Numerical Hydrodynamic and Mooring Optimization of a Wave Energy Converter for the Mexican Coast

Abstract

This study presents a hydrodynamic assessment of a toroidal wave energy converter (WEC) operating under low-energy conditions of the west coast of Mexico. Performance analysis incorporates the coupling surge, heave, and pitch motions. To investigate mooring–device interaction, two mooring configurations were examined: (A) a single catenary system and (B) a catenary system with a surface-floating buoy. The WEC was evaluated under operational conditions, operational conditions with a constant surface current, and extreme seas. The results show that under operational conditions, the WEC-mooring B configuration achieves higher energy capture than the WEC-mooring A configuration, with performance peaks at 13 s and 11 s, respectively. The presence of a surface current does not significantly influence absorbed power. Under extreme conditions, mooring B reduces mooring-line stresses but causes greater horizontal foundation forces and increased floater drift compared to mooring A. When mooring effects are included, mooring A’s performance is advantageous because it shifts peak energy capture toward the dominant sea states at the study site. This maintains better station-keeping capability and achieves a maximum capture width ratio (CWR) of approximately 0.5.

1. Introduction

A significant number of projects are currently being investigated by both industry and academia to develop commercially viable wave energy converter concepts. Several numerical and laboratory experiments have been conducted to assess WEC performance, with a limited number being tested in real conditions. There is an abundance of concepts available in the literature [1].

Table 1. Wave energy converters classification by working principles [1].

Working Principle	Characteristic
Point absorber	Point absorbers are floating or submerged devices significantly smaller than the incident wavelength. These devices have the capacity to absorb energy from all wave directions. Examples of such devices include AquayBuOY, AWS, Pontoon, and Wavebob.
Oscillating wave surge	Oscillating wave surge converters consist of flaps that oscillate around a hinged shaft in response to the wave action, thereby harvesting energy from horizontal water motion. These devices are designed for operation in intermediate and shallow water, for instance, WEC Langlee.
Oscillating water columns	Oscillating water columns are open-chamber structures that enclose the water column and a trapped air pocket above it. The waves induce oscillatory motion in the water inside the chamber; this motion pushes the air back and forth through a bidirectional turbine. This turbine then generates electricity. An example of this technology is the OEbuoy.
Attenuators	Attenuators are floating structures aligned with the wave direction, featuring a horizontal extension comparable to the wavelength. Examples of this technology include the Pelamis and SeaPower systems.

shows concept classification based on the working principle. More detailed descriptions and discussions are available in recent reviews [2,3].

The prevailing trend in technological development has been toward systems that are optimized for high-energy-density sites. However, in those places, the storm conditions require the deployment of robust WECs. The high-energy wave climate results in high costs for construction, installation, maintenance, and operation, potentially leading to substantial investment risks. By contrast, wave energy conversion in low-energy seas, where the mean available wave power is below 15 kW/m², could entail minor economic risks due to less aggressive predominant wave conditions. Examples include the Baltic and Mediterranean seas (3 kW/m² [4] and 4 kW/m² [5], respectively) in Europe; the Indian coast (5–25 kW/m² [6]) in Asia; the Mexican Pacific coast (10 kW/m²) in America [7,8].

Recent studies have focused on downscaling WEC devices (adjusting WEC dimensions to the most common sea states) by applying Froude’s scaling law to operational low-energetic conditions, with a particular focus on the Baltic and Mediterranean seas. In [1], the downscaling of eight WECs (AquaBouy, AWS, Langlee, OEBuoy, Pelamis, Pontoon, SeaPower, and Wavebob) along the entire Mediterranean coastline was investigated. The findings showed an average downscaling factor of 0.25, with the SeaPower achieving a maximum annual energy output of 50 MWh. However, there is limited information on the adaptation of WECs’ mechanical aspects, which represents a significant knowledge gap. Conversely, the pursuit of wave energy conversion for low-energy sea states has prompted the development of novel WEC designs capable of operating under such conditions. For instance, PeWEC [9] is a multi-degree-of-freedom WEC comprising an electric generator excited by a resonant pendulum located inside a floating hull. FWEC [10] presents a concept for operation under low-energy conditions that comprises a central floating body and four hinged floaters, with the floater and main body connected via hydraulic pistons that drive the PTO (power take-off) system.

The primary objective of a mooring system is to maintain the wave energy converter in position. The design of the mooring system is strongly influenced by site conditions and the operational principles of WECs [11]. Theoretical studies suggest that the cost of the mooring system can represent up to 30% of the total expenditure associated with the construction of a WEC structure [12]. Therefore, it is essential to integrate the mooring system into the overall design process of WECs, considering its impact on both structural feasibility and device performance. In the literature review presented in [13], studies on several mooring system configurations, including single-cable mooring with a buoy element, and combinations of diverse mooring materials such as chains and ropes, are reported.

In addition to hydrodynamic optimization, recent studies emphasize the need to integrate techno-economic evaluations, for example, [14]. Although such analyses are beyond the scope of the present work, accurate predictions of device dynamics and energy capture are the first steps toward techno-economic evaluations.

This work extends the findings of a prior study of the WEC system presented in [15], in which a toroidal WEC was proposed to operate efficiently under wave conditions in Ensenada Bay, Mexico. The optimization study of WEC dimensions and the optimal number of PTO units was conducted in both the frequency and time domains. The hydrodynamic coefficients were obtained with Nemoh [16], while the time domain analysis of the heave-pitch-surge coupling response and energy capture was performed using the WEC-Sim [17] model under irregular wave conditions.

Other numerical approaches have been implemented to study WECs’ performance. Recently, Sandoval Munoz [18] employed a high-fidelity Large Eddy Simulation method in a 3D numerical wave tank to investigate overtopping wave energy converters. Such an approach provides detailed insight into turbulent flow; however, it requires substantial computational power, making it less suitable for extensive parametric studies.

The objective of the present work is to design/investigate the most effective PTO damping and mooring system to enhance the performance of the WEC from a hydrodynamic perspective. The following structure is adopted in this paper: Section 1 introduces the study area; Section 2 outlines the WEC concept and numerical modeling setup; Section 3 presents the WEC performance analysis for two mooring configurations; Section 4 discusses the results; Section 5 provides the conclusion.

Study Area

Several studies identify the Mexican west coast as an area with wave energy potential, estimated at approximately 10 kW/m [8]. The potential increases from south to north, with the upper values concentrated in the northern part of Baja California. Most of the existing literature for this region has focused on Todos Santos Bay, Ensenada. The city of Ensenada is equipped with an electrical substation that will facilitate the integration of ocean wave-generated electricity into the national grid [19]. These factors combine to make Todos Santos Bay an optimal location for studying wave energy harvesting.

Todos Santos Bay is situated on the northwest coast of Mexico, between latitudes 31.7 and 31.9 N and longitudes 116.6 and 116.8 W. It is a semi-sheltered system, connected to the Pacific Ocean (see Figure 1). The wave characteristics are the result of the interaction of swell from the North and South Pacific and local winds. The north swell dominates in autumn and winter, while the stronger south swell occurs during the summer months.

As demonstrated in [20], the significant wave height (H_s) outside the bay is approximately 1 m during the summer months and ranges from 1.5 m to 2 m during the winter period. The peak period (T_p) is between 12 s and 15 s. With regard to the available energy, the majority of wave power in the bay is concentrated between H_s = 0.75 m and 1.5 m, and T_p = 10 to 16 s, as illustrated in Figure 2. This figure depicts the joint probability of occurrence of H_s and T_p (Figure 2a) and the cumulative probability curve (Figure 2b) in Figure 1.

2. Materials and Methods

2.1. Wave Energy Converter Concept

The WEC consists of a torus-shaped buoy that is anchored to the seabed by cylinder structures [15]. These structures function as pistons, moving along the local vertical axis. A spherical joint is provided at both sides of the piston to allow six degrees of motion (see Figure 3). Despite the buoy being able to move in six degrees of freedom, the PTOs extract energy from the linear displacement of the piston. Gonzalez et al. [15] prove that including 6 DoF can increase the energy capture from the only-heave response. In this study, the PTO system is not presented in detail; instead, it is idealized as a linear damping system.

Table 2. Geometric and hydrostatic properties.

Properties	Value
Buoy diameter (m)	30.0
Draft (m)	2.5
Displacement (ton) 1	948.4
Inertial Moment, Ixx, (kg −m2)	1.0771 × 108
Inertial Moment, Iyy, (kg −m2)	1.0771 × 108
Inertial Moment, Izz, (kg −m2)	2.12524 × 108
Gravity center [x, y, z] (m) 2	[0, 0, 0]
Buoyancy center [x, y, z] (m)	[0, 0, −1.05]
Hydrostatic stiffness [z] (N/m)	4.7 × 106
Restoring moment [x] (N/°)	5.3 × 108
Restoring moment [y] (N/°)	5.3e × 108

Note(s): ¹ Displacement is the weight of displaced water divided by the submerged body volume equal to the body’s total weight, which includes the weight of structural elements, equipment, and ballast. ² At this design stage, the gravity center is kept at position [0, 0, 0] due to the lack of information about real weight distribution.

presents the geometric and hydrostatic properties of the WEC. In the initial phase of this study, the mooring system is replaced by a stiffness matrix applied to the buoy’s gravity center. The stiffness value was iteratively estimated to restrict buoy motion and stabilize the numerical simulation under the most challenging conditions.

This study focuses on energy capture optimization, where damping and mooring configurations are addressed to increase energy capture, and structural implications are not included.

2.2. Numerical Model Description

2.2.1. WEC-Sim Model

WEC-Sim (Version v5.0.1) is an open-source code developed by the National Renewable Energy Laboratory (NREL) and Sandia National Laboratories (Sandia), with funding from the U.S. Department of Energy’s Water Power Technologies Office. The code can solve both single-body and multi-body dynamics in the time domain, and incorporates external forces generated by PTO and mooring systems. The code has been developed in MATLAB/Simulink (Simscape-Fluids) (Version R2020b) using the Simscape Multibody solver. A summary of the model equation is presented below; however, a complete description is available on the WEC-Sim website [17]. The WEC-Sim numerical model has been validated against other numerical models [21] and experimental data [22], showing reliable results.

The numerical model solves the system’s dynamic response by Equation (1) as presented by [17], which employs linear wave theory and assumes that waves are the sum of incident, radiated, and diffracted components. The terms in Equation (1) represent a vector comprising six forces, three translational and three rotational.

$$\begin{array}{c}m\ddot{X}=F_{exc}\left(t\right)+F_{rad}\left(t\right)+F_B\left(t\right)+F_{md}\left(t\right)+F_{pto}\left(t\right)+F_v\left(t\right)+F_{me}\left(t\right)+F_m\left(t\right)\end{array}$$

(1)

The left side represents the force on the body, which is equal to the product of the body’s mass and acceleration. The right side is the sum of the acting force over the body, wave excitation force ($F_{exc}$), radiation force ($F_{rad}$), buoyancy force ($F_B$), mean drift force ($F_{md}$), PTO force ($F_{PTO}$), damping force ($F_v$), Morison Element force ($F_{me}$) and mooring force ($F_m$).

The first three terms depend on the hydrodynamic coefficients; in this work, they are calculated with the frequency-domain BEM solver Nemoh (Version v.2.0) [16]. The drag forces due to the waves and ocean currents are introduced by $F_{me}$ term, and calculated with the Morison equation [23]. The Morison equation is a function of fluid velocity, body velocity, drag coefficient, added mass coefficient and displacement volume ($v,\dot{X},C_d,C_a,\forall$). This approach is valid when the quotient of the structure diameter $D$, and wavelength, $\lambda ,$ is between 0.1 and 0.2. Ocean current profiles could be included in three forms: (1) constant profile, (2) following a 1/7th power law, or (3) varying linearly with depth.

2.2.2. MoorDyn Model

MoorDyn (Version v2) is a lumped-mass mooring dynamic open-source model, as outlined in reference [24]. The model subdivides the mooring line into segments and connects them via nodes where mass, buoyancy, hydrodynamic forces, and reaction forces are concentrated. MoorDyn categorizes forces into two distinct groups: internal and external. The internal loads are weight, buoyancy, internal stiffness, and damping, whereas the external forces, including added mass and drag, are solved using the Morison equation. MoorDyn is integrated with WEC-Sim, enabling the exchange of forces and responses between the two models at each time step.

2.3. Numerical Model Setup

Figure 4 presents the WEC-Sim setup. The torus buoy (blue block) is a rigid hydrodynamic body, which means the hydrodynamic coefficients are used to solve the body response. Yellow blocks correspond to the spherical connections; these types of connections allow the torus buoy to respond in 6 degrees of freedom. The PTO is simplified by a damping coefficient and included in red blocks. Due to the PTO block placed between two spherical connections, it reacts to 6 degrees of freedom, but works exclusively along its local “Z” axis. Non-hydrodynamic bodies were connected between the PTO and the Spherical Joint blocks. The role of these bodies is to ensure the stable flow of information between the PTO and Joint blocks; however, their hydrodynamic properties are not included in the motion-response equations. The mooring system is included in the Mooring block, where the first approximation is a single coefficient in the X direction with a value of 1 MN/m. While a second approximation, it is a subsystem with the MoorDyn library, compiled in WEC-Sim by [24].

The simulations were performed with a time step of Δt = 0.01 s and a total simulation duration of 4100 s, including an initial ramp period of 500 s to avoid transient effects. Drag forces were included through Morison Element.

Hydrodynamic Coefficients

The hydrodynamic coefficients were calculated from BEM Solver Nemoh over a frequency range from 0.1 to 3 rad/s and a frequency step of 0.01 rad/s. A mesh sensitivity analysis with 4 resolutions was carried out over the frequency range 0.1–3 rad/s, with a frequency step of 0.01 rad/s. Resolutions were based on the average size element, with coarse (2 m), two medium (1.0 and 0.75 m), and a finer (0.50 m) resolution. Figure 5 and Figure 6 show the results of the sensitivity analysis, in which major differences are observed at coarse resolution. With mediums and fine resolution, the results do not show significant differences; however, computational time increases by three orders of magnitude between 1 m and 0.5 m resolutions. Full-frequency hydrodynamic coefficients were estimated at 0.75 m resolution and incorporated into the WEC-Sim model.

2.4. Power Take-Off

The PTO system is a linear damping coefficient. A stroke length of ±3 m has been provided to ensure that the buoy is free to oscillate and no additional forces are applied to the system. The stroke length was determined through an iterative process, accounting for the most challenging wave conditions expected at the site.

The aim was to optimize the PTO damping coefficient to increase the energy captured by the WEC. As stated in [25], the optimum damping for heave motion is approximately linear with respect to the wave period, but independent of the wave height. The period ranges from 8 to 16 s, based on Figure 2. The PTO damping ranges from 0.1 to 2.9 MN/s, divided into eight intervals.

2.5. Energy Capture Width Ratio

The WEC performance is assessed by the captured width ratio ($CWR$). The $CWR$ is the ratio of the power captured by the device and the total power available along a characteristic length of the device [26]. Equation (2) shows the formulation of the $CWR$, where P is the absorbed power, J is the wave power, and B is the device’s characteristic dimension. According to [27], B is calculated as $B=\sqrt{4A_w/\pi },$ with $A_w$ being the horizontal cross-sectional area.

$$\begin{array}{c}CWR={\displaystyle \frac{\mathit{P}}{\mathit{J}\mathit{B}}} \end{array}$$

(2)

$$\begin{array}{c}J={\displaystyle \frac{\mathit{\rho }{\mathit{g}}^2}{16\mathit{\pi }}}{\mathit{H}}_{\mathit{S}}^2{\mathit{T}}_{\mathit{p}}{\mathit{C}}_{\mathit{g}}\end{array}$$

(3)

$$\begin{array}{c}{\mathit{C}}_{\mathit{g}}={\displaystyle \frac{1}{2}}\left[1+{\displaystyle \frac{2\mathit{k}\mathit{d}}{\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(2\mathit{k}\mathit{d}\right)}}\right]\end{array}$$

(4)

where $C_g$ is the group celerity, k is the wave number, and d is the water depth.

Once the optimal layout is found, the Mean Annual Energy Production (MAEP) is calculated. The MAEP results from multiplying bin by bin the power matrix by the site operation’s scatter diagram [27]. The MAEP is represented by the following:

$$\begin{array}{c}MAEP= 8760{\sum }_{\mathit{i}=1}^{\mathit{i}=\mathit{N}}{\mathit{P}}_{\mathit{i}}·{\mathit{F}}_{\mathit{i}}\end{array}$$

(5)

where P is the power matrix, F is the occurrence probability matrix, and 8760 is the mean hours per year; the power matrix contains information on the captured or produced energy for each sea state.

2.6. Mooring System Configurations

The design of a mooring system is a multifaceted process, influenced by numerous factors including the working principle, environmental conditions, natural frequency, and the deployment site’s characteristics (water depth, soil restrictions, flora, fauna, etc.). Xu, Sheng [13] presents a systematic approach for the design of a mooring system for wave energy converters, as outlined in Figure 7 and utilized in this study. The fundamental objective of the mooring design procedure is to fulfill the structural criteria, namely the ultimate strength limit (USL), the allowable strength limit (ALS), and the fatigue limit state (FLS). DNVGL-OS-E301 [28] defines USL as the ultimate limit state that ensures each mooring line has sufficient strength to withstand the load imposed by extreme environmental actions. The accidental limit state (ALS) is defined as the maximum load that the mooring system can withstand in the event of the failure of one mooring line. Finally, the fatigue limit state (FLS) refers to the analysis of fatigue. There are numerous structural codes for the design of mooring elements, including ABS, DNV, IEC, ISO, and API. The preponderance of these codes pertains to the oil and gas industry, with a focus on various stages of the design process, including environmental loads, floating structure response, and materials.

This study limits the forces in the mooring chains to accomplish the structural safety factor stipulated in Recommended Practice DNV-RP-C205 [29] for the ULS check. The safety factor is the ratio of the yield stress (σ_stress) to the maximum stress (σ_max), where, in mooring design, they are commonly replaced by the breaking load and the maximum force. The objective of this study is to define an optimal mooring configuration that enhances the performance of the WEC and incorporates it into the final energy capture estimate. However, the optimization and full structural design of the mooring elements are future research.

As mentioned before, the first mooring approximation is a linear spring along the “X” axis. However, to investigate the WEC response and energy capture under a more realistic mooring system, the linear spring was replaced with a mooring line system by coupling the WEC-Sim and MoorDyn models. Two configurations were investigated: a single catenary (mooring A) and a catenary with a floating buoy (mooring B), as illustrated in Figure 8. The mooring system utilizes an R3 chain-grade stud with a 102 mm diameter, a weight of 227 kg/m, and a breaking load of 8336 kN. In the case of mooring B, the material employed between the anchor and the floating buoy is identical to that used in mooring A. The section between the buoy and the main floater comprises an R3 chain-grade stud, with a diameter of 76 mm, a weight of 126.5 kg/m, and a breaking load of 4884 kN. A fully mooring design must comply with the stress requirements and several other factors, including fabrication, installation, maintenance, station-keeping performance, fatigue evaluation, and decommissioning. However, this study limits itself to the WEC hydrodynamic performance and leaves the optimization of the mooring line for future research.

3. Results

This section is organized as follows: first, the optimal PTO number and damping estimation are presented; second, the power matrix, CWR, and MAEP are analyzed; third, the mooring system design is described; finally, the performance matrix is estimated for the optimized mooring layout.

3.1. Number of PTOs

To determine the optimal number of PTOs, three different arrays were analyzed. In each array layout, the PTOs are uniformly distributed along the circumference of the torus. Energy capture is estimated assuming irregular waves, with the following parameters: H_s = 1 m, T_p = 8–16 s, and damping = 0.1–2.9 MNs/m. Figure 9 illustrates the matrices for the layouts mentioned above, with colors representing the hourly average energy capture.

It can be observed that the maximum energy capture value remains relatively consistent between the layouts, although the energy capture distribution of the plot varies with the number of PTOs. For four PTOs, the maximum energy capture is more spread across damping and peak period, whereas for six and eight PTOs, it concentrates at lower damping values. The six PTO layout shows more consistent energy capture around 0.9 MNs/m, and thus represents the optimal layout for estimating the WEC power matrix.

To understand the decrease in energy captured from 4 to 8 PTO, and along the damping coefficient, heave response spectrum for damping = 0.9 and 2.0 MNs/m, at T_p = 12 s, and unitary wave height are plotted in Figure 10. It is evident that with a damping coefficient of 2.0 MNs/m, the torus floater becomes more rigid, and the heave response decreases. The 4 PTO arrangement shows a bigger response than the 6 and 8 PTO arrays for both damping coefficients. This fact is also observed in Figure 7, where 4 PTO energy captured shows higher values and a wider spread distribution. On the other hand, the 8 PTOs arrangement shows two-frequency peaks, with the smaller peak at the spectral peak frequency ω_peak = 0.5236 rad/s (T_p = 12 s) and the bigger around 1.4 rad/s (T_p ~ 4.5 s). In other words, increasing the PTO number and operating at high damping, the heave resonance frequency shifts dramatically, thereby affecting WEC efficiency.

3.2. Power Matrix, CWR, and MAEP

From Figure 9, it can be seen that the 6 PTOs layout with damping equal to 0.9 MNs/m represents the optimal configuration for the predominant wave conditions at the site of interest. Figure 8a depicts the power matrix for this layout. It can be observed that maximum power is placed between T_p of 8–15 s. Power matrices with a similar energy capture distribution have been identified in previous studies of heave absorbers [25]. Figure 11b illustrates the CWR contours for the power matrix considering the wave conditions at the site of interest, indicating CWR between 0.20 and 0.50 over the T_p range at the site of interest.

Figure 12 shows the performance matrix for the study area; the mean power absorption ranges from 2 to 14 MWh/y, mainly driven by T_p. The total annual power absorption is equal to 585 MWh/y.

3.3. Mooring System

As previously stated, the primary objective of the mooring system is to maintain the WEC within specified limits, regardless of the incoming environmental conditions. This section presents the findings of the evaluation of the mooring configurations. This analysis was conducted considering the following assumptions:

• It is assumed that the bottom can support the vertical and horizontal loading in the PTO foundations and anchor points. Consequently, the design of those structural elements is not included in the present analysis.
• The wind forces are not considered.
• The speed of the ocean current is imposed on the surface and decays exponentially with depth, aligning with the direction of the wave.

As stated in [30], the mean velocity of the surface current within the bay is 0.15 m/s, with a maximum value of 0.30 m/s. Reference [31] documented the occurrence of anticyclonic eddy circulation during local wind events, with an average velocity of 0.10 m/s. Based on the literature review, the performance of the WEC has been studied in the context of wave–current interaction, with typical speed-current of 0.3 m/s, H_s = 1 m, and T_p = 8 and 13 s. The total simulation time was 4100 s, which included a 500 s ramp period and a time step of 0.01 s.

Figure 13 and Figure 14 present the floater response and energy capture for WEC-Moorings A and B during T_p of 8 and 13 s. The dashed line represents the results without ocean currents, while the solid line represents those with currents. The presence of surface currents has a negligible impact on the heave and pitch responses of WEC-Mooring A, with amplitudes of approximately 0.4 m and 0.04 radians, respectively. Conversely, the surge response shows greater temporal variation, although this does not extend to the amplitude response, which averages approximately 3 m. Regarding energy capture, a brief reduction is observed in the presence of ocean currents. With a peak period of 8 s, the average energy capture increases from 31.43 kWh to 34.02 kWh when the simulation includes the surface current. Conversely, a slight reduction is observed when T_p is 13 s, from 86.93 kWh to 79.80 kWh.

A similar trend is observed for WEC-Mooring B, where the torus response shows no significant changes with the inclusion of surface current interaction. However, some observations can be made:

• With T_p = 8 s, the surge response reaches values of 8 m, while at 13 s, the surge response is 4 m, but with a permanent offset from the original position.
• The heave motion shows a similar amplitude as configuration A for both wave peak periods.
• Pitch motions decrease in comparison with WEC-Mooring A and can be associated directly with weaker restrictions imposed by the mooring line.
• The average energy capture is less influenced by the current, and the values remain around 32.69 and 97.43, 8 and 13 s, respectively.

In Figure 15, the surge response of the torus floater, total mooring force, and foundation force for mooring A and B are presented. The floater surge response generally exhibits a greater amplitude with WEC-mooring B than with A. Its mean position is displaced by approximately 2 m in the wave direction. Furthermore, it was observed that the floater response with mooring B has a longer oscillating period in comparison with mooring A. This behavior is illustrated in Figure 16, where the surge mooring force PSD at the GC shows that mooring A reacts in the angular frequency range of 0.2 to 0.6 rad/s (31 to 10 s), whereas mooring B reacts at angular frequencies lower than 0.2 rad/s. Additionally, mooring B forces are three orders of magnitude smaller than those of mooring A. The PTO foundation forces show values in the same order for both configurations; however, when the WEC experiences the larger displacement, the foundation forces exhibit a considerable increase, e.g., mooring B, T_p = 13 s around t = 935 s and t = 1100 s. These indicate that mooring configuration B is incapable of absorbing the hydrodynamic forces leading to substantial surge oscillation, requiring the contribution of PTO foundations to limit the surge response, and therefore, mooring B will demand a robust PTO foundation to withstand high lateral forces and a substantial PTO stroke length to handle the surge response and avert additional impact forces.

As shown in Figure 13 and Figure 14, the ocean current does not significantly affect WEC performance; therefore, it is not included in the rest of the study. Energy capture is estimated over the entire operational range of the peak period, and H_s = 1 m, see Figure 17. Energy capture is similar for both mooring configurations when T_p < 11 s. Over 11 s of wave peak period, the curves separate, reaching their maximum at 13 s with 82.93 and 96.83 kW/h for WEC-mooring A and B, respectively. After 13 s, both energy captures decrease with a similar trend.

3.4. Extreme Operational Conditions

To assess the mooring forces under severe operational conditions, H_s is set to 3 m, and PTO damping to 0.9 MNs/m. A numerical simulation was carried out from T_p = 5 s to 20 s with a constant increment of 1 s. According to Figure 18a, the anchors at mooring A experience similar forces over the range T_p = 6–10 s. For T_p > than 10 s, the lift force curves diverge with bigger values in anchor 1. It can be observed in Figure 19a,b that under extreme operational conditions, the lines of mooring A reach their maximum extension, resulting in massive horizontal and vertical anchor forces. This operational condition demands a non-conventional anchor system, for instance, a suction anchor system. On the other hand, the mooring B anchored forces are significantly lower (Figure 18b), which can be attributed to the lines not reaching maximum extension (Figure 19c,d); therefore, the energy is absorbed by the PTO foundation.

Figure 20 shows the forces of the 0° foundation, which experiences the highest forces during extreme operational conditions. The vertical force does not vary significantly for moorings A and B, with a peak value around T_p = 8 s of about 450 kN. On the other hand, the horizontal force on mooring B is about twice that on mooring A, suggesting that the PTO foundation plays an essential role in counteracting the surge response.

As mentioned in [32], buoys attached to the lines could reduce the tension, as observed in

Table 3. Mooring line tensions.

Line	Mooring A		Mooring B
	Force (kN)	Tp (s)	Force (kN)	Tp
Line 1	1169.3	13	L1 = 430.68 L2 = 331.92	13 12
Line 2, 3	971.59	12	L1 = 79.77 L2 = 133.88	12 12

. For mooring A, the maximum tension occurs with T_p of 13 s in line 1, while for mooring B, the tension is reduced by around 60%. Tension lines are below the breaking limit, ensuring safe operation. However, as mentioned before, this arrangement releases the floater to experience large displacements and increases foundation forces.

3.5. WEC Performance with Mooring System

This section shows the WEC performance exclusively with the mooring A system, a configuration selected for its ability to provide a more stable floater response. This configuration may offer operational advantages, such as reducing hydraulic pistons or eliminating the need for a robust end-stop installation, which could otherwise increase structural demands. Figure 18a presents the power matrix, and Figure 21b illustrates the CWR. In comparison with the power matrix estimated in Section 3.2, the following differences can be observed:

• The high-energy capture contours concentrate around T_p = 13 s, corresponding to the high occurrence probability;
• The maximum energy capture increases from 200 to 500 kW;
• The CWR maintains around 0.5 but occurs between T_p 10–14 s instead of 7–10 s.

Finally, Figure 22 shows the performance matrix, which clearly shows that energy capture increased by 10 MWh/y compared to the performance matrix in Section 3.2. The annual energy capture is 657 MW, an increase of 12% over the previous estimate.

4. Discussion

This study evaluated the hydrodynamic performance of a toroidal WEC adapted for Mexican coastal conditions, with particular focus on the interaction between the device and its mooring system. The mooring design was required to limit excessive surge motion while avoiding any detrimental impact on power absorption.

The preliminary analysis included determining the number of PTO units and their corresponding optimal damping coefficients, assuming a simplified mooring representation as a linear spring and a unit wave height. The optimal damping coefficient was set to 0.9 MN·s/m with six PTO units, as this configuration provided stable performance across the T_p range. The corresponding CWR reached 0.50, consistent with values reported in the literature. Although local sea states are dominated by T_p of 10–14 s, the maximum CWR occurred between 7 and 10 s, resulting in two distinct performance peaks in the annual power matrix.

To evaluate the mooring system’s influence on energy capture, the linear spring approximation was replaced with two explicit mooring configurations: (A) a single catenary system and (B) a catenary system incorporating a surface buoy. For T_p below 11 s, both systems yielded comparable performance; however, for longer periods, WEC-Mooring B produced up to 20% higher energy capture.

The results of including a surface current align with [33]: absorbed power, heave, and pitch motions remain largely unaffected. In contrast, surge motion under current conditions was altered for mooring A, particularly near T_p = 8 s, where surge amplitude decreased (Figure 23).

Under extreme operational conditions (H_s = 3 m), both configurations exhibited maximum line tensions near T_p = 12–13 s (Table 3), though their mechanical behavior differed substantially. Configuration B reduced peak line tensions but allowed greater horizontal drift of the floater, leading to reaction forces on the PTO foundation up to three times larger than those observed with configuration A. While mooring A improves station-keeping and reduces foundation loading, its anchoring requirements are more demanding and warrant further investigation.

5. Conclusions

After defining the mooring configuration, the power matrix, CWR, and performance matrix were recomputed. The refined model produced a 12% increase in annual mean energy capture, reaching 657 MWh/y, while the maximum CWR remained at 0.50. The improvement occurred primarily within the dominant operational range of 10–14 s. These results indicate that the simplified stiffness-matrix approach provides a useful first estimate of WEC performance; however, accurate prediction of hydrodynamic response and mooring loads requires explicit modeling of mooring–device interactions.

These results demonstrate the importance of including a full-coupling mooring system during the hydrodynamic and captured energy evaluation performance of WECs. In low and moderate-energy seas, due to resource limitations, accurately estimating relevant parameters such as energy captured, structural loads, and mooring response becomes crucial to the success of energy conversion projects. Although the current work focuses on the Ensenada Bay area, future investigations will extend the results to other Mexican regions, like the Caribbean Sea, the Gulf of Mexico, and the Gulf of Tehuantepec. On the other hand, further steps include physical experiments and PTO design.

Despite the insights presented, several limitations should be acknowledged. The numerical simulations rely on linear wave theory and do not explicitly include viscous effects; therefore, they do not capture nonlinear wave–structure interaction effects. Additionally, the PTO system was replaced by ideal damping coefficients that ignore real issues such as technological limitations. Although the wave climate data set represents long-term sea conditions, there is a lack of information on extreme events that would enable more reliable estimations. These simplifications are common in early-stage WEC performance assessments; however, they will be addressed in future work to increase the robustness of the results.