ABSTRACT
The power absorption capability of a floating oscillating body depends on a proper wave-structure interaction. A wave energy converter (WEC) captures wave power when the system radiates waves in opposite phase to the incident waves. Hence, determination of the radiation force is crucial for the initial design and optimal control of point absorber WECs. In this paper, hydrodynamic coefficients of a heaving vertical-cylinder buoy are obtained using ANSYS AQWA, and the equation of motion is analyzed based on the Cummins integro-differential equation. Numerical solution of this equation is computationally demanding due to the convolution integral associated with the radiation force vector. Therefore, a MATLAB code is developed and the convolution term is replaced by a linear state-space model. Frequency domain results in the range of 1–2 rad/sec, corresponding to the predominant wave conditions of the Persian Gulf offshore region, are implemented. The proposed open-source model, compared to direct calculation of the convolution integral, provides highly accurate prediction and maintains system stability. The model is also computationally cost-effective for higher degrees of freedom.

1. INTRODUCTION
With the advancement of research on wave energy converters (WECs), the need for more accurate modeling methods has become increasingly evident. WECs, including point absorbers, are typically designed as oscillating systems, and the transfer of wave energy to these systems is described by the magnitude and damping characteristics of the excitation force of the incident wave. Therefore, the hydrodynamic problem in these devices is usually considered as a combination of diffraction and radiation problems (Cruz, 2008). Forces acting on marine structures due to waves can be divided into inertial (mass) forces and viscous forces. Inertial forces, assuming the fluid is inviscid, can be calculated using potential theory (Sarpkaya, 2010).
Thus, the dynamic response of the structure is generally analyzed using first-order wave potential theory in the frequency domain. If nonlinear effects such as viscous forces or mooring are considered, the equation of motion must be analyzed nonlinearly in the time domain (Li and Yu, 2012). In the field of wave energy extraction by oscillating systems, numerous efforts have been made, often focusing on the analysis of the Cummins equation. In this integro-differential equation, a term related to the radiation force (also known as the fluid memory effect) complicates the mathematical analysis of WECs. Taghipour et al. (2008) investigated various methods to identify a suitable replacement model, where the memory effect is generally estimated using a state-space model. However, this research only addresses ship motion analysis and does not provide information on the specific analysis conditions of a WEC. Nolte and Ertekin (2014) conducted numerical modeling of a heaving point absorber and performed time-domain analysis for irregular waves. To analyze the equation of motion, the HYDRAN program was developed, but it is not open-source. Medeiros and Brizzolara (2018) presented a mathematical framework for time-domain simulation of WECs using a proprietary program called AEGIR, which is also not available. Xiao et al. (2021) investigated new prototype samples of such converters, which can increase the power received from sea waves up to two times. Engström et al. (2020) investigated the possibility of creating an array of 200 point absorbers in the Swedish Special Economic Zone. Zhou et al. (2020) focused on the geometry of the absorber buoy, considering different ratios of diameter to draft for cylindrical buoys with conical, flat, and hemispherical bottoms.
In this paper, a linear time-domain method based on the Cummins equation is used to model the point absorber buoy. The term related to the fluid memory effect includes a convolution integral, and determining the system motion response by numerically solving this term can be very time-consuming (Taghipour et al., 2008). Since convolution is a linear operator, an equivalent linear system can be identified in the form of a transfer function and then converted to a state-space model. System identification aims to find the lowest possible order for numerical simulation while ensuring stability (Ljung, 2010).
2. MATERIALS AND METHODS
Assuming that the buoy is constrained to move only in heave and that the incident wave has negligible effect on the extracted power (Eriksson et al., 2007), the device can be modeled as a simple one-degree-of-freedom system. The buoy is considered as a vertical cylinder with radius *r* and draft dr in water depth *d*. The equation of heave motion based on Newton's second law states that the total mass of the device multiplied by its vertical acceleration is equal to the sum of the excitation force, radiation force, hydrostatic restoring force, and the power take-off (PTO) reaction force. The excitation force is proportional to the incident wave amplitude. The radiation force is expressed in terms of the radiation damping and the added mass of the body. The hydrostatic restoring force is proportional to the vertical displacement of the buoy, the water density, gravitational acceleration, and the waterplane area. The PTO reaction force, assuming a rigid connection between the buoy and the PTO system, is expressed in terms of the PTO inertia, a linear damping coefficient, and a spring constant. By applying the Fourier transform, the frequency-domain equation of motion is obtained, which yields the response amplitude operator (RAO) of the device. Applying the inverse Fourier transform leads to the Cummins integro-differential equation, in which the radiation force is represented by a convolution integral involving the radiation force impulse response function (IRF) and the heave velocity history, together with the added mass at infinite frequency (Falnes and Kurniawan, 2020).
The fluid memory effect can be estimated through discrete time estimation, but this approach is time-consuming and cannot be implemented in simulation tools such as MATLAB/Simulink. The common method for representing linear systems in the time domain is the state-space model. The fluid memory effect is therefore estimated based on a single-input-single-output state-space model, in which the state vector, the input excitation corresponding to the heave velocity, and the constant state, input, and output matrices define the system dynamics. The impulse response function is obtained from the state-space matrices, and the corresponding transfer function is derived by taking the Laplace transform. The Laplace transform of the convolution term is equal to the product of the transfer function and the Laplace transform of the heave velocity, where the transfer function evaluated at s = iω is equivalent to the frequency response function of the radiation impedance. Therefore, by determining the hydrodynamic parameters and identifying a rational transfer function, the convolution term can be converted to an equivalent state-space model.
A floating cylindrical buoy with radius *r* = 2 m, draft dr = 1.5 m, and height *h* = 3 m was selected, and its hydrodynamic parameters were obtained using the AQWA module in ANSYS. The buoy is located at a water depth of 25 m. Since the cylindrical buoy is symmetric about its vertical axis, the incident wave is considered in only one direction. The frequency range of analysis is from 0.05 to 3 rad/sec for 80 frequencies. Meshing was performed automatically, and the maximum element size was selected based on the highest frequency of analysis.
Using the obtained hydrodynamic coefficients, a rational transfer function was estimated by frequency-domain regression. The least squares (LS) method was selected for a single-input-single-output system. The transfer function is considered as a ratio of two polynomials, where the roots of the numerator and denominator polynomials are the zeros and poles of the system, respectively. The order of the denominator polynomial determines the order of the system, and the difference between the degrees of the numerator and denominator polynomials is the relative degree. According to the initial value theorem, the relative degree must be equal to one, and the minimum order of the transfer function must be two. The coefficients were estimated using the least squares method with the Sanathanan-Koerner linear approximation, in which the denominator polynomial from each iteration is used as a weighting function for the next iteration.
3. RESULTS
The hydrodynamic coefficients for the heaving cylindrical buoy were obtained. The added mass and radiation damping were calculated and are shown in Figure 4 of the main text. The transfer function was identified using frequency-domain regression, and its absolute value and phase were compared with the calculated frequency response function K(ω), as shown in Figure 5. The impulse response function k(t) was also calculated and compared with the identified transfer function, as shown in Figure 6. The identified transfer function was converted to a state-space model, and the matrices A, B, and C were obtained. The state-space model was then substituted into the equation of motion, and the heave response of the device was simulated in Simulink under two harmonic waves with amplitude A = 1 m and frequencies ω = 1.25 and ω = 2 rad/sec. The results were compared with the heave response obtained from the RAO, as shown in Figure 7.
4. DISCUSSION AND CONCLUSION
In this paper, an open-source model for estimating the radiation force in the equation of motion of a heaving point absorber WEC was presented. The Cummins integro-differential equation was analyzed, and the convolution term related to the fluid memory effect was replaced by a linear state-space model. The state-space model was identified using frequency-domain regression based on hydrodynamic coefficients obtained from ANSYS AQWA. The accuracy of the identified transfer function was evaluated by comparing it with the frequency response function of the system. The results confirm the applicability of the identified transfer function. The proposed model, while ensuring system stability, provides an acceptable estimate of the radiation force in the buoy equation of motion with significantly reduced computation time. The model can be easily extended to higher degrees of freedom and facilitates the analysis of nonlinear effects such as mooring. The open-source nature of the model addresses the limitations of existing proprietary codes and provides a practical tool for the initial design and optimal control of point absorber WECs.
ACKNOWLEDGEMENT
The authors would like to acknowledge the support of the Persian Gulf University for providing the computational resources required for this study.
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