**1. Introduction**

Wave energy converters (WEC), a new type of energy extractor with little pollution, are expected to be a reliable alternative to the current generation method. There are two stages for an oscillating body WEC transforming wave energy into other forms of energy like electricity. A floating body is firstly required to capture the wave energy induced by a wave's motion. Then the moving floating body drives a generator to generate power. An intact oscillating body WEC system is generally composed of a moving floating body, a power take-off (PTO) system, and an anchor chain, etc. At present, the conversion efficiency of WECs is relatively low, so the main research objective is to improve the power generation of a specific device.

One method is to design a different floating body's shape, and the shape is usually irregular curved surface. McCabe [1] researched the optimization of the shape of a wave energy collector to improve energy extraction by genetic algorithms, and a benchmark collector shape was identified. Colby [2] used evolutionary algorithms to optimize the ballast geometry and achieved 84% improvement in power output. Fang [3] designed a mass-adjustable float, and a new optimization calculation method was proposed. Multifreedom buoys have been also proposed in [4–6]. They can translate or rotate in more than one freedom, so more wave energy can be captured. Another means is to design an innovative PTO system. Reabroy [7] proposed a novel floating device integrated with a

**Citation:** Wang, W.; Liu, Y.; Bai, F.; Xue, G. Capture Power Prediction of the Frustum of a Cone Shaped Floating Body Based on BP Neural Network. *J. Mar. Sci. Eng.* **2021**, *9*, 656. https://doi.org/10.3390/ jmse9060656

Academic Editor: Spyros A. Mavrakos

Received: 7 May 2021 Accepted: 10 June 2021 Published: 13 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

fixed breakwater. The simulations and experiments proved that installing a breakwater can greatly improve the conversion efficiency. Liang [8] designed a novel PTO system which is inside the buoy with a mechanical motion rectifier (MMR). This mechanism can convert the bidirectional wave motion into unidirectional rotation of the generator by two one-way bearings. Li [9] improved this device by substituting one-way bearings for two one-way clutches. Chen [10] proposed a new point-absorber WEC with an outer-floater and a built-in power take-off mechanism. Besides, array-type WECs, integrated with many buoys and PTO systems, are also researched to achieve large scale power generation. The typical one is WaveNET [11], developed by Albatern in Scotland. Sun [12] proposed an array-type energy-capturing mechanism integrated with marine structures. Liu [13] proposed an array-type WEC combined with oscillating buoy.

The factors that affect the power generation have also been studied recently. Zou [14] analyzed the effects of spring force, mass force, and damping force on energy conversion efficiency based on a 3D wave tank model. Yu [15] and Wu [16] discussed the influence of the floating body's shape, PTO damping coefficient, system stiffness coefficient, and geometry parameters on power generation. Zheng [17] established an optimization model of energy conversion performance via genetic algorithm. Ma [18] researched the two-body floating point absorber and the results showed that stiffness coefficient had less effect on the power generation than damping coefficient. Ji [19] proved that PTO damping coefficient and submerged body volume were the most important parameters that affect the output power, and that the significant wave height had little influence on conversion efficiency. Tongphong [20] analyzed the effects of wave frequency, PTO damping coefficient, and structure form (floating or fixed) on capture factors.

Wave load and hydrodynamic parameters are vital factors in the analysis process of floating body's motions. Numerical simulations are widely used in hydrodynamic performance analysis to obtain these parameters. Ma [21] used ANSYS-AQWA software to assess the hydrodynamic performance and energy conversion of a pitching float WEC and analyzed key factors' influences on the performance. Amiri [22] presented a numerical simulation scheme for a point wave absorber and analyzed its performance. Yu [23] applied Reynolds-Averaged Navier-Stokes (RANS) computational method for analyzing the hydrodynamic heave response of a specific WEC device.

In addition to the traditional physical model [24,25], novel methods and models based on big data and machine learning have also been presented. Law [26] carried out wave prediction over a large distance downstream using artificial neural network, introducing machine learning algorithm into ocean engineering. Desouky [27] utilized non-linear autoregressive with exogenous input network to predict the surface elevation with the help of an ahead located sensor. Kumar [28] used the Minimal Resource Allocation Network (MRAN) and the Growing and Pruning Radial Basis Function (GAP-RBF) network to predict the daily wave heights based on real marine data. Some elevating measurements are also proposed to assess the performance of predictions in [29]. Avila [30] combined Fuzzy Inference Systems (FIS) and Artificial Neural Networks (ANN) to forecast wave energy in Canary Islands. Wang [31] predicted power outputs of a WEC in shallow water, taking bottom effects into accounts. Halliday [32] utilized Fast Fourier Transform (FFT) to predict wave behavior in short term based on real marine data. Davis [33] used a nonlinear Extended Kalman Filter to estimate the wave excitation force based on experimental wave tank data. Ni [34] combined the Long Short-Term Memory (LSTM) algorithm and the principal component analysis (PCA) together to predict the power generation of a WEC.

Different from traditional mathematical model, this paper presents an agent model using BP neural network to determine the complex non-linear reflection between design variables and power generation. Power predictions are the foundation of multi-objective optimizations of a floating body. Accurate power prediction can provide a guide for the electricity consumption, allocation, and distribution in power grid. Through the prediction, the unknown generation power becomes measurable, so reasonable manners can be arranged to increase the grid capacity.

The remainder of this paper is organized as follows: Section 2 develops the mathematical model of the oscillating float-type WEC. In Section 3, a sample database is established by LHS method, and a simple feature study is conducted. The geometric model and simulations of each sample are done in ANSYS-AQWA (developed by ANSYS company, based in Canonsburg, Pennsylvania, USA) in Section 4. Section 5 designs a BP neural network model and it is used to predict the capture power. Results and discussion are given in Section 6 and conclusions are presented in Section 7.

#### **2. Mathematical Model**

A schematic diagram of the oscillating body WEC is shown in Figure 1. To simplify the study, some assumptions are made as below:


**Figure 1.** Mechanical model.

Under three assumptions, the following forces act on the floating body: hydrodynamic forces (excitation and radiation force); hydrostatic buoyancy; PTO damping force; rigid restoring force. According to the theory of fluid mechanics and Newton's law, the governing equation of the floating body can be expressed as follows:

$$M\_0 \ddot{z}(t) = f\_\mathcal{E} - f\_\mathcal{R} - f\_\mathcal{S} - f\_{\text{PTO}} - f\_\mathcal{K} \tag{1}$$

where *M*<sup>0</sup> represents mass; *z*(*t*) represents the heave displacement; *fE* represents excitation force; *fR* represents radiation force; *fS* represents hydrostatic buoyancy; *fPTO* represents PTO damping force; *fK* represents rigid restoring force.

The excitation force imparts on the floating body by the incoming wave. It is the summation of the Froude-Krylov force *fFK* and the diffraction force *fD*, so it can be written as follows:

$$f\_E = f\_{FK} + f\_D \tag{2}$$

The radiation force is induced by the floating body's motion and can be decomposed into an added mass term and a radiation damping term [25], so it can be expressed as follows:

$$f\_{\mathbb{R}} = A\_M \ddot{\bar{z}}(t) + B\_{\mathbb{C}} \dot{\bar{z}}(t) \tag{3}$$

where *AM* and *BC* are the added mass and radiation damping in the vertical direction, respectively.

The hydrostatic buoyancy, induced by seawater static pressure, is the resultant force of gravity and buoyancy. It is a force that restores the structure to hydrostatic equilibrium and is linear with the heave displacement of the floating body. It can be written as:

$$f\_{\mathbb{S}} = \rho \lg A\_{\mathbb{W}} z(t) \tag{4}$$

where *ρ* is seawater density; *g* is acceleration of gravity; *AW* is water cross area of the floating body. In this paper, the value of PTO damping is relatively large, and the heave displacement of the floating body is small. As a result, it is assumed that the water cross area of the floating body does not change. It is the section where the water line is located when the floating body is in still water. Therefore, the hydrostatic buoyancy can be expressed as:

$$f\_{\mathbb{S}} = \frac{1}{4}\pi\rho g D^2 z(t) \tag{5}$$

where *D* is the diameter of the floating body.

The energy conversion system can be simplified to a linear spring damping system, so the PTO damping force is

$$f\_{\rm PTO} = c\dot{z}(t) \tag{6}$$

where *c* is the damping coefficient of the PTO system.

The rigid restoring force is proportional to the heave displacement, and it can be written as follows:

$$f\_k = kz(t) \tag{7}$$

where *k* is stiffness coefficient.

Reformulate Equation (1) through Equations (2), (3), (5)–(7):

$$\dot{z}\left[M\_0 + A\_M\right]\ddot{z}(t) + (B\_\mathbb{C} + c)\dot{z}(t) + (\rho g A\_W + k)z(t) = f\_\mathbb{E}(t)\tag{8}$$

Apply Fourier transform to Equation (8) and obtain another governing equation in the frequency domain. It is

$$\left[ (\rho g A\_W + k) + j\omega (B\_\mathbb{C} + c) - \omega^2 (M\_0 + A\_M) \right] Z(\omega) = F\_\mathbb{E}(\omega) \tag{9}$$

where *ω* is the wave frequency; *j* is imaginary unit; *Z*(*ω*) and *FE*(*ω*) are functions of displacement and excitation force in the frequency domain, respectively.

In the frequency domain, the excitation force can be expressed by the product of the unit excitation force and the incident wave amplitude [35]. It is

$$F\_{\rm E}(\omega) = F\_{\rm unit}(\omega) A(\omega) \tag{10}$$

Equation (9) can be rewritten as follows:

$$\left[ (\rho g A\_W + k) + j\omega (B\_\mathbb{C} + \varepsilon) - \omega^2 (M\_0 + A\_M) \right] Z(\omega) = F\_{unit}(\omega) A(\omega) \tag{11}$$

Formula (11) is a typical damped and forced vibration equation, so the natural frequency and damping factor can be expressed as below:

$$
\omega\_n = \sqrt{\frac{\rho g A\_W + k}{M\_0 + A\_M}} \tag{12}
$$

$$\beta\_n = \frac{B\_\mathbb{C} + \mathfrak{c}}{2(M\_0 + A\_M)} \tag{13}$$

From Equations (12) and (13), the natural frequency and damping factor of a given WEC change over added mass and added damping.

According to Equation (11), the heave response in the frequency domain is

$$Z(\omega) = \frac{F\_{unit}(\omega)A(\omega)}{(\rho g A\_W + k) + j\omega (B\_\mathbb{C} + \varepsilon) - \omega^2 (M\_0 + A\_M)} \tag{14}$$

The average power in one wave period, captured by the floating body with heave motion, can be written as the product of damping force and vertical velocity. The work done by damping force is the energy absorbed by the floating body, so the mean capture power is

$$\begin{array}{ll} P\_{\text{mean}} &= \frac{1}{T} \int\_{0}^{T} f\_{\text{PTO}} \dot{z}(t) dt \\ &= \frac{1}{T} \int\_{0}^{T} c \dot{z}(t)^{2} dt \\ &= \frac{1}{2} \omega^{2} c \big| Z(\omega) \big|^{2} \\ &= \frac{1}{2} c \frac{\omega^{2} \big|\_{\mathcal{E}} \big|^{2}}{\left[ -\omega^{2} \big( \mathcal{M}\_{\mathbb{C}} + \mathcal{A}\_{\text{M}} \big) + k + \rho\_{\text{A}} \mathcal{A}\_{\text{M}} \big]^{2} + \omega^{2} \big( \mathcal{B}\_{\text{C}} + c \big)^{2}}} \\ &= \frac{1}{2} c \frac{\big|\_{\mathcal{E}} \big|^{2}}{\left[ \frac{-\omega^{2} \big( \mathcal{M}\_{\mathbb{C}} + \mathcal{A}\_{\text{M}} \big) + k + \rho\_{\text{A}} \mathcal{A}\_{\text{M}}} \right]^{2} + \left( \mathcal{B}\_{\text{C}} + c \right)^{2}} \end{array} \tag{15}$$

The mean capture power reaches the maximum when the following conditions are met.

$$k = \omega^2 (M\_0 + A\_M) - \rho \lg A\_W \tag{16}$$

$$\mathcal{L} = \begin{cases} \begin{array}{c} B\_{\mathsf{C}\prime} B\_{\mathsf{C}} > 0 \\ -B\_{\mathsf{C}\prime} B\_{\mathsf{C}} < 0 \end{array} \end{cases} \tag{17}$$

This stiffness and damping are called the best stiffness and the best damping, respectively. When *Bc* > 0, the natural frequency, damping factor, and displacement are

$$
\omega\_n = \omega \tag{18}
$$

$$
\beta\_{\rm II} = \frac{B\_{\rm C}}{M\_0 + A\_M} \tag{19}
$$

$$Z(\omega) = -\frac{jF\_{\text{unit}}(\omega)A(\omega)}{2\omega B\_{\odot}}\tag{20}$$

The max capture power is

$$P\_{\text{max}} = \frac{|F\_E|^2}{8B\_{\text{C}}} \tag{21}$$

#### **3. Design of Experiments (DOE) Method**

#### *3.1. Latin Hypercube Sampling*

The sampling method is of great importance in experimental design. A good sampling method can result in more reasonable sample distribution, leading to a better model with higher accuracy. In this paper, a Latin hypercube sampling (LHS) method is utilized to generate sample points. Different from random sampling, LHS has a high efficiency of space filling by maximizing the stratification of each edge distribution, which improves the uniformity.

According to Equation (15), the factors that determine the capture power under given wave conditions are PTO damping coefficient *c*, system stiffness coefficient *k*, wave exciting force *FE*, float mass *m*, added mass *AM*, and added damping *BC*. Added mass, added damping, and wave exciting force are related to the geometry and submergence depth of the floating body. The geometric features of the floating body depend on radius *R*, semivertical angle *α*, and mass *m*. As a result, four main geometric parameters, including radius *R*, semi-vertical angle *α*, mass *m*, and submergence depth *d*, plus two system parameters, PTO damping coefficient *c* and stiffness coefficient *k*, are selected as key variables that affect the capture power.

The sample space of six key variables are defined as follows:

$$\begin{cases} \quad d \in [2, 3] \\ \quad R \in [1.5, 3] \\ \quad m \in [7000, 8000] \\ \quad \alpha \in [5, 25] \\ \quad c \in [10, 000, 30, 000] \\ \quad k \in [3000, 6000] \end{cases} \tag{22}$$

A database covering 100 sample points is established (see in Appendix A) and scatter diagrams of these samples are shown in Figure 2.

**Figure 2.** Sample scatter diagrams. (**a**) semi-vertical angle; (**b**) radius; (**c**) submergence depth; (**d**) mass; (**e**) damping; (**f**) stiffness.

In Figure 2, each variable fills the whole sample space and the standard deviation of the value is small. It can reflect the relationship between the factor and the response in the six spaces.
