1. Introduction
The idea of extracting electricity from wave energy is attracting the attention of developers and researchers once again due to its abundance as well as its low environmental impact. In particular, the development of new energy sources is unavoidable in an era of challenging global issues such as climate change and rising levels of CO2. However, there also exist economic and technical challenges that have to be overcome for commercialization. Furthermore, commercial competitiveness with other sources of renewable energy is vital if these new energy sources are to survive. Several efforts have been taken to resolve the expensive installation and power transmission costs, as well as the unpredictability of the ocean environment. Recently, several efforts have included combined power generation, where at least two power sources among wind, wave, and current power are used for a single supporting structure; direct consumption of generated power at the site; and the storing of generated power into hydrogen fuel cells without a transmission cable.
There are a large number of concepts and technologies related to wave power generation. In particular, over 1000 wave energy conversion techniques have been patented in Japan, North America, and Europe however, the number of wave energy converters (WECs) that approach the commercialization stage is limited. Developed WECs can be classified into three types: attenuator, point absorber, and terminator. Attenuators lie parallel to the predominant wave direction and ‘ride’ the waves. An example of an attenuator WEC is the Pelamis. A point absorber is a device that has small dimensions relative to the incident wavelength. They can be floating structures that heave up and down on the water surface or can be submerged below the water level to rely on pressure differentials. Because of their small size, wave direction is not important for these devices. Power buoy and Wavebob are point absorber types. Terminator devices are positioned at right angles to the propagating direction of incident waves; they have a flap, arm, and are pivoted to allow forward and backward movement as the waves pass by. A historical device of this type is the nodding duck designed by Salter [
1].
For the first-stage conversion from wave energy to mechanical energy, the WECs have to transform the wave energy into hydraulic, pneumatic, and potential energy. The oscillating water column (OWC) uses airflow to drive the pneumatic turbine, which is directly coupled to a generator. The oscillating flux at the internal surface induces oscillating air pressure, rotates the turbine by pushing air through it via pressure difference, and finally produces electricity from the generator. The typical pneumatic turbines used at OWC are the Wells turbine and the Impulse turbine. They are self-rectifying turbines, which deliver a uni-directional torque for bi-directional airflow. Since Masuda [
2] proposed the OWC device first, numerous researchers have studied this interesting topic sustainedly through analytical, numerical, and experimental approaches [
3,
4,
5,
6,
7]. The OWC devices are presently installed and operated in many regions throughout the world, as the technology has been stabilized. The Land Installed Marine Powered Energy Transformer (LIMPET) is known as the world’s first wave power generation system that is operated through connection with an existing power grid [
8]. It was set up on the island of Islay offshore of the Scottish west coast, started operations in November 2000, and presently supplies power to the UK. The Korea Research Institute of Ships and Ocean Engineering (KRISO) recently constructed an OWC pilot plant with a capacity of 500 kW. The chamber has dimensions of 31.2 × 37.0 m. It was installed on a seabed with a water depth of 15 m in a test site for prototype WECs near Chagwido Island, Jeju. It is currently operating to evaluate its performance [
9].
A new concept OWC device, modified from the conventional OWC system, was proposed by Boccotti [
10]. It is called a U-shaped oscillating water column device, which adds a bottom-mounted vertical barrier in front of the existing OWC device. Consequently, wave energy is not transmitted directly into the internal fluid region of the air chamber, and the incident wave induces oscillatory motion of the U-shaped water column formed by the vertical barrier as shown in
Figure 1. Conclusively, the air turbine is operated by the airflow in the chamber, which is caused by the internal surface displacement at one end of the U-shaped column. Investigations of hydrodynamic performance for a U-OWC device have been performed theoretically, numerically, and experimentally by many researchers since the concept was published. Boccotti [
10] used the unsteady Bernoulli equation to establish the basis for a theoretical model of the oscillation of a U-shaped column. Later, Boccotti et al. [
11] conducted a model experiment to verify the theoretical model. Malara and Arena [
12] developed a hybrid numerical model applying the eigenfunction expansion method based on linear potential theory to the outer region of the U-OWC and the unsteady Bernoulli equation to the inner region. Malara et al. [
13] derived the boundary integral equation based on the linear potential theory and developed a three-dimensional numerical model of the U-OWC device. Malara and Arena [
14] proposed a semi-analytical approach for the estimation of hydrodynamic efficiency when multiple U-OWCs are arranged. It was found that an arrangement of several U-OWC devices reduces the hydrodynamic efficiency compared to independent U-OWC devices. Ning et al. [
15] examined the effects of a vertical barrier on hydrodynamic performance using the higher-order boundary element method (HOBEM) based on the nonlinear potential theory including nonlinear free-surface boundary conditions.
An ANN model that is made up of multiple hidden layers allows the computational model to learn the correlation between features in the dataset [
16]. ANN models are making crucial advances in solving problems that have held out against the artificial intelligence (AI) community for a long time. It has turned out to be outstanding at discovering complicated structures in high-dimensional data [
17] and is therefore relevant to many fields of science, business, and government. Lawrence et al. [
18] developed a hybrid neural network model for face recognition. Their system combines a local image sampling, a self-organizing map (SOM) neural network, and a convolutional neural network. Dahl et al. [
19] proposed a novel context-dependent (CD) model for large-vocabulary speech recognition (LVSR) that leverages recent advances in the use of deep belief networks for phone recognition. They used a pre-trained deep neural network hidden Markov model (DNN-HMM) hybrid architecture that trains the DNN to produce a distribution over senones (tied triphone states) as its output. Rifaioglu et al. [
20] reviewed deep learning-based identification of the interaction between drugs and their targets, something known as virtual screening. Recently, researchers have applied different machine learning techniques to optimize WECs systems. Sarkar et al. [
21] used a machine learning algorithm to obtain the optimal layout of WECs in arrays. Deberneh [
22] used wave data from nearshore floating buoys to train different machine learning regression models to predict the optimal site for nearshore wave energy harvesting. Masoumi [
23] classified regions in the United States to improve decision-making in the design of wave-wind hybrid systems. They used an unsupervised K-means clustering algorithm based on wave height, wave period, and wind speed.
In the present study, the hydrodynamic performance of a 2D U-OWC is investigated in irregular waves using an analytical model. Then, an artificial neural network (ANN) model belonging to a supervised machine learning algorithm is used to obtain the optimal geometry of a U-OWC for maximizing power generation. As an analytical model, a MEEM based on linear potential theory is used to obtain hydrodynamic parameters (flow rates, air pressure, etc.). A sample set of input features comprising geometry of a vertical barrier and submergence depth of a chamber is created using the Latin hypercube sampling (LHS) method. Using the analytical model, output features such as conversion efficiency and wave force are calculated for given input features and wave conditions. Finally, a database including all the features is established for each energy period and a detailed feature study is conducted on the database to obtain correlations between the features. To search for the optimal geometric features for maximizing extracted power, an artificial neural network (ANN) model is chosen. After preprocessing the database, an ANN model is designed and trained to predict output features like conversion efficiency and wave forces in irregular waves. Using the well-trained ANN model, predictions are made for a very large sample set (4000 samples) and from these predictions, optimal design parameters of a U-OWC are determined.
This paper is organized as follows:
Section 2 describes the analytical model of a 2D U-OWC device. In
Section 3, the numerical calculation is performed using a developed analytical model.
Section 4 designs an ANN model and predicts the conversion efficiency and wave force from irregular waves. Conclusions are presented in
Section 5.
2. Analytical Model
A U-shaped OWC device installed at a constant water depth (
h) is used as the analytical model. A U-OWC device is composed of an air chamber of length (
a) and height (
H) and a chamber wall that is submerged at
below the water surface. The airflow in the chamber escapes through the turbine installed at the top. A bottom-mounted vertical barrier with a height of
is placed apart at a distance of
b from the chamber (see
Figure 1). By adding a vertical barrier in front of the chamber, it looks like an oscillating U-tube, different from a conventional OWC. A two-dimensional Cartesian coordinate system is introduced, the origin is positioned at the location where the chamber wall and water surface meet, and the positive direction of the
z-axis is set vertically downward. The energy of the incident wave going in the positive x-direction is partially reflected and some enters into the chamber and oscillates the water surface there. Electricity is produced by rotating the turbine installed on the top of the chamber.
It is assumed that the fluid is incompressible and inviscid, and the flow is irrotational so that linear potential theory can be used. The fluid particle velocity can then be described by the gradient of a velocity potential . Assuming harmonic motion of frequency , the velocity potential can be written as , where A is the incident wave amplitude. Similarly, we can write down the wave elevation and oscillating air pressure in the chamber.
Under the assumption of linear potential theory, the velocity potential can be expressed as the sum of the scattering potential (
), i.e., the sum of incident potential (
) and diffraction potential (
), and the radiation potential (
) due to oscillating air pressure in the chamber in the absence of an incident wave, as shown in Equation (1).
The diffraction
and radiation
potentials satisfy the following boundary-value problem
with the following boundary conditions
where
,
is the Kronecker delta defined by
if
, and
if
.
2.1. Matched Eigenfunction Expansion Method
For applying a MEEM, the fluid domain is divided into three regions, as shown in
Figure 1. Region (1) is defined by
, region (2) by
, and region (3) by
. By the method of separation of variables, the velocity potentials in each region can be written as:
where
represents the propagating mode, while each
corresponds to evanescent modes. The eigenvalues
are the solutions of the dispersion equation given by
.
The vertical eigenfunctions are given by with The vertical eigenfunctions form a complete orthogonal set in [0,h]:
where
is the Kronecker delta.
The unknown coefficients
in Equations (7)–(9) can then be determined by invoking the continuity of potential and horizontal velocity across
.
where
are the unknown horizontal fluid velocities at the openings.
After substituting Equations (7)–(9) into Equation (11), then multiplying both sides by
and integrating over
, we obtain the following equations
Using the above equations, four unknowns
can be expressed by two unknowns
and then applying the continuity of the velocity potential given by Equation (12), the following equations are readily derived
Following Evans and Porter (1995), we can expand the horizontal fluid velocities
and
at the openings as a series of Chebyshev polynomials
where
are the unknown expansion coefficients.
, satisfying the free-surface boundary condition and the square-root singularity at the edge of the plate, are given by
where
is the 2nd-order Chebyshev polynomial of the second kind.
Substituting Equation (16) into Equations (14) and (15), the coupled algebraic equations are obtained as follows:
where
By multiplying both sides of Equation (18) by
, respectively, and integrating over
and
, we can obtain the final algebraic equations by truncating
n and
q after
N and
M terms.
By solving the coupled algebraic Equations (19) and (20), the unknown expansion coefficients can be readily determined. Subsequently, the unknowns in each region can be obtained from Equations (13) and (16).
2.2. Flux at the Internal Surface
The flux (
) at the internal surface can be obtained by integrating the scattered and radiated potentials over the internal surface.
where
is the flux at the internal surface due to incident wave, and
due to the oscillating pressure inside the chamber.
is in phase with the flux acceleration and has a similar characteristic of the added mass (i.e., radiation admittance).
is in phase with the flux velocity having an attribute of radiation damping (i.e., radiation conductance).
The horizontal wave forces
on the vertical barrier and chamber wall can be found by integrating pressure differences.
The reflection coefficient of a U-OWC is written by
2.3. Oscillating Air Pressure
To determine the oscillating air pressure
in the chamber, a continuity equation is used, for which the rate of change of mass inside the chamber is equal to the mass flux through the turbine. It is assumed that the air in the chamber is a compressible fluid, and the compression and expansion follow an adiabatic process.
where
is the air density;
the specific heat for an adiabatic situation;
atmospheric pressure and;
the volume of the air chamber. The flux
through the turbine is assumed to be proportional to the air chamber pressure.
is the function of the damping coefficient (
) that is dependent on the shape of a turbine, turbine diameter (
), and rotational velocity (
) of a turbine.
In Equation (24),
is the linearized rate of total upward displacement of the water surface inside the chamber and is equal to the upward flux
at the internal surface. For simple harmonic motion,
can be written by
When combining Equation (25) with Equation (24), we obtain the chamber pressure
2.4. Extracted Power
The power output is the time-averaged rate of work done by the chamber pressure pushing air through the turbine.
Thus, the conversion efficiency can be calculated by dividing the incident wave power
per unit width of a regular wave with amplitude
A.
where
is the group velocity.
There is an optimal turbine constant that maximizes the conversion efficiency at each wave frequency. The optimal turbine constant,
, can be obtained by imposing
, which yields
2.5. Extracted Power in Irregular Waves
The variance of the oscillating chamber pressure in irregular waves can be written by using Equation (26).
where
is the incident wave spectrum. In the present study, a Pierson–Moskowitz (PM) spectrum is used
where
is the significant wave height and
the energy period and
the
n-th order moment of the area under the spectral curve.
When the irregular wave is incident to the OWC, the maximum power per unit width is given by
As the optimal turbine constant is a function of wave frequency, in irregular waves, we use the value at the particular wave frequency that may be equal to the resonance frequency of piston-mode or energy frequency.
The maximum conversion efficiency of OWC attainable in irregular waves is
where the denominator
in Equation (33) is the irregular wave power per unit width and given by
Both the wave force spectrums on the vertical barrier (
i = 1) and chamber wall (
i = 2) in irregular waves are defined as
The significant wave forces can be obtained from integrating the above spectrums
5. Conclusions
Using a developed analytical model (MEEM) based on linear potential theory, it is found that there exist two resonance peaks in the power-extraction (or conversion efficiency) curve of a U-OWC device. The first resonance is associated with the piston-mode resonance inside the chamber, while the second spike-like resonance is caused by the similar fluid motion between the barrier and chamber wall. The combination of two peaks helps to expand the range of high wave-energy conversion. The corresponding wave forces on the front barrier are also computed from the analytic solutions. It is shown that a U-OWC with a barrier height of = 2.0 m has a maximum conversion efficiency of 0.70, which is higher than that (=0.56) of a conventional OWC with no-barrier ( = 0 m) in irregular waves.
Subsequently, the optimal design of the U-OWC is determined by using a machine learning method. In this regard, three input features, the vertical barrier geometry (height, distance) and submergence depth of the chamber wall, and two output features, the conversion efficiency and wave forces on a front barrier, are chosen. Using the analytical model described in
Section 2, the database from randomly distributed input and output features is created using Latin hypercube sampling (LHS). From the feature study on the database, it is observed that there exists a strong correlation between barrier height and the output features. The present ANN model is shown to be trained well with
score of 0.95 even if a database is comparatively small in size (200 samples). Using the trained ANN model, the optimal geometric values of
,
, and
b are predicted to be 2.51 m, 2.65 m, and 2.18 m from a large dataset (4000 samples) in irregular waves with
= 5 s.
The developed ANN model can supply guidelines in determining the optimal design values of a U-OWC suitable to wave conditions at the installation site. The present design tool based on an ANN model is especially valuable when the number of input features is large.
The analytical model is based on a potential theory, therefore, it cannot take into account energy dissipation by the formation of vortices at the ends of the barrier and chamber wall. Therefore, the conversion efficiency and wave forces may overestimate the actual values. To consider the viscous effects in the design of a U-OWC, either the numerical simulations by CFD codes or physical experiments need to be paralleled as a supplementary measure of the present analytical model.