_R+F_{PTO}$$

(2)

where M is the buoy’s mass. Assuming that the buoy does not affect the wave field, the pressure exerted on the buoy by the water is denoted as $F_p$, and the buoy weight is $F_g$. The scattering force, $F_S$, when combined with the Froude–Krylov force, is called the diffraction force, $F_D$ [38], and captures the effect of wave distortion around a fixed buoy [8]. The radiation force, $F_R$, is due to the energy transfer between the buoy and the water as the buoy moves. Finally, $F_{PTO}$ is the power take-off force that adds or removes buoy energy.

The two model variants shown in Equation (3) are distinguished by how the $F_p+F_g$ term of Equation (2) is treated. In one case, they are replaced by the Froude–Krylov force, $F_{fk}=F_P+F_g$, and in the other by the buoyancy force, $F_b=F_P+F_g$.

$$\begin{array}{cc}\hfill M\ddot{\zeta }& =F_{fk}+F_S+F_R+F_{PTO}\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill M\ddot{\zeta }& =F_b+F_S+F_R+F_{PTO}\hfill \end{array}$$

(3b)

In both cases, the $F_P+F_g$ term will be expressed in a parametric form where a wide range of buoy shapes can be represented using only three independent quantities. In addition, the advantages and disadvantages of the $F_{fk}$ and $F_b$ approaches are discussed. The most significant distinction between them is the assumed motion of the buoy. The $F_{fk}$ derivation assumes small motions, while the $F_b$ approach does not.

3. Derivation of F_fk

The derivation of $F_{fk}$ begins by developing the pressure field for irregular waves. The parameterization of the buoy shape is considered next, followed by a representation of $F_{fk}$ in compact form.

3.1. Pressure

The FK term of Equation (3a) is the integration of the water pressure field, $\overrightarrow{P}(x,z,t)$, over the instantaneous wetted surface of the buoy, S, illustrated by the blue shaded region of Figure 2,

$$F_{fk}=F_g-\int \int \overrightarrow{P}(x,z,t)·\widehat{n}dS,$$

(4)

where $\widehat{n}$ is the normal unit vector pointing outwards from the buoy’s surface, $dS$ is the infinitesimal surface area of the buoy, and the red arrows show the pressure field at a few points. In this snapshot, the buoy’s draft line is below the free elevation, indicating that the buoy is moving relative to the wave.

To derive a closed-form expression for $F_{fk}$, it is necessary to obtain an expression for $\overrightarrow{P}(x,z,t)$ that can be integrated. The approach used here is the same as that introduced by Giorgi et al. [29] but with three modifications.

1.

Consideration of irregular waves of Equation (1).

2.

Inclusion of flat-bottomed buoys.

3.

A three-parameter buoy shape.

The pressure applied to the buoy is derived using linear wave theory in the appendix and given by Equation (5), where $\rho$ is the water density and g is the gravitational acceleration. Equation (5) assumes that the fluid is incompressible, inviscid, and irrotational with small particle velocities. Furthermore, it is valid only for the portion of the buoy below the still-water line (SWL), unrelated to the wave elevation.

$$P(x,z,t)=\rho \sum _{i=1}^n{\eta }_i(x,t){\displaystyle \frac{{\omega }_i^2}{{\chi }_i}}{\displaystyle \frac{\cosh \left({\chi }_i(z+h_w)\right)}{\sinh {\chi }_i{h}_w}}-\rho gz$$

(5)

The dispersion relationship between the wave number, frequency, and depth, ${\omega }^2=g\chi \tanh \left(\chi h_w\right)$, is also derived in Appendix A and valid under the above-mentioned assumptions. If we further assume deep water, $\chi h_w\gg 1,$ it can be approximated as ${\omega }^2=g\chi$. When applied to Equation (5), it yields the compact form of Equation (6).

$$P(x,z,t)=\rho g\sum _{i=1}^n{\eta }_i(x,t)e^{{\chi }_{i}z}-\rho gz$$

(6)

If the wave is regular with a single amplitude, frequency, and phase shift, $\varphi ={\displaystyle \frac{\pi }{2}}$, Equation (6) reduces to the equation used by Giorgi et al. in [29].

$$P(x,z,t)=\rho g{e}^{\chi z}A\cos (\omega t-\chi x)-\rho gz$$

(7)

Finally, it is assumed that the pressure is uniform in x near the buoy, letting us set $x=0$ in Equation (6). This is important since it allows for closed-form integration to obtain the FK forces and gives the final form of the pressure field of Equation (8).

$$P(z,t)=P_d(z,t)+P_s\left(z\right)=\rho g\sum _{i=1}^n{\eta }_i\left(t\right)e^{{\chi }_{i}z}-\rho gz$$

(8)

where $P_d(z,t)$ and $P_s\left(z\right)$ are the dynamic and static pressure components, respectively.

3.2. Pressure Integration and Buoy Shape Parameterization

The approach for generating a closed-form expression for $F_{fk}$ is the same as [29], namely, define the buoy shape by a radius $r\left(z\right)$, which varies vertically, and then integrate the pressure field of Equation (8) along the radius using cylindrical coordinates. The three additions to the procedure mentioned in Section 3.1 are discussed below. In preparation for a later comparison of FK and buoyancy models, we will write $F_{fk}$ with the static and dynamic terms of Equation (9).

$$F_{fk}=F_{fk,s}+F_{fk,d}$$

(9)

using the “s” and “d” subscripts of Equation (8).

Figure 3 illustrates the notation used in the derivation, where $r\left(z\right)$ is the revolution profile of the symmetry axis as a function of z that defines the shape of the buoy, $\theta$ is the angle of revolution, $\zeta \left(t\right)$ is the vertical displacement of the buoy’s draft line relative to the SWL at $z=0$, and h defines the draft line relative to a fixed reference of the buoy, in this case, at the bottom of the buoy.

Unlike previous models [29], Equation (10) is also supplemented by the pressure acting on the base of the buoy, calculated as the product of the base pressure and its area. The cylindrical coordinate rotation limits, $[0,2\pi ]$, and the vertical limits, $[z_1,z_2]$, are determined by the instantaneous wetted surface below the SWL, or $z_1=-(h-\zeta )$ and $z_2=0$. The buoy’s non-uniform cross-sectional plane area is $S\left(z\right)=\pi r^2\left(z\right)$. Based on these definitions, and utilizing Equation (4) with considering only the vertical component of $P(z,t)$ due to heave-only-motion, the $F_{fk}$ forces can be calculated as:

$$F_{fk}=F_g+\left[P_d(z_1,t)+P_s\left(z_1\right)\right]S\left(z_1\right)+{\int }_0^{2\pi }{\int }_{z_1}^{z_2}\left[P_d(z,t)+P_s\left(z\right)\right]r\left(z\right)r^{\prime }\left(z\right)dz$$

(10)

Equation (10) could be rearranged to obtain the components of the Froude–Krylov (FK) forces as

$$\begin{array}{cc}\hfill F_{fk}=& F_g+S\left(z_1\right)P_s\left(z_1\right)+{\int }_0^{2\pi }{\int }_{z_1}^{z_2}{P}_s\left(z\right)r\left(z\right)r^{\prime }\left(z\right)dz+\hfill \\ \hfill & S\left(z_1\right)P_d(z_1,t)+{\int }_0^{2\pi }{\int }_{z_1}^{z_2}{P}_d(z,t)r\left(z\right)r^{\prime }\left(z\right)dz\hfill \end{array}$$

(11)

Substituting Equation (8) into Equation (11) and splitting it into static and dynamic components yields Equation (12).

$$\begin{array}{cc}\hfill F_{fk,s}& =F_g-\rho g\left(S\left(z_1\right)z_1+{\int }_0^{2\pi }{\int }_{z_1}^{z_2}zr\left(z\right)r^{\prime }\left(z\right)dzd\theta \right)\hfill \\ \hfill F_{fk,d}& =\rho g\left(S\left(z_1\right)\sum _{i=1}^n{\eta }_i\left(t\right)e^{{\chi }_i{z}_1}+{\int }_0^{2\pi }{\int }_{z_1}^{z_2}\sum _{i=1}^n{\eta }_i\left(t\right)e^{{\chi }_{i}z}r\left(z\right)r^{\prime }\left(z\right)dzd\theta \right)\hfill \end{array}$$

(12)

Closed-form FK force expressions can be created for a wide range of buoy shapes by introducing a special case of a quadric surface given by Equation (13), where R is the midline buoy radius and $\alpha$ is the inverse square of the profile slope’s asymptote in the vertical direction. Equation (13) limits the buoy shapes to being axisymmetric with circular cross sections.

$$x^2+y^2-\alpha z^2=R^2$$

(13)

Evaluation of Equation (12) requires the profile equation $r\left(z\right)$ obtained from Equation (13) by shifting the z axis by the buoy displacement, $\zeta$, and solving for x or y while setting the other variable to zero, as shown in Equation (14).

$$r\left(z\right)=\sqrt{R^2+\alpha {(z-\zeta )}^2}$$

(14)

The area of the circular cross-section is

$$S=\pi r^2=\pi \left(R^2+\alpha {(z-\zeta )}^2\right)$$

(15)

and is used to describe the pressure effect on buoy shapes with flat bottoms in Equation (12).

Table 1. Buoy shapes and corresponding parameters.

Shape	$\mathit{\alpha }$	R	h
Spheroid, Oblate	<$-1$	Free	${\displaystyle \frac{R}{\sqrt{-\alpha }}}$
Sphere	$-1$	Free	R
Spheroid, Prolate	$(-1,0)$	Free	${\displaystyle \frac{R}{\sqrt{-\alpha }}}$
Cylinder	0	Free	Free
Hyperboloid (Hourglass)	>0	Free	Free
Cone–Cone	>0	0	Free

summarizes the classes of shapes possible with the parameterization of Equation (14). The value of $\alpha$ lets the shape range from an oblate spheroid to the hourglass shape of a hyperboloid. The conical shape is a limiting case with zero radius at the midline. This has little practical application for a buoy but provides insight into the effect of the “necking” of the hyperboloid on nonlinear terms in the final differential equation model. Although h is not part of the parameterization of $r\left(z\right)$, it is an important parameter to define the vertical height of the buoy.

Three $n\times 1$ arrays, $\mathit{\eta }$, $\mathit{ϵ}$, and $\mathit{\kappa }$, whose elements are given in Equation (16), are introduced to make the closed form $F_{fk}$ expression more compact. Note that $\mathit{\eta }$ contains the components ${\eta }_i$, while $\eta$ of Equation (1) is the sum of the components or the free elevation.

$${\eta }_i=A_i\sin ({\varphi }_i-{\omega }_{i}t),{ϵ}_i=e^{z_1{\chi }_i},{\kappa }_i={\int }_{z_1}^{z_2}{e}^{z{\chi }_i}r\left(z\right)r^{\prime }\left(z\right)dz$$

(16)

Substituting Equation (14) into ${\kappa }_i$ and carrying out the integration gives the closed-form expression for ${\kappa }_i$ of Equation (17).

$${\kappa }_i={\displaystyle \frac{\alpha }{{\chi }_i^2}}\left[{ϵ}_i(1+h{\chi }_i)-{\chi }_i\zeta -1\right]$$

(17)

Substituting $\mathit{\eta }$ and $\mathit{ϵ}$ from Equation (16) and $\mathit{\kappa }$ of Equation (17) into Equation (12) provides compact expressions for the static and dynamic FK force, which are combined in Equation (19).

$$\begin{array}{cc}\hfill F_{fk,s}& =-{\displaystyle \frac{1}{3}}\pi \rho g\zeta \left[3R^2+\alpha {\zeta }^2\right]\hfill \\ \hfill F_{fk,d}& =\pi \rho g{\mathit{\eta }}^T\left[(R^2+\alpha h^2)\mathit{ϵ}+2\mathit{\kappa }\right]\hfill \end{array}$$

(18)

$$F_{fk}={\displaystyle \frac{1}{3}}\pi \rho g\left[3{\mathit{\eta }}^T\left((R^2+\alpha h^2)\mathit{ϵ}+2\mathit{\kappa }\right)-\zeta (3R^2+\alpha {\zeta }^2)\right]$$

(19)

While this work focuses on closed-form expressions, it is important to note that $r\left(z\right)$ could be any set of points describing an axisymmetric buoy shape. The integrals of Equation (12) could be computed numerically in real time as the buoy moves. The form of the FK force would be the same except for a modification of the ${\kappa }_i$ term.

4. Derivation of F_b

Before developing the $F_b$ model, note that $F_{fk,s}$ of Equation (18) is indeed a buoyancy force, but it neglects submersion due to wave elevation. This effect is included in the $F_b$ model below.

The total buoyancy force acting on the buoy is

$$F_b=F_g+\rho g{V}_{sub}$$

(20)

where $V_{sub}$ is its submerged volume, calculated as the integral of its varying cross-sectional area $S\left(r\right(z\left)\right)$ over the wetted surface.

$$V_{sub}={\int }_{z_1}^{z_2}S\left(z\right)dz$$

(21)

To account for wave elevation, the area function, $S\left(z\right)$, and the lower integration bound, $z_1$, are modified from those of Section 3 to include the instantaneous wave elevation and are shown in Equation (22):

$$\begin{array}{cc}\hfill S\left(z\right)& =\pi \left(R^2+\alpha {(z+\eta -\zeta )}^2\right)\hfill \\ \hfill z_1& =-h+\zeta -\eta \hfill \end{array}$$

(22)

Substituting Equation (22) into the volume integral of Equation (21), the buoyancy, $F_b$, is given in Equation (23).

$$F_b={\displaystyle \frac{1}{3}}\pi \rho g(\eta -\zeta )\left[3R^2+\alpha {(\eta -\zeta )}^2\right]$$

(23)

5. Model Form Comparison

The difference between Equation (23) and $F_{fk}$ of Equation (19) is how wave elevation, $\eta$, is used. The $F_{fk}$ expression uses the pressure field containing ${\eta }_i$. However, the boundary conditions of linear wave theory, used to solve Laplace’s equation, are applied to the still-water line instead of the free elevation, $\eta$. This means that the pressure field of Equation (8) is only valid up to $z_2=0$, which is fine as long as both $\eta$ and $\zeta$ are small. In contrast, the $F_b$ model can be applied to scenarios where both waves and buoy movement are large, though it omits dynamic pressure effects.

Another way to compare $F_{fk}$ and $F_b$ is to write $F_b$ in two parts, where the first term is identical to $F_{fk,s}$ of Equation (18),

$$F_b=-{\displaystyle \frac{1}{3}}\pi \rho g\zeta \left[3R^2+\alpha {\zeta }^2\right]+{\displaystyle \frac{1}{3}}\pi \rho g\eta \left[3R^2+\alpha ({\eta }^2-3\eta \zeta +3{\zeta }^2)\right],$$

(24)

and the second term is an alternate representation of $F_{fk,d}$.

To illustrate the differences in the models, Equations (19) and (23) are applied to four sample buoy geometries in

Table 2. Buoyancy model, $F_b$, and Froude–Krylov, $F_{fk}$, forces for different buoy shapes in regular waves.

Shape	Force Expression
	$\begin{array}{c}{F}_b=\pi \rho g{R}^2(\eta -\zeta )\\ F_{fk}=\pi \rho g{R}^2\left[\eta e^{-\chi (h-\zeta )}-\zeta \right]\end{array}$
	$\begin{array}{c}{F}_b={\displaystyle \frac{1}{3}}\pi \rho g\alpha {(\eta -\zeta )}^3\\ F_{fk}={\displaystyle \frac{1}{3}}\pi \rho g\left[3\eta \left(\alpha h^2{e}^{-\chi (h-\zeta )}+{\displaystyle \frac{2\alpha }{{\chi }^2}}\left((1+h\chi )e^{-\chi (h-\zeta )}-\chi \zeta -1\right)\right)-\alpha {\zeta }^3\right]\end{array}$
	$\begin{array}{c}{F}_b={\displaystyle \frac{1}{3}}\pi \rho g(\eta -\zeta )\left[3R^2-{(\eta -\zeta )}^2\right]\\ F_{fk}={\displaystyle \frac{1}{3}}\pi \rho g\left[{\displaystyle \frac{6}{{\chi }^2}}\eta \left(1+\chi \zeta -(1+R\chi )e^{-\chi (R-\zeta )}\right)-\zeta \left(3R^2-{\zeta }^2\right)\right]\end{array}$
	$\begin{array}{c}{F}_b={\displaystyle \frac{1}{3}}\pi \rho g(\eta -\zeta )\left[3R^2+\alpha {(\eta -\zeta )}^2\right]\\ F_{fk}={\displaystyle \frac{1}{3}}\pi \rho g\left[3\eta \left(\left(R^2+\alpha h^2\right)e^{-\chi (h-\zeta )}+{\displaystyle \frac{2\alpha }{{\chi }^2}}\left((1+h\chi )e^{-\chi (h-\zeta )}-\chi \zeta -1\right)\right)-\zeta \left(3R^2+\alpha {\zeta }^2\right)\right]\end{array}$

for regular waves, where $\eta$ has one component. The $F_b$ expressions illustrate the effect of the buoy shape on the nature of the nonlinear contribution to the differential equation model of Equation (3b). For example, the cylinder is linear in $\eta -\zeta$, whereas the other shapes have a cubic effect. Compared to the linear term, the relative effect of the nonlinearity depends on R and, to some extent, $\alpha$. The limiting case of the cone–cone shape has no linear term, which means that it has zero stiffness at its equilibrium position, with a draft line at its apex. The hourglass shape approximates this behavior for small R. While the vertical height, h, is explicitly used in the FK formulation due to the pressure integration process, it does not appear in the buoyancy-based model, $F_b$, since the buoyancy force depends only on the submerged volume at any given instant, irrespective of the total height of the body.

As seen, both $F_b$ and $F_{fk}$ can be decomposed into a static and dynamic component, and the static portion is analytically identical in both models. Importantly, the static part is independent of the wave elevation, $\eta$, and hence cannot contribute to discrepancies in force prediction. Therefore, the difference in model accuracy, especially at high wave frequencies, arises from the dynamic components. These components scale with $\eta$ and reflect how each model handles wave-induced pressure effects.

Quantitatively, this distinction becomes evident when comparing predicted and measured forces in the experiments (see Section 6). For instance, while the static contribution to force remains the same, the $F_b$ model increasingly deviates from experimental data as frequency increases, indicating that its simplified treatment of dynamic wave pressure is insufficient at higher wave numbers. In contrast, the $F_{fk,d}$ term, which explicitly accounts for exponential depth-dependent pressure profiles through $e^{\chi z}$, better captures the increased hydrodynamic force associated with high-frequency waves. This explains the significant improvement observed with the FK model over $F_b$ in dynamic conditions.

6. Experimental Validation

The two models of Equation (3) are evaluated below using regular wave experiments. The objective was to compare $F_b$ with $F_{fk}$ without buoy motion, $\zeta =0$, and then examine the effect of the scattering force $F_S$.

6.1. Experimental Setup

The tests were carried out at MTUWave, shown in Figure 4, consisting of a 10 m long, 3 m wide, and 1 m deep concrete and glass basin. A spherical buoy with a 10 cm radius was mounted to a dynamometer with a Sensing Systems load cell to measure the vertical force exerted by incoming waves. The range of the load cell was $\pm 222$ N, with a maximum error of $\pm 0.556$ N, and it was sampled at 100 Hz. The buoy was positioned to have a midline draft, and the load cell output was adjusted to zero Newtons.

Two Edinburgh Designs resistance wave gauges, sampled at 128 Hz, were placed on either side of the buoy, laterally aligned with the center of the buoy. The wave gauges were calibrated at five discrete vertical positions, where each gauge was moved between −2 cm, −1 cm, 0 cm, 1 cm, and 2 cm, relative to the still-water level. The uncertainty in wave elevation measurement due to calibration and device precision was approximately ±0.5 mm. Geometric tolerances from 3D printing introduce an uncertainty of ±1 mm in the draft and ±0.5 mm in the buoy radius. These small deviations were determined to have negligible impact on the force comparisons between the experimental and model results.

Wave excitation was generated using eight independently controlled paddles mounted at one end of the tank, enabling both regular and irregular wave fields to be produced. A dSPACE MicroLabBox was used to log the load cell and provide a synchronization signal to start the wave maker paddles and wave gauge logging. This allowed for precise alignment of the data collected across multiple acquisition platforms.

The test conditions are shown in

Table 3. Regular wave characteristics used for model comparison experiments. The first two columns are the independent parameters, while the remaining columns are for reference.

Frequency f, Hz	Amplitude A, mm	Number $\mathit{\chi }$, 1/m	Length $\mathit{\lambda }$, m	Steepness
0.2	[10, 15, 20]	0.4	15.2	[0.0005, 0.0008, 0.001]
0.6	[10, 15, 20]	1.6	4.0	[0.005, 0.007, 0.009]
1.0	[10, 15, 20]	4.0	1.6	[0.013, 0.019, 0.026]

and were chosen to exercise the model while not exceeding the capabilities of the wave tank. The nine tests included three different frequencies and amplitudes, with wave steepness—the ratio of wave amplitude to wavelength—provided for reference.

Each test ran for 50 s, during which the first 10 s were discarded due to transient effects caused by paddle startup. From the remaining period, a 20 s segment was extracted for force and wave elevation analysis. The wave gauges operated at a sampling rate of 128 Hz, ensuring high-resolution capture of the wave field.

6.2. Data Comparison

Wave elevation measurements, $\eta$, were used to calculate $F_{fk}$ and $F_b$ using Equation (25), taken from the sphere entry of Table 2, where $\zeta =0$, $\rho =1000\mathrm{kg}/{\mathrm{m}}^3$, $g=9.81\mathrm{m}/{\mathrm{s}}^2$, and $R=0.1\mathrm{m}$.

$$\begin{array}{cc}\hfill F_{fk}& ={\displaystyle \frac{2\pi \rho g}{{\chi }^2}}\eta \left[1-(1+R\chi )e^{-R\chi }\right]\hfill \\ \hfill F_b& ={\displaystyle \frac{1}{3}}\pi \rho g\eta (3R^2-{\eta }^2)\hfill \end{array}$$

(25)

The calculated and measured forces are compared in Figure 5, Figure 6 and Figure 7. Both models perform well at low frequencies, as shown in the uppermost plot in Figure 5, Figure 6 and Figure 7. As the frequency of the wave increases, both models overpredict the force; however, the $F_{fk}$ approach outperforms the $F_b$ model.

The comparison of $F_{fk}$ and $F_b$ with the measured buoy forces ignores the contribution of the scattering force, $F_S$, of Equation (3). This force is generally considered negligible when the buoy diameter is significantly smaller than the wavelength [9]. Neglecting $F_S$ aligns well with the 0.2 Hz frequency data, where $\lambda /R\approx 150$ and the difference between the measured and calculated forces is less than 0.5 N for the 20 mm amplitude case. However, neglecting $F_S$ for the 0.6 Hz and 1.0 Hz cases where $\lambda /R$ is about 40 and 16, respectively, is a possible explanation for the increase in the deviation between the measured and computed forces in the second and third plots in Figure 5, Figure 6 and Figure 7. The maximum deviation for $F_b$ is 1.7 N, and for $F_{fk}$, it is 0.8 N, both occurring in the 1.0 Hz case of Figure 7.

The total diffraction force, $F_D$, is defined as the sum of the scattering and Froude–Krylov forces [39].

$$F_D=F_{fk}+F_S$$

(26)

While the Froude–Krylov component can be obtained as described in Section 3, the scattering force can be extracted from the numerical solution of the diffraction problem [9]. The complex scattering force coefficients, $F_S\left(\omega \right)$, were obtained using the boundary element method solver, WAMIT, applied to the spherical buoy using a 972-panel mesh from $\omega =0.01-15$ rad/s in increments of 0.01 rad/s. The real part of $F_s\left(\omega \right)$ is shown in Figure 8, and the time domain force used in the model was created using Equation (27),

$$F_S\left(t\right)=\mathrm{Re}\left[F_S\left(\omega \right)\right]\eta \left(t\right),$$

(27)

for the nine cases of frequencies and amplitudes considered. Equation (27) is the special case, $n=1$, of the irregular wave of Equation (28).

$$F_S\left(t\right)=\sum _{i=1}^n\mathrm{Re}\left[F_{\mathrm{S}}\left({\omega }_i\right)\right]A_i\sin ({\varphi }_i-{\omega }_{i}t)=\mathrm{Re}\left[F_S^T\left({\omega }_i\right)\right]\mathit{\eta }\left(t\right)$$

(28)

Figure 8 shows the frequency-dependent behavior of the real component of the scattering force coefficient, $\mathrm{Re}\left[F_S\left(\omega \right)\right]$, for a 0.1 m spherical buoy. The red markers indicate the angular frequencies corresponding to the experimental test cases, 0.2, 0.6, and 1.0 Hz, which correspond to $\omega =1.3$, $3.8$, and $6.3$ rad/s, respectively. These values align with the central frequencies used in the regular wave tests shown in Table 3. As the figure illustrates, the magnitude of the scattering coefficient increases significantly with frequency, from approximately $-3$ N/m at 1.3 rad/s to nearly $-49$ N/m at 6.3 rad/s. This trend highlights the increasing importance of incorporating scattering forces in the excitation force model as wave frequency increases. While the scattering force contribution is minimal in the long-wavelength regime (low $\omega$), it becomes non-negligible in the shorter wavelength regime, where the buoy size becomes more comparable to the incident wavelength. Therefore, accurate force modeling—especially for mid-to-high frequency waves—requires the inclusion of $F_S$, as its omission would lead to underprediction of total excitation forces and model mismatch, particularly visible in the higher-frequency experimental cases.

Figure 9 compares the effect on the maximum force error of using the $F_s$ expression of Equation (27) in the model. Each $3\times 3$ matrix shows the maximum force error between the model and the experiment for three frequencies and three amplitudes. The top two matrices are generated from the data of Figure 5, Figure 6 and Figure 7, and the bottom two matrices show maximum errors between the experiments and the model with $F_s$. As expected, as the frequency of the wave increases, $F_s$ has a larger effect. It is also readily observed that the $F_{fk}$ model estimates force better than the $F_b$ model as frequency increases. The $F_{fk}$ force is about twice as accurate as the $F_b$ force at 0.6 Hz and four times as accurate at 1.0 Hz.

As an example, Figure 10 shows the time history comparison of the measured and modeled forces when including $F_s$ and is a direct comparison to Figure 7.

7. Conclusions and Future Work

This study presented a parameterized nonlinear model of the Froude–Krylov (FK) excitation force for heaving-point-absorber wave energy converters (WECs), with the aim of enabling accurate yet computationally efficient force predictions for real-time control applications. A key contribution was the derivation of a closed-form FK force expression using only three parameters to represent a broad class of buoy geometries. This formulation was extended to account for irregular wave conditions and flat-bottomed buoy profiles, enhancing the model’s applicability to realistic ocean environments. Comparison with a standard buoyancy force model demonstrated that the proposed FK model offers significantly improved accuracy, particularly at higher wave frequencies.

The inclusion of scattering forces further improved model fidelity. The frequency-dependent scattering force coefficients, obtained from boundary element method simulations, showed increasing importance at higher wave frequencies, where neglecting scattering leads to the underestimation of the total excitation force. The experimental validation confirmed that incorporating both FK and scattering contributions yields the lowest prediction errors across all test cases, reducing discrepancies in the force amplitude to under 0.5 N at low frequencies and around 0.8 N at high frequencies.

In practical terms, the ability to represent a wide range of buoy shapes using a small set of parameters enables efficient shape optimization and real-time control implementation, which are crucial for the economic viability of WECs. By bridging the gap between high-fidelity hydrodynamic models and computational efficiency, this work contributes to the development of more effective model-based control strategies in ocean energy systems.

Future research will focus on addressing the small remaining discrepancies observed between the model and the experimental data. In particular, more complex revolution profiles beyond the current parameterization could be explored to better capture nonlinear geometrical effects. Furthermore, extending the model to accommodate more sophisticated representations of ocean wave spectra, such as directional spreading, wave superposition, and phase randomness, will improve applicability under realistic sea states. Ultimately, integrating these advancements into real-time control architectures could significantly enhance the energy yield and robustness of WEC deployments in diverse marine environments.