Oscillating body wave energy conversion efficiency based on Simulink simulation training

With the development of the economy and society, we are faced with the dual challenges of energy demands and environmental pollution. As one of the important marine renewable energies, wave energy is widely distributed and abundant, and has considerable applications in various fields. Oscillating body wave energy converter (WEC) is most popular among WECs; its energy conversion efficiency is thus one of the key issues for practical large‐scale applications. This article establishes a new model of the float and vibrator under different motion conditions, conducts simulation training, and then solves the displacement and velocity at different times, and finally optimizes the maximum output power and optimal damping coefficient. Simulink calculation is one of the most effective tools including neural networks, which allows us to achieve intelligent training and simulation. The obtained data imply that the maximum output power could be 8.102 W when the corresponding linear damping is 21,000 N s/m and the rotation damping is 95,000 N s/m. The article takes into account the pitching motion of the float, providing a more realistic model and obtaining optimal power absorbing efficiency. The obtained results could be the potential reference data in practically setting up the wave energy conversion systems.

the waves in the sea is about 10 13 W, 3 which can reach 60% of the world's energy consumed (16 TW) per year. 4n addition, it also benefits from the wide distribution, so wave energy has been extensively studied, 5,6 particularly by coastal countries.
To better use the energy in ocean waves, wave-energy converters (WECs) have been developed, which can be divided into mechanical transmission, low-pressure hydraulic transmission, high-pressure hydraulic transmission, and pneumatic transmission according to the way of energy conversion. 7In recent years, much literature has been reported on the optimization problem of WECs with several modifying algorithms. 8,9Gomes et al. 10 carried out the study on the floating oscillating water column (OWC), and twoheaving body model was formulated based on linear forces.Strati et al. 11 also proposed a control strategy to maximize the performance of the OWC device in a variety of environmental conditions.In recent years, much research focused on floating point-absorbing WECs based on modeling.Liu et al. 12 employed an array of floating point-absorbing WECs for extracting ocean wave energy efficiently.The WECs in regular waves were proposed when considering the power-take-off (PTO) damping type, array layout, separating distance, and incident wave direction.Peng et al. 13 established a scale model in the laboratory and used experimental data to model the average output power and wave energy conversion efficiency, which are 0.13 W and 32.6%, respectively.Schmitt and Elsaesser 14 used computational fluid dynamics (CFD) to simulate the wave traces and experimented to assess the accuracy of the solver.Josset et al. 15 described a numerical model of the second-generation WECs called SEAREV using MATLAB.To achieve the maximum generated power using WECs layout and PTO optimization play important roles.However, it is a big challenge due to the complicated hydrodynamic interactions among converters.In this case, a more effective tool with integrated algorithms is needed to solve the current problem.
Simulink is an extension of MATLAB and is one of the most effective simulation tools.The Simulink block library is vast and includes tools for control systems, neural networks, fuzzy logic, DSP, signal creation and processing, visualization, and many more specialized toolboxes. 16It is reported that Simulink as a modeling method has been widely used in many engineering fields, but its application in WEC modeling is still very limited.In this article, a new model of an oscillating body WEC device consisting of a floater, vibrator, and PTO system was established based on the Simulink simulation.The oscillating motion of a floating body is converted into the flow of a liquid at high pressure by using a system of hydraulic rams.At the other end of the hydraulic circuit, a hydraulic motor is employed to drive an electric generator.The heaving motion and pitch motion of the oscillator and float in seawater are considered.First, the force analysis of the float and the vibrator is carried out and the corresponding motion equation and moment of inertia equation are established, and then the simulation is carried out by Simulink software.Based on the model, the heaving displacements, heaving velocity, pitch angular displacement, and pitch angular velocity of the oscillator at 10, 20, 40, 60, and 100 s are obtained.Second, the model of power calculation is added, and the maximum output power, the corresponding linear damping coefficient, and the rotational damping coefficient are obtained.

| Assumptions and notation
It is assumed that the fluid in the combined wave-current field is an ideal one that is uniform, inviscid, and incompressible, and its motion in the wave field is vortexless.The incident wave is described by a linear micro-amplitude wave theory, moreover, atmospheric pressure to the sea level is constant and there is no wind resistance by ignoring the friction at the joints in the system.Assuming that the wave excitation force is vertical, the wave excitation force in the horizontal direction completely provides the vertical moment.All the symbols used in the text are explained in Table 1.
T A B L E 1 The explanation of symbols used in the text.

Symbol
Explain Micro-amplitude wave theory is a linear wave theory, which mainly uses potential functions to study the motion law of waves.When the liquid is in an ideal situation, that is to say, under the action of gravity, the motion of seawater is irrotational and cannot be compressed.Then, in this case, the wave fluctuation can be studied through the velocity potential function φ x z t ( , , ), and the coordinate system could be set before the study, as shown in Figure 1.
In the potential motion, the flow velocity potential φ x z t ( , , ) is generally used to represent the continuity equation of the fluid, and the following equation satisfies the Laplace equation, When the wave is a nonconstant potential wave, the integral formula of the fluid motion equation can be expressed by the Lagrangian integral formula, that is, the Bernoulli equation, At the water surface z η = 0 , the boundary conditions of the potential flow field can be divided into two parts: dynamic boundary conditions and motion boundary conditions.Since term u v ( + ) be negligible as it is inappreciable compared to the other components under the assumption that the motion of the water spot in the traveling wave is very slow and the amplitude of the wave is small.Dynamic boundary conditions of the free water surface: z η = 0 , p = 0, so Equation ( 2) could be simplified as Motion boundary conditions for the free water surface: Getting the water quality points on the free water surface together to form the free water surface, so the movement speed of each water quality point also reflects the movement speed at any point on the free water surface, that is, Thus, the motion boundary conditions of the free water surface can be obtained as, Suppose that the propagation velocity of the wave is c, the wave height is H, the wavelength is L, and the wave period is T, therefore the motion equation of the wave on the free water surface is When the water depth is finite, that is, when z = −d, the potential function φ of the propulsive wave can be expressed as

| Motion analysis of the oscillators and floats
Considering the entire oscillator and float system as a whole, and then analyzing the motion of the entire system separately, as shown in Figure 2.
The undulating motion of the floating body with the waves, in addition to the action of buoyancy, will also be affected by radiation damping and viscous damping.In the case of ignoring the damping, the force of the wave on the floating body is where ρ is the seawater density, g the gravitational acceleration, and A the cross-sectional area of the floating body.
Let the mass of the floating body be m, when the floating body floats on the sea surface and is in a static state; the seawater buoyancy on the floating body is equal to the gravity on the floating body, that is, Therefore, in the case of ignoring damping, the vertical force on the floating body is When the floating body moves up and down with the waves, it will also be affected by the damping force, which mainly includes two parts: linear radiation damping force and nonlinear viscous damping force.Considering that the seawater is inviscid and swirling, the nonlinear viscous damping force is ignored here.
The linear damping force can be expressed as Therefore, in the case of considering damping, the vertical force on the floating body is When the oscillator and float are under consideration of only heave motion, according to Equations ( 5)-( 8), it can be seen that where When the oscillator and float perform heave motion and pitch motion, the force analysis of pitch motion should be added based on heave motion (Figure 3).
According to the force analysis, the mathematical model of the motion of the vibrator and the float and the model of the moment of inertia are established.
In the formula, where I 1 , I 2 are the moment of inertia of vibrator and float, θ β ̈, ̈-θ β , the angular acceleration, α-α the angular velocity, and L L L , , e r h the moment of inertia of wave excitation force, radiation force, and hydrostatic restoring force, respectively.
Considering that the float is composed of a cylindrical shell and a conical shell with uniform mass distribution when calculating the moment of inertia of the float, the vertical distance from the center of mass of the float to the rotating shaft is required.In this article, it is considered that the geometric bodies that constitute the float are all regular geometric bodies, which are respectively composed of a top circle, a side cylinder, and a bottom cone.When obtaining the center of mass of this regular geometry, the center of the bottom circle is used as the coordinate origin to establish a threedimensional coordinate system; therefore, the coordinate system is established according to Figure 4.
In Figure 4, ① represents the top circle, ② the lateral cylinder, and ③ the bottom cone.To calculate the center of mass of the float, the mass of each part needs to be obtained first.Since the float is composed of a cylindrical shell and a conical shell with uniform mass distribution, this article adopts the method of area component mass to solve the mass of each component of the float.If we assume that the total mass of the float is 4866 kg, the radius of the float bottom is 1 m, the height of the cylindrical part is 3 m, and the height of the conical part is 0.8 m, thus we have , respectively.It can be seen from Figure 4 that the barycentric coordinates of each component are G 1 (0, 0, 3), G 2 (0, 0, 1.5), G 3 (0, 0, −0.2).Considering that the coordinates of the gravity center of each component of the float on the x-axis and the y-axis are both 0, to simplify the calculations, only the coordinates in the z-axis direction will be calculated, According to Equation ( 17), the coordinates of the mass center of the float are calculated as M (0, 0, 1.42), so the vertical distance from the center of mass of the float to the axis of rotation is 1.42 m.

| SOLVING THE MODEL
Based on the established motion model of the vibrator and float, using the implementation of Simulink software, the block diagram of the simulation program is obtained.
In the Simulink software simulation training, the heaving displacement and angular velocity of the oscillator and float with the action of wave excitation force and wave excitation moment could be assigned as: fcosω, Lcosω, where, f is the wave excitation force amplitude, L is the wave excitation moment amplitude, and ω is the wave frequency, which is calculated respectively with a time interval of 0.2 s in the first 40 wave cycles.According to Equations ( 15) and ( 16), in the simulation training of oscillator and float in Simulink software, the simulation of oscillator and float under the action of wave excitation force and wave excitation moment fcosω, Lcosω respectively in the first 40 wave cycles.Based on the above mathematical model, the S-Function module of MATLAB is used to calculate the vertical wave excitation moment and the longitudinal wave excitation force of the float and the vibrator based on Equations ( 15) and (16).Using the modules in the Simulink toolbox, after setting the corresponding parameters, then according to the internal transmission | 1415 sequence of the system, the modules are therefore connected, and thus a complete system simulation model was established.Further, the processing steps of the calculations mentioned above are presented in Figure 5.The heave displacement and velocity, pitch angular displacement, and angular velocity data of the oscillator and the float at a time interval of 0.2 s are calculated.The corresponding results are shown in Figures 6 and 7.Under the action of the wave excitation force fcosω, the heaving amplitudes of the vibrator and float gradually decrease until they stabilize at a fixed heave amplitude.The angular velocity of the pitching motion also increases with time, while the pitch amplitude gradually decreases.The modeling simulates the motion state of WEC in the wave, which can be used as a condition to continue modeling.
Combining the heave displacement and velocity curves, the pitch angular displacement and angular velocity curves of the vibrator and float, the wave excitation force, and as well as the wave excitation moment of the vibrator fcosωt, Lcosωt (f is the amplitude of the wave excitation force, L is the amplitude of the wave excitation moment, ω is the wave excitation force when the heave displacement and velocity and pitch angular displacement and angular velocity with a time interval of 0.2 s in the first 40 wave cycles under the action of frequency), considering that the float only does vertical swing and pitch in the wave, a mathematical model is established to determine the optimal damping coefficients of the linear damper and the rotary damper when the damping coefficients of the linear damper and the rotary damper are both constant.Both the damping coefficients of the linear damper and the rotary damper are within the region [0, 100,000].The provided parameter values (wave frequency = 1.9806 s −1 ) were used to solve the displacement and velocity in two directions of freedom within 10, 20, 40, 60, and 100 s periods.The results are shown in Tables 2 and 3. Furthermore, the maximum output power and the corresponding maximum damping coefficient are calculated.

| Maximum power and optimal damping coefficient
Under certain conditions, the WEC system has an optimal linear damping constant, which maximizes the system's overall power generation.Based on the model established above, according to Aurélien's description of the linear PTO system, the presence or absence of a spring constant in a linearized PTO system depends on the actual situation.For the linearized PTO model established in this article, the obtained power of the system can be calculated by the square of the difference between the damping of the PTO and the speed of the The processing steps of the calculation model.float and the vibrator, and the obtained power Pcap of the PTO system can be further calculated as The PTO damping value is denoted here by c.The velocity of the device as a result of the energy spectral component is given as the constant x.This method produces a unique optimal damping value for the incident spectrum. 17Therefore, in this article, the optimal PTO value is obtained, so that the energy output efficiency can reach the maximum value, which could be simulated using Simulink software.When the damping coefficients of the linear damper and the rotary damper are both within the interval [0, 100,000], the simulation results are shown in Figure 8 (3D diagram).
In the diagram, the x-axis represents the linear damping coefficient, the y-axis is the rotational damping coefficient, and the z-axis is the output power.It can be seen that when the damping coefficients of the linear damper and the rotary damper are both in the interval [0, 100,000], the maximum output power is 8.102 W, the corresponding linear damping coefficient is 21,000 N s/m, and the rotary damping coefficient is 95,000 N s/m.

| Uncertainties
Uncertainties in modeling can be classified into two main categories: aleatory uncertainties (inherent variability) and epistemic uncertainties (lack of knowledge). 18In this article, there are several sources of uncertainties that need to be considered.The first is parameter uncertainty; the parameters of the mathematical model such as damping coefficients, spring constants, and moment of inertia are likely to be subject to uncertainties due to measurement errors or variability in the physical properties of the system.The second is the uncertainty of the model, this arises from the simplifications and assumptions made in the mathematical model.For instance, the article assumes that the fluid is ideal, and the excitation force of the wave is vertical, but these assumptions are not ideal in reality.Finally, numerical uncertainty is associated with the use of numerical methods for solving the equations of motion and for the optimization of the damping system.Numerical methods introduce approximations and discretization errors.To evaluate and mitigate these uncertainties, several methodologies could be employed.Sensitivity analysis, for instance, can be instrumental in determining which parameters exert the most significant influence on the model's outcome, thereby proving particularly beneficial in addressing parameter uncertainty. 19Monte Carlo simulations could be utilized to propagate uncertainties throughout the model, furnishing a probabilistic representation of the model output. 20,21

| Model promotion evaluation
Compared with the previous model, the model established in this article has several improvements. 22,235][26][27] This allows for a more accurate representation of the motion dynamics in real-world wave conditions.In addition, the model includes linear radiation damping and nonlinear viscous damping forces, which are often neglected in other models.9][30][31] The model also considers the fluid in the combined wave-current action field, an aspect that is often overlooked in existing models.This provides a more realistic evaluation of the wave energy converter's performance.These advancements collectively lead to a more accurate and reliable prediction of the WEC's performance.Further, the model's robustness to uncertainties makes it a valuable tool for designing and optimizing WECs under varying operating conditions.While the proposed methods in this article provide a comprehensive approach to the problems at hand, several limitations should be taken into account: 1.The model assumes that the fluid is ideal, that is, it is uniform, nonviscous, and incompressible.In reality, water does have viscosity and compressibility, which may influence the damping and power capture characteristics of the system.The ideal fluid assumption simplifies the calculations but may result in inaccuracies in the results.2. The incident wave is described by the linear microamplitude wave theory.However, ocean waves are typically nonlinear and have larger amplitudes.The use of linear wave theory may limit the model's applicability in high-wave environments.3. The model assumes no friction at the system joints, which is often not the case in real-world scenarios.
Friction can significantly affect the dynamics of the system, especially in the presence of moving parts, such as the rotating shaft, damper, and torsion spring.
These limitations suggest that though the model provides a solid foundation for understanding and predicting the system's behavior, improvements could be made.Future work could involve incorporating more | 1419 realistic wave and fluid dynamics, taking into account friction effects, and exploring the impact of more complex geometries on the system's performance.

| CONCLUSIONS
A wave linearization numerical model and the model of floats and oscillators in heaving and pitch motions based on the Simulink calculations were established.On this basis, the maximum output power has been gained as 8.102 W when the corresponding linear damping is 21,000 N s/m, and the rotation damping is 95,000 N s/m.The study provided a numerical method for analyses of the WEC system and an insight into the optimal damping factor and efficiency by using Simulink.According to the simulation results, the optimal energy output efficiency has been obtained.In further research, random waves will be introduced in the fluid model and the dynamic influence of PTO will be considered in the motion of oscillators and floats.Therefore, the model could be used to create energy conversion equipment at sea level so that natural resources are fully utilized and a clean environment is achieved.

1 2 2 2
is nonlinear, to facilitate the analysis, the nonlinear problem needs to be transformed into a linear one.The value of u v

YIN ET AL. | 1413 F
I G U R E 2 System model in a wave.L is the length of the float, d the initial draft of the float, A the water cross-sectional area of the floating body, z is the distance from the draught surface of the floating body to the horizontal surface, x the distance from the wave surface to the water surface, y the distance from the top of the floating body on the wave surface.F I G U R E 3 Force analysis of the oscillator and float for heave and pitch motion.The float is the β angle, the vibrator is the α angle relative to the float, and the total angular displacement of the vibrator is α + β.

F
I G U R E 6 (A) Oscillator heave displacement curve, (B) oscillator heaving velocity curve, (C) float heave displacement curve, and (D) float heave velocity curve.

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I G U R E 7 (A) Vibrator pitch angle displacement curve, (B) vibrator pitch angular velocity curve, (C) float pitch angle displacement curve, and (D) float pitching angular velocity curve.T A B L E 2 Vibrator's heave displacement, velocity, pitch angular displacement, and angular velocity.

T A B L E 3 8
Float's heave displacement, velocity, pitch angular displacement, and angular velocity.Simulation results of maximum output power and damping coefficient.YIN ET AL.