## 2 Oscillating Water Column (OWC) Wave Energy Converter (WEC) Benchmark Test  
The efficiency of the OWC WEC depends greatly on the amount of air trapped in the device, and the amount of wave energy that causes the movement of the WEC. To develop our prediction models, we will use the experimental dataset in [33] where the hydrodynamic efficiency of fixed OWC wave energy devices are measured under various wave conditions and variable geometric characteristics of the devices. In Figure 1, a sketch of the experimental setting is shown which was designed to investigate the effect on the hydrodynamic efficiency of five variables. The variable \(x_{1}\) is the unitless wavenumber of the incident waves with constant amplitude of 0.03 m, which were generated by the wave-maker. The still water depth \(h\) was 0.8 m during the experiment and therefore \(x_{1}=K=(2\pi h)/\lambda=(1.6\pi)/\lambda\) for a wave with wavelength \(\lambda\) [m]. The variables \(x_{2}=B,x_{3}=d\), \(x_{4}=D\), and \(x_{5}=\theta\) describe the geometry of the OWC WEC device (see Figure 1). The names, the measurement units, and experimental values of the independent variables are given in Table 1. For simplicity, the input variables can be organized in 5-dimensional column vector of independent input \(\overrightarrow{x}=(x_{1},x_{2},x_{3},x_{4},x_{5})^{T}=(K,B,d,D,\theta)^{T}\) where \({}^{T}\) stands for transpose. The dependent variable \(y\) being the efficiency of the OWC WEC device was calculated as the ratio of the hydrodynamic energy absorbed by the waves for time \(\tau\) and the energy contained in the incident waves for time \(\tau\), where \(\tau\) is one wave period. We will use a data set consisting of 126 records where the \(i\)th record contains the measurement, \(y_{i}\), of dependent variable (the efficiency) under experimental conditions determined by the values of the independent variables \(x_{i1}\), \(x_{i2}\), \(x_{i3}\), \(x_{i4}\), and \(x_{i5}\). Since the latter form a vector of independent input \(\overrightarrow{x}_{i}=(x_{i1},x_{i2},x_{i3},x_{i4},x_{i5})^{T}\), the \(i\)th record in the dataset is the couple \(\left\langle\overrightarrow{x}_{i},y_{i}\right\rangle\). Refer to [33] for more details about the experimental setting of the wave flume and about the dataset formation. The 126 records in the data set are divided into 9 groups of 14 records each. The experimental group \(k\) (for \(k=1,\,2,\,\ldots,\,9\)) contains the records \(i=14k{-}13,\,14k{-}12,\,\ldots,14k{\rm{. }}\) In each group the 14 records have the consecutive unitless values of \(x_{i,1}=K\) as shown in the last column of Table 2. The values of \(x_{i,2}=B\) are 0.7 m for group 2, 0.85 m for group 3, and 0.55m otherwise. The values of \(x_{i,3}=d\) are 0.17 m for group no. 4, 0.20 m for group no. 5, and 0.14 m otherwise. The values of \(x_{i,4}=D\) are 0.04 m for group no. 6, 0.08 m for group no. 7, and 0.06 m otherwise. The values of \(x_{i,5}=\theta\) are 10\({}^{\circ}\) for group no. 8, 20 \({}^{\circ}\) for group no. 9, and 0 \({}^{\circ}\) otherwise. The measured values of the hydrodynamic efficiency, \(y_{i}\), (acquired as in [34]) are shown in the first nine columns of Table 2. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Input** & **Symbol** & **Definition** & **Tested Values/Range** & **Unit** \\ \hline \(x_{1}\) & \(K\) & Unitless wavenumber & 0.85-3.6 & [-] \\ \(x_{2}\) & \(d\) & Submerged front wall length & 0.14, 0.17 and 0.20 & [m] \\ \(x_{3}\) & \(B\) & Width of the chamber & 0.55, 0.70 and 0.85 & [m] \\ \(x_{4}\) & \(D\) & Diameter of the orifice & 0.04, 0.06 and 0.08 & [m] \\ \(x_{5}\) & \(\theta\) & Slope angle of the bottom & 0, 10 and 20 & [\({}^{\circ}\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the independent variables.