Numerical and Experimental Power Output Estimation for a Small-Scale Hinged Wave Energy Converter

Abstract

Wave energy converters (WECs) integrated into breakwaters present a promising solution for combining coastal protection with renewable energy generation, addressing both energy demands and environmental concerns. Additionally, this integration offers cost-sharing opportunities, making the overall investment more economically viable. This study explores the potential of a hinged point-absorber WEC, specifically designed as a floating hinged half-sphere, by assessing the device’s power output and comparing two different breakwater configurations. To evaluate the device’s performance, a comprehensive numerical and experimental approach was adopted. Numerical simulations were carried out using a radiation-diffraction model, a time domain tool for analyzing wave–structure interactions. These simulations predicted average power outputs of 25 kW for sloped breakwaters and 18 kW for vertical breakwaters located at two strategic sites: the Port of Leixões and the mouth of the Douro River in Portugal. To validate these predictions, a 1:14 scale model of the WEC was constructed and subjected to testing in a wave–current flume, replicating different sea-state conditions. The experimental results closely aligned with the numerical simulations, demonstrating a good match in terms of relative error and relative amplitude operator (RAO). This alignment confirms the reliability of the predictive model. These findings support the potential of integrating WECs into breakwaters, contributing to port energy self-sufficiency and decarbonization.

1. Introduction

Energy is crucial to modern life, particularly in coastal areas where a significant portion of the global population resides. From transportation to medicine, energy is currently used in a significant number of applications that shape the way humans live nowadays. The majority of developed nations heavily depend on energy to fulfill various aspects of the economy. However, the growing demand of energy and the increase of global temperature require new and cleaner ways of meeting the energy demand worldwide. Thus, developing new technologies and innovative solutions to meet those challenges can help to improve energy efficiency, reduce greenhouse gas emissions, and enhance the reliability, affordability, and security of energy systems, as well as increase life quality. Moreover, addressing the European target of climate neutrality by 2050 will also require significant improvements in the energy sector.

Conventional renewable energy resources, such as wind and solar, have some limitations for on-land large scale development. On the other hand, marine renewable energy resources are promising and still a very much unexplored option for meeting those needs. Marine renewable energy sources (tidal, thermal, offshore, and wave) have the potential to supply up to 20% of the world’s electricity demand by 2050 [1]. Furthermore, wave and tidal energy have several advantages, including high energy density [2] and low visual impact due to offshore installations. Their output is predictable [3], with wave power levels forecastable 1–2 days in advance, which is beneficial for grid balancing. Additionally, the seasonal wave resource correlates with electricity consumption in the northern hemisphere, making it a reliable energy source especially in winter [4].

Wave energy is one of the types of marine energy that can be explored, and research gained widespread attention in the 1960s and 1970s during the global energy crisis, but interest in wave energy conversion technology has fluctuated over the years. Despite some progress, WECs remain relatively new, with only a few designs having undergone real-sea testing. Currently, wave energy is not yet economically competitive with other renewable resources, such as wind and solar energy. To become a viable source of renewable energy, capital and operating expenditures must be reduced by 45%, and power production must increase by 200% [5]. Therefore, a significant improvement, rather than just incremental progress, is necessary for wave energy to achieve economic viability. Thus, strategies to reach commercialization are required. One possibility is to combine wave energy with coastal or other renewable energy structures to take advantage of cost-sharing benefits.

The global potential of wave energy is enormous, with estimates of over 2 TW [6]. However, despite its vast potential, wave energy remains underdeveloped largely due to the high costs associated with it. The testing and deployment of WECs can be particularly expensive due to the harsh marine environment conditions, the need for significant transportation efforts [7], and the high labor requirements. Over 1000 different concepts for WECs were already invented and studied, but only a few of them have advanced to the testing phase [8]. Numerical modeling plays a fundamental role in this sense, being a tool able to predict the behavior of the WEC even before building a physical model.

A hinged WEC device is a technology for harvesting electricity from waves, which is based on the conversion of energy from the movement of a floating body connected to an articulated arm fixed to a coastal structure (e.g., breakwater port, jetty). Several different configurations were already tested: Zhang et al. [9] studied a WEC consisting of two rafts hinged at each raft end and one pendulum hung at the joint of the rafts and showed that the performance of the device with the hanging pendulum is much better than a conventional raft-type WEC in terms of capture width ratio and wavelength bandwidth. Coiro et al. [10] studied a WEC that consists of an oscillating point pivoted floater with an electromechanical generator based on ball-bearing screw, conducting to a real-scale model for a preliminary design of a 60 kW wave generator. Giannini et al. and Rosa-Santos et al. [11,12] studied a WEC that was based on the oblique motion of two floating lateral mobile modules, and it was shown that 10 and 40% of the incident wave power can be absorbed. Those are only a few examples of studies that were supported by numerical simulation and physical modeling to consistently predict the power output of near-shore devices.

Under the scope of device performance, Pastor et al. [13] suggest that larger diameter WECs are generally more effective for wave energy absorption, provided that an economically viable method for power conversion can be implemented. However, there has been limited research on deploying wave energy converters (WECs) on pre-existing structures like breakwaters and jetties. Given the specific characteristics of hinged WECs, understanding the influence of factors such as structure type, slope, roughness, and other environmental conditions is crucial for optimizing performance and conducting feasibility studies for WEC deployment at particular locations.

Hence, this study aims to investigate, optimize, and experimentally validate a hinged WEC technology for producing electricity from waves, consisting of a single floater attached to a movable arm, as well as to compare and understand the effects of different breakwaters’ geometries in the device performance. To achieve that, two breakwaters’ configurations (sloped and vertical) associated with two existing locations in Porto, Portugal, were analyzed and discussed. The first breakwater is of the sloped type and was based on the north breakwater of the Port of Leixões and the second is of the vertical type, based on the north breakwater of the Douro River (Figure 1). Both locations present good resources for wave energy harvesting, and significant interest has been shown in those locations in recent years [14]. Furthermore, numerical simulations were made to support the performance assessment by using proper simulation software; by doing so, it was possible to predict the expectable power output of the device. Additionally, an experimental model in the scale 1:14 was built to test and validate the numerical simulations in a wave current flume tank. Finally, a statistical study was developed to better understand the relation between the factors determining the system response, namely, the significance wave height, wave direction, and wave period. Figure 2 displays a flowchart with the overall workflow followed in this study.

2. Case Study

2.1. Wave Resource

Before proceeding to any device performance estimation, it is crucial to understand the resource available for wave energy generation. Both breakwaters under study are relatively close to one another (2.5 km apart); therefore, the same resource was considered as valid for both locations.

The Laboratório Nacional de Engenharia Civil (LNEC) performed a series of studies in 2017 due to the extension of the north breakwater of the Port of Leixões. One of those studies was related to the sea states’ characterization that was used to describe the wave resource for this study. The values are represented according to the significant wave height (Hs), the zero-crossing wave period (Tz), and the mean wave direction in degrees. The relation between the different wave periods commonly used in WEC studies are shown in Equation (1) [15]. Where, Te (energy period) represents the time it takes for wave energy to pass a certain point, Tp (peak period) is the interval between the highest wave peaks, and Tz (zero-crossing period) is the average time between zero up-crossings. Having multiple wave periods is necessary because each of these parameters conveys different aspects of the wave’s behavior, and, together, they provide a comprehensive understanding of the ocean wave environment.

Tp = 1.12 Te = 1.29 Tz

(1)

Table 1 and Table 2 display the values for the wave resource under the location of interest. Overall, the wave characteristics of the given dataset indicate that the significant wave heights range between 0 and 8.98 m, with an average value of 1.63 m [16]. The majority of the recorded values lie between 0 and 4 m, with the most common range being between 1 and 3 m. Similarly, the average zero-crossing periods range from 3.9 to 17.3 s, with an average value of 8.7 s. Most of the recorded values fall within the range of 6 to 12 s. Lastly, the average wave directions fall between 182° and 342°, with an average value of 295°, with the most common ranges being between 280 and 315°, corresponding mainly to the NW direction [17].

The concept of SST (sea state) refers to the combination of specific sea conditions, typically defined by three key components: wave period, wave height, and wave direction. Together, these parameters describe a particular sea condition, which can vary in likelihood depending on the location. Maritime buoys are commonly used to monitor and record these sea states, providing valuable data on the prevailing conditions in a given area. Based on these data, the most frequent sea states for a region can be identified and analyzed. Therefore, for performing the simulations, only the most common SSTs were selected due to the necessity of decreasing the number of SST to be analyzed. However, the ranges that were selected according to what was previously discussed encompass more than 7100 h per year (81%), thus ensuring representativeness. Table 3 displays all the 42 simulations that were built according to the three parameters (Direction, Hs, and Tz).

2.2. Hinged WEC

As the name already describes, hinged, articulated, or pivoted WECs rely somehow on hinged motion [18]. This is, these WECs absorb the energy associated with the relative motion of the adjacent bodies, and, through a power take-off (PTO) system, they produce electricity. In the case of this study, the adjacent floating body, for simplicity, is a half sphere floater, and the hinged motion happens to be relative to the breakwater. The natural oscillation of the incoming waves makes the body move up and then move down. The up and down movement drives the hinged system, making the arm move together with the floater that in turn drives the PTO. The PTO, however, can vary in nature, with the most common types being pneumatic and hydraulic systems, although mechanical systems have also been employed.

Table 1. Hs and Tz distribution [17].

Wave Resource		Tz (s)												Total
		4	5	6	7	8	9	10	11	12	13	14	15
Hs (m)	6	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
	5	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
	4	0%	0%	0%	0%	0%	1%	1%	1%	1%	1%	1%	0%	6%
	3	0%	0%	0%	0%	1%	3%	4%	4%	3%	2%	1%	0%	18%
	2	0%	0%	2%	6%	9%	10%	8%	6%	3%	1%	0%	0%	45%
	1	0%	1%	6%	8%	7%	4%	1%	0%	0%	0%	0%	0%	27%
	0	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
Total		0%	1%	8%	14%	17%	18%	14%	11%	7%	4%	2%	0%		96% *
														98% *

* Some totals are not 100% due to decimals next to zeros that were unfortunately not available in the database.

Table 1. Hs and Tz distribution [17].

Wave Resource		Tz (s)												Total
		4	5	6	7	8	9	10	11	12	13	14	15
Hs (m)	6	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
	5	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
	4	0%	0%	0%	0%	0%	1%	1%	1%	1%	1%	1%	0%	6%
	3	0%	0%	0%	0%	1%	3%	4%	4%	3%	2%	1%	0%	18%
	2	0%	0%	2%	6%	9%	10%	8%	6%	3%	1%	0%	0%	45%
	1	0%	1%	6%	8%	7%	4%	1%	0%	0%	0%	0%	0%	27%
	0	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
Total		0%	1%	8%	14%	17%	18%	14%	11%	7%	4%	2%	0%		96% *
														98% *

* Some totals are not 100% due to decimals next to zeros that were unfortunately not available in the database.

Table 2. Hs and mean wave direction distribution [17].

Wave Resource		Mean Wave Direction (°)									Total
		180	203	225	248	270	293	315	338	360
Hs (m)	6	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
	5	0%	0%	0%	0%	0%	1%	0%	0%	0%	1%
	4	0%	0%	0%	0%	1%	4%	1%	0%	0%	6%
	3	0%	0%	0%	1%	2%	9%	6%	0%	0%	18%
	2	0%	0%	0%	1%	3%	14%	21%	6%	0%	45%
	1	0%	0%	0%	0%	1%	3%	8%	16%	0%	28%
	0	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
Total		0%	0%	0%	2%	7%	31%	36%	22%	0%		98% *
											98% *

* Some totals are not 100% due to decimals next to zeros that were unfortunately not available in the database.

Table 2. Hs and mean wave direction distribution [17].

Wave Resource		Mean Wave Direction (°)									Total
		180	203	225	248	270	293	315	338	360
Hs (m)	6	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
	5	0%	0%	0%	0%	0%	1%	0%	0%	0%	1%
	4	0%	0%	0%	0%	1%	4%	1%	0%	0%	6%
	3	0%	0%	0%	1%	2%	9%	6%	0%	0%	18%
	2	0%	0%	0%	1%	3%	14%	21%	6%	0%	45%
	1	0%	0%	0%	0%	1%	3%	8%	16%	0%	28%
	0	0%	0%	0%	0%	0%	0%	0%	0%	0%	0%
Total		0%	0%	0%	2%	7%	31%	36%	22%	0%		98% *
											98% *

* Some totals are not 100% due to decimals next to zeros that were unfortunately not available in the database.

Table 3. SST selected for analysis.

SST	Hs (m)	Tp (s)	Dir. (°) *	Freq. (%)	Hours (h)	SST	Hs (m)	Tp (s)	Dir. (°) *	Freq. (%)	Hours (h)
1	3	9	280–300	3%	263	22	2	10	280–300	8%	701
2	3	9	290			23	2	10	290
3	3	9	315			24	2	10	315
4	3	10	280–300	4%	351	25	2	11	280–300	6%	526
5	3	10	290			26	2	11	290
6	3	10	315			27	2	11	315
7	3	11	280–300	4%	351	28	2	12	280–300	3%	263
8	3	11	290			29	2	12	290
9	3	11	315			30	2	12	315
10	3	12	280–300	3%	263	31	1	6	280–300	6%	526
11	3	12	290			32	1	6	290
12	3	12	315			33	1	6	315
13	2	7	280–300	6%	526	34	1	7	280–300	8%	701
14	2	7	290			35	1	7	290
15	2	7	315			36	1	7	315
16	2	8	280–300	9%	789	37	1	8	280–300	7%	614
17	2	8	290			38	1	8	290
18	2	8	315			39	1	8	315
19	2	9	280–300	10%	877	40	1	9	280–300	4%	351
20	2	9	290			41	1	9	290
21	2	9	315			42	1	9	315
Total (h)											7100
Annual (h)											8766
Representativeness (%)											81%

* The directions 280–300°, 290°, and 300° are in accordance with the usual waves’ rose diagram, where north corresponds to 0°. Thus, those values can be interpreted as 80°, 90°, and 115° perpendicular to the breakwater.

Table 3. SST selected for analysis.

SST	Hs (m)	Tp (s)	Dir. (°) *	Freq. (%)	Hours (h)	SST	Hs (m)	Tp (s)	Dir. (°) *	Freq. (%)	Hours (h)
1	3	9	280–300	3%	263	22	2	10	280–300	8%	701
2	3	9	290			23	2	10	290
3	3	9	315			24	2	10	315
4	3	10	280–300	4%	351	25	2	11	280–300	6%	526
5	3	10	290			26	2	11	290
6	3	10	315			27	2	11	315
7	3	11	280–300	4%	351	28	2	12	280–300	3%	263
8	3	11	290			29	2	12	290
9	3	11	315			30	2	12	315
10	3	12	280–300	3%	263	31	1	6	280–300	6%	526
11	3	12	290			32	1	6	290
12	3	12	315			33	1	6	315
13	2	7	280–300	6%	526	34	1	7	280–300	8%	701
14	2	7	290			35	1	7	290
15	2	7	315			36	1	7	315
16	2	8	280–300	9%	789	37	1	8	280–300	7%	614
17	2	8	290			38	1	8	290
18	2	8	315			39	1	8	315
19	2	9	280–300	10%	877	40	1	9	280–300	4%	351
20	2	9	290			41	1	9	290
21	2	9	315			42	1	9	315
Total (h)											7100
Annual (h)											8766
Representativeness (%)											81%

* The directions 280–300°, 290°, and 300° are in accordance with the usual waves’ rose diagram, where north corresponds to 0°. Thus, those values can be interpreted as 80°, 90°, and 115° perpendicular to the breakwater.

The dimensions of the body for the numerical model followed pre-existing devices [19]. The body is a 3 m diameter hollow half sphere, connected through a 6 m rigid arm to the breakwater. The shell is made of steel and has a thickness of 18 mm, ensuring good floatability and endurance. From the mean water level (MWL), the arm is 6° inclined horizontally, and the freeboard of the floating body is considered to be circa half of its radius (0.7 m). Finally, the vertical breakwater considered in this study has an approximate height of 5 m from the MWL (Figure 3), and the bathymetry in that region suggests that the depth is about 8 m at the hydrographic zero of Leixões (ZHL) [17]. The sloped breakwater tested has the same height and was placed over the same bathymetry; however, it follows a trapezoidal shape, sloping 56° with the horizontal plane.

3. Numerical Simulation

3.1. Software Parameters

The software used for the numerical simulations was ANSYS^® (v2022.R2), specifically the AQWA module, which is tailored for analyzing wave loading and responses of floating and fixed structures. The 3D model was created in SolidWorks^® (v2023.SP2.0) and then imported into the ANSYS^® environment. Consequently, all relevant parameters such as inertia moments and densities were seamlessly transmitted along with the 3D model. The software operates by solving the linearized potential flow theory equations to predict the hydrodynamic behavior of structures subjected to wave action. The AQWA module divides the problem into frequency and time domains. In the frequency domain analysis, it computes the hydrodynamic coefficients such as added mass, damping, and wave excitation forces by solving the boundary value problem of the fluid flow around the structure. These coefficients are then used in the time domain to simulate the dynamic response of the structure to time-varying wave loads. The dynamic problem can be described as Equation (2).

$$\mathrm{m}\mathsf{\alpha }={\mathrm{f}}_{\mathrm{h}\mathrm{s}}+{\mathrm{f}}_{\mathrm{e}\mathrm{x}\mathrm{c}}-{\mathrm{f}}_{\mathrm{p}\mathrm{t}\mathrm{o}}-{\mathrm{f}}_{\mathrm{r}\mathrm{a}\mathrm{d}}$$

(2)

where $\mathrm{m}$ is the mass of the WEC plus its added mass, $\mathsf{\alpha }$ is the WEC acceleration, ${\mathrm{f}}_{\mathrm{h}\mathrm{s}}$ is the hydrostatic restoring force, ${\mathrm{f}}_{\mathrm{e}\mathrm{x}\mathrm{c}}$ are the wave forces, ${\mathrm{f}}_{\mathrm{p}\mathrm{t}\mathrm{o}}$ is the PTO force, and ${\mathrm{f}}_{\mathrm{r}\mathrm{a}\mathrm{d}}$ is the wave radiation force. By coupling and calculating those parameters, the software is able to describe the body acceleration based on the waves’ loads.

Software parameter calibration was necessary in order to give the right input to the moving body. Initially, joints were defined to establish connections between the body and the breakwater, together with the boundary conditions, governing the degrees of freedom for body movement. A hinged joint was employed to permit only vertical (Z-axis) movement, while a rigid joint ensured the immovability of the breakwater structure. Mesh generation was customized to suit the varying scales of the body and breakwater. Thus, separate meshes were designated, with finer resolution allocated to the body (more than 1.000 elements) and a coarser representation for the breakwater, ensuring computational efficiency without sacrificing accuracy. Furthermore, the material properties were assigned based on the Archimedes floating principle coupled with standard values for steel.

Hydrodynamic diffraction analysis encompassed a wave direction range from −180° to 180° at intervals of 45°. Data on hydrodynamic coefficients were acquired for seven distinct wave directions, with frequencies spanning from 2.4 to 65 s. Time domain simulations of the hinged WEC were performed over a duration of 1000 s with a time-step of 0.05 s. The JONSWAP wave spectra was utilized with a gamma value of 3.3, which is suitable for the study area.

3.2. Best Damping Coefficient

The damping coefficient (Cd) represents the amount of resistance applied by the PTO system in a wave energy converter. It controls how much the WEC’s motion is “damped” or restricted. An optimal damping coefficient allows the WEC to move in sync with the waves [20], maximizing energy absorption without being overly restricted or too free to move.

The procedure adopted to find the optimal damping coefficient, i.e., Cd value, consisted of running 26 different simulations for different Cd values. The SST used for running those simulations was the most representative one, i.e., 2 m of Hs and 9 s of Tp. The Cd value ranged from 1 to 130 kNsm/°, and the method used was the bisection method. This method involves repeatedly narrowing down the range by selecting the midpoint and adjusting the range based on the results, thereby gradually finding the optimal value. With the numerical model established, it was possible to easily change the damping value and predict the power output based on the device velocity. Finally, Figure 4 shows the power curve obtained for this study after its normalization. It is graphically possible to observe that the best Cd value is somewhere between 20 and 30 kNsm/° for the vertical breakwater and between 37 and 42 kNsm/° for the sloped breakwater. Hence, the Cd value that was chosen to be used in the simulations was 25 kNsm/° for vertical breakwaters and 40 kNsm/° for sloped breakwaters.

3.3. Power Matrix

By the end of the simulations, data were processed to obtain power values. Table 4 and Table 5 display the power matrix (PM) generated after running all different SST simulations. The value displayed for each SST is the average for the three directions analyzed, calculated as the function of the z-movement of the body and the damping coefficient applied, and the cells without results (-) were not analyzed due to few occurrences throughout the year. The equation that allows the estimation of the power output for a point absorber body is shown below (Equation (3)). Where P represents the power output, Cd represents the damping coefficient and V the body velocity (in this case, towards the Z-axis).

P = Cd × V²

(3)

Table 4. PM obtained for the device in a vertical breakwater.

Numerical PM Vertical (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	38.9	35.6	32.4	28.9
	2	-	15.2	16.7	17.7	15.6	13.7	11.8
	1	4.1	4.0	4.1	4.0	-	-	-
	0	No waves

Table 4. PM obtained for the device in a vertical breakwater.

Numerical PM Vertical (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	38.9	35.6	32.4	28.9
	2	-	15.2	16.7	17.7	15.6	13.7	11.8
	1	4.1	4.0	4.1	4.0	-	-	-
	0	No waves

Table 5. PM obtained for the device in a sloped breakwater.

Numerical PM Sloped (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	51.7	50.6	50.0	47.5
	2	-	18.3	20.8	23.8	23.1	22.3	21.2
	1	4.5	4.7	5.2	5.8	-	-	-
	0	No waves

Table 5. PM obtained for the device in a sloped breakwater.

Numerical PM Sloped (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	51.7	50.6	50.0	47.5
	2	-	18.3	20.8	23.8	23.1	22.3	21.2
	1	4.5	4.7	5.2	5.8	-	-	-
	0	No waves

As can be seen, higher values of Hs lead to higher values of power, with the inverse happening to the Tp value in which a lower value leads to a higher power. This is strictly related to the natural period of oscillation of the floater: when the frequency of the incident waves is synchronized with the natural oscillation frequency of the device, the amplitude of the movement will be greater, consequently, the velocity and ultimately the power will also be greater [21]. Therefore, it can be said that the hinged WEC under study has a natural oscillation period of about 8 to 10 s, since those are the period values that the WEC produces more power for a given Hs. Figure 5 shows how the power value is always higher for frequencies between 0.1 and 0.12 Hz (8 to 10 s) for all the three Hs analyzed. The discontinuity observed in the 1 m and 3 m Hs curves is due to the agitation regime that is either excessively intensive (storm protection mode) or excessively calm (no waves).

Active damping optimization (ADO) control systems can be applied but, to present, have not yet been investigated. ADO aims to maximize the power output of a WEC by applying the best Cd for each SST that is currently happening. In this study, the Cd used is the best for only one SST, which is the most frequent SST; however, for other SSTs, this value would certainly vary. Much research and development has been done in this field, and the enhancement can be over 300% [22]. Moreover, for WECs like the one in this study, increasing the diameter of the buoy leads to a greater amount of wave energy that can be captured [23]. Consequently, the power output demonstrated in this study could potentially be enhanced by utilizing larger buoy bodies, allowing for more energy to be extracted from the waves.

One interesting way of graphically showing the power matrix is using a surface; this way, it is possible to display all the three variables (Hs, Tp, and Power) in the same figure. The steeper the surface, the more sensible is the device to handle different SSTs. Thus, an optimal surface should not resemble a uniformly high plateau where all SSTs yield consistently high-power output. In regions with lower available wave power, this would result in an excessively high capture-width ratio (CWR), which is theoretically unfeasible. The WEC under study showed itself to be a sensible WEC due to a large difference in the power output for small differences in the SST. Figure 6 displays the power matrix surface for the WEC under study.

3.4. Efficiencies and Breakwater Type Comparison

The capture width ratio (CWR) is a measure of how effectively a WEC captures wave energy relative to the available wave power in a given wave front. It is calculated as the ratio of the power captured by the device to the total wave power available across a certain width of the wave front. For estimating the CWR (efficiency, i.e., the relative capture width shown in Equation (5)), the available wave power along the length (Pw) was estimated according to Equation (4). Moreover, the characteristic length (L) of the device is the diameter of the floater.

$${\mathrm{P}}_{\mathrm{W}}={\displaystyle \frac{{\mathrm{g}}^2\mathsf{\rho }{\mathrm{H}}_{\mathrm{S}}^2{\mathrm{T}}_{\mathrm{E}}}{64\mathsf{\pi }}}$$

(4)

$$\mathrm{C}\mathrm{W}\mathrm{R}={\displaystyle \frac{\mathrm{P}}{{\mathrm{P}}_{\mathrm{W}}\times \mathrm{L}}}$$

(5)

Table 6 and Table 7 display the CWR values for both breakwaters. In this study, the value was almost constant for Hs variations, while for Tp variations, the CWR varied significantly, averaging 34% for vertical breakwaters and 45% for sloped breakwaters. Those variations can be graphically seen in Figure 7.

For both breakwater configurations, the results show that the lower the significant wave height, the higher the efficiency. The relationship with the wave period is also similar, matching findings on devices of the same category [11]. Additionally, although away from the natural oscillation period, SSTs with lower wave heights and lower periods will have larger values of CWR due to less power available.

One interesting aspect to be discussed is the differences between the two breakwater configurations. The sloped one showed itself to be a more effective power producer than the vertical one in every simulation. This was even expected as vertical breakwaters, despite their capability to reflect nearly all wave energy with minimal dissipation, experience significant energy losses in non-useful ways, such as splashing [24]. Conversely, in the numerical model, sloped breakwaters allowed waves to enter more smoothly, resulting in more energy being converted in a useful way, specifically through the WEC’s PTO. However, it is important to note that real-life sloped breakwaters are not smooth structures, they are designed to be rough and permeable to dissipate wave energy. This numerical model did not account for energy dissipation phenomena such as porosity, roughness, and turbulence, which are present in actual sloped breakwaters and might lead to lower power values compared to vertical breakwaters. More detailed and complex numerical models would be required to evaluate these effects accurately.

Additionally, the breakwater configuration plays a significant role in influencing the damping coefficient. The sloped layout resulted in a Cd that was 60% higher than that of the vertical layout, allowing for more power to be extracted even under similar wave conditions, as shown in Equation (3). This increase aligns better with the device’s natural oscillation frequency, as the sloped breakwater allows for greater vertical (Z-axis) movement of the buoy. Consequently, a higher damping coefficient is required to counterbalance the increased force, enabling the system to capture more energy. This highlights the improved synergy between the WEC and breakwater design, which leads to enhanced energy conversion efficiency.

Besides that, CWR values averaged around 40% for the WEC under study. This value is higher than the expectable with others similar heaving devices, in which the CWR value ranged between 4 to 36% [22]. This is probably due to the energy dissipation phenomena that were not considered by the model as mentioned before.

Table 6. CWR values obtained in a vertical breakwater.

Numerical CWR Vertical		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	37%	30%	25%	20%
	2	-	41%	40%	37%	30%	24%	19%
	1	52%	43%	39%	34%	-	-	-
	0	No waves

Table 6. CWR values obtained in a vertical breakwater.

Numerical CWR Vertical		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	37%	30%	25%	20%
	2	-	41%	40%	37%	30%	24%	19%
	1	52%	43%	39%	34%	-	-	-
	0	No waves

Table 7. CWR values obtained in a sloped breakwater.

Numerical CWR Sloped		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	49%	43%	38%	34%
	2	-	50%	49%	50%	44%	39%	34%
	1	58%	51%	50%	49%	-	-	-
	0	No waves

Table 7. CWR values obtained in a sloped breakwater.

Numerical CWR Sloped		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	4	Storm protection mode
	3	-	-	-	49%	43%	38%	34%
	2	-	50%	49%	50%	44%	39%	34%
	1	58%	51%	50%	49%	-	-	-
	0	No waves

4. Experimental Tests

4.1. Setup

The experimental study was carried out in the Hydraulics Laboratory of the Faculty of Engineering of the University of Porto, Portugal. The body concept for the physical model followed the idea of a disc-brake bicycle wheel (Figure 8a), where the bicycle fork would act as the WEC support, providing zero lateral movement, and the hydraulic disc brake would be the damping.

The floater itself is a crafted structure, originally a spherical marine signal buoy that has been modified by cutting it in half and adding a wooden top. For the arm, two metal bars were used, which were cut to 44 cm in length and screwed to the floater using six screws and nuts (Figure 8c). The angle of the arm with the floater was established using a piece of wood cut with the necessary inclination.

For the damping, a hydraulic brake system was used where a known preload would be applied to the brake, allowing to keep the damping coefficient constant. The bike fork purchased was already designed to have brake support, so the brake disc would be perfectly aligned and without friction. The brake lever was connected to a metal bar, and a system of threads and screws was made so that a specific damping value could be applied depending on how tight the brake was (Figure 8b).

Finally, the vertical breakwater used was 1 m by 1 m and had to be drilled in the middle so that the disc would fit into the structure. Two lateral wooden supports were also built to support the force of incident waves. Reinforced concrete blocks (2500 kg/m³) were used to give strength and robustness to the breakwater to stand the waves generated (Figure 9b). More than 750 kg of blocks were placed behind the breakwater.

The whole model was scaled according to Froude (Willian Froude 1810-1879) scaling laws, applying a 1:14 geometrical scale. This scale was determined by the diameter of the floater in comparison to the same diameter used in the numerical model. Equation (6) shows the relation between the numerical and physical models. SF represents the scaling factor, which can assume different values depending on the magnitude of the parameter [25].

$${\mathsf{\lambda }}_{\mathrm{R}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}}={\mathsf{\lambda }}_{\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{D}\mathrm{o}\mathrm{w}\mathrm{n}}\times {14}^{\mathrm{S}\mathrm{F}}$$

(6)

The channel where the tests were performed is a large wave–current flume that works in a closed circuit, located within the Hydraulics Laboratory of FEUP. The channel has a bottom with a slope of 0.5%, length of 32.3 m, and rectangular cross-section of 1 m wide and 1.33 m high. The test section has 7 glass windows, 2 m long and 1 m high, to allow an easy visualization of the flow and measurements. Downstream, a vertical flat gate, acting as a discharger, allows to regulate the height of the flow [26].

The physical model damping characterization was achieved by creating a digital twin of the pendulum in MATLAB^® (R2023a). By doing so, it was possible to determine the time that the system was required to completely stop when using a certain value of damping coefficient. Thus, after scaling down the Cd, it was possible to get the same value on the physical model with only a stopwatch.

Regarding the equipment used and the files generated, there were essentially three:

(1)

Infrared motion capture system (QUALISYS^®) to extract the exact position of the floater while being hit by the waves, generating files with the floater motions along the test;

(2)

Wave probes for measuring the water free surface elevation and consequently obtaining the period and height of the waves, generating files with the different water levels along the test;

(3)

Wavemaker, which allows for the generation of waves with the required characteristics (Figure 9a), generating files with some technical information about the test, such as paddle velocity and paddle displacement, as well as some other parameters.

4.2. Experimental Testing

Finally, two groups of tests were defined that encompassed the 14 SSTs analyzed in the numerical study. One of the groups admitted that waves were regular, and the other group considered the waves to be irregular.

Table 8. Experimental tests’ Hs and Tp values.

SST	Waves	Time (s)	Numerical		Experimental		SST	Waves	Time (s)	Numerical		Experimental
			Hs * (m)	Tp * (s)	Hs (m)	Tp (s)				Hs (m)	Tp (s)	Hs (m)	Tp (s)
2	Reg.	60	3.00	9.00	0.22	2.44	23	Reg.	60	2.00	10.00	0.15	2.71
	Irre.							Irre.
5	Reg.	60	3.00	10.00	0.22	2.71	26	Reg.	60	2.00	11.00	0.15	2.98
	Irre.							Irre.
8	Reg.	60	3.00	11.00	0.22	2.98	29	Reg.	60	2.00	12.00	0.15	3.25
	Irre.							Irre.
11	Reg.	60	3.00	12.00	0.22	3.25	32	Reg.	60	1.00	6.00	0.07	1.62
	Irre.							Irre.
14	Reg.	60	2.00	7.00	0.15	1.90	35	Reg.	60	1.00	7.00	0.07	1.90
	Irre.							Irre.
17	Reg.	60	2.00	8.00	0.15	2.17	38	Reg.	60	1.00	8.00	0.07	2.17
	Irre.							Irre.
20	Reg.	60	2.00	9.00	0.15	2.44	41	Reg.	60	1.00	9.00	0.07	2.44
	Irre.							Irre.

* Hs and Tp values are only meaningful for irregular waves due to their random behavior. For the regular waves, those values were adjusted for normal amplitude and wave period.

shows the values of Hs and Tp used in the 28 tests.

4.3. Power Matrix

Table 9 and Table 10 display the power matrix obtained after running the tests in the channel and after processing the data for both regular and irregular wave tests. As can be seen, regular waves are showing themselves to be better power productors than irregular waves. These differences are significantly linked to the short duration of the tests in the channel (60 s). The channel, characterized as a closed structure, features a vertical breakwater at the downstream end where the WEC is positioned and another vertical “breakwater” at the upstream end where waves are generated. This configuration leads to significant wave reflection, with reflection rates exceeding 80%, resulting in a “ping-pong” effect of waves within the channel. Therefore, it becomes impossible to carry out long tests under these conditions, as from a given moment the waves no longer behave as expected due to the high reflection rates. That is why the tests had to be short.

However, short-term tests may not be representative of irregular waves, unless they are meticulously defined. Irregular waves have varied behavior, and their analysis must be conducted with long time series so that random effects are disregarded and only the average values are analyzed. Combining this fact with the short duration of the tests carried out, it can be concluded that there was not enough time for the irregular SST to develop; thus, the power value was compromised. A lengthy analysis would be necessary to truly understand the device’s performance under irregular waves.

This effect can be seen in the power output of irregular waves under 3 m Hs and 12 s Tp. According to what was seen previously, the power should decrease while increasing the wave period; however, in this case, the opposite happens, highlighting the variability and somewhat the randomness of short duration irregular waves tests.

Nevertheless, it is worth mentioning that given the same wave height and period, regular waves have twice as much available power than irregular waves. This happens because regular waves follow a single frequency spectrum, while, for irregular waves, there are many frequencies happening simultaneously [27], apart from the fact that Hs represents the average of the upper third of the wave heights. Therefore, the denominator of Equation (4) becomes 32π instead of 64π. Hence, the CWR for irregular waves might be even larger than the ones calculated to regular waves once the available power is less. In the case of this study, it averages 18% for irregular waves and 16% for regular waves.

Table 9. PM obtained from regular waves.

PM Reg. Waves Obtained (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	3	-	-	-	38.6	24.9	28.7	28.4
	2	-	15.2	16.7	17.7	15.6	13.7	11.8
	1	4.1	4.0	4.1	4.0	-	-	-

Table 9. PM obtained from regular waves.

PM Reg. Waves Obtained (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	3	-	-	-	38.6	24.9	28.7	28.4
	2	-	15.2	16.7	17.7	15.6	13.7	11.8
	1	4.1	4.0	4.1	4.0	-	-	-

Table 10. PM obtained from irregular waves.

PM Irreg. Waves Obtained (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	3	-	-	-	14.5	12.7	7.2	14.5
	2	-	9.1	7.7	5.3	4.9	4.5	3.9
	1	3.9	6.3	1.2	1.0	-	-	-

Table 10. PM obtained from irregular waves.

PM Irreg. Waves Obtained (kW)		Tp (s)
		6	7	8	9	10	11	12
Hs (m)	3	-	-	-	14.5	12.7	7.2	14.5
	2	-	9.1	7.7	5.3	4.9	4.5	3.9
	1	3.9	6.3	1.2	1.0	-	-	-

4.4. Validation of Numerical Model

Numerical model validation takes place by comparing the values obtained using the physical setup built and tested with the values obtained previously in the software. Thus, it is possible to test the quality of the computer estimations that are obviously way faster and less time-consuming.

Known by the acronym of RAO, response amplitude operators are usually obtained from models of proposed floater designs tested in a model basin or from running specialized CFD computer programs, or often both. Comparing the RAO behavior for both numerical and experimental models can also be a practical way of validating the results obtained. In the purpose of this study, the parameter is defined as the ratio between the heave motion and the wave amplitude (Equation (7)).

$$\mathrm{R}\mathrm{A}\mathrm{O}={\displaystyle \frac{\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{v}\mathrm{e} \mathrm{M}\mathrm{o}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\mathrm{W}\mathrm{a}\mathrm{v}\mathrm{e} \mathrm{A}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}}}$$

(7)

$$\mathrm{E}\mathrm{r}\mathrm{r}(\mathrm{\%})={\displaystyle \frac{\left|{\mathrm{P}}_{\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}-{\mathrm{P}}_{\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}}\right|}{{\mathrm{P}}_{\mathrm{N}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}}}\times 100$$

(8)

Using the wave probes’ water surface elevation data and the CSV files obtained with the numerical modeling, it was possible to compare both results. Figure 10 displays the comparison between the numerical values (shown in curves) and the experimental values (shown in points).

Furthermore, relative errors between numerical and experimental simulations were also calculated according to Equation (8). With the relative error averaging 21%, together with a good matching of the RAO, it was possible to conclude that the experimental model was a good representation of the numerical model simulated in ANSYS^®. Therefore, both models are matching and are a good representation between computational and physical studies. Lower values for relative errors could be achieved if some issues were considered and addressed, as already discussed. Further research should be able to incorporate those issues in order to get even more precise values.

Table 11. Power values for numerical and experimental models under regular and irregular waves.

Regular Waves					Irregular Waves
Simu.	PN (kW)	PE Scale Down (W)	PE Scale Up (kW)	Err.	Simu.	PN (kW)	PE Scale Down (W)	PE Scale Up (kW)	Err.
2_Reg	38.90	4.11	38.55	1%	2_Irreg	38.90	1.55	14.50	63%
5_Reg	35.56	2.66	24.88	30%	5_Irreg	35.56	1.35	12.67	64%
8_Reg	32.38	3.07	28.73	11%	8_Irreg	32.38	0.77	7.24	78%
11_Reg	28.93	3.03	28.42	2%	11_Irreg	28.93	1.54	14.47	50%
14_Reg	15.21	1.09	10.20	33%	14_Irreg	15.21	0.97	9.07	40%
17_Reg	16.67	1.45	13.62	18%	17_Irreg	16.67	0.82	7.69	54%
20_Reg	17.66	1.81	17.01	4%	20_Irreg	17.66	0.57	5.35	70%
23_Reg	15.64	1.16	10.86	31%	23_Irreg	15.64	0.52	4.88	69%
26_Reg	13.68	0.65	6.11	55%	26_Irreg	13.68	0.48	4.51	67%
29_Reg	11.80	1.24	11.62	2%	29_Irreg	11.80	0.41	3.89	67%
32_Reg	4.11	0.43	4.01	2%	32_Irreg	4.11	0.41	3.89	5%
35_Reg	3.96	0.20	1.86	53%	35_Irreg	3.96	0.67	6.28	58%
38_Reg	4.08	0.21	1.93	53%	38_Irreg	4.08	0.12	1.15	72%
41_Reg	4.04	0.44	4.08	1%	41_Irreg	4.04	0.10	0.95	76%
			Average	21%				Average	60%

displays the predicted power value for numerical (PN) and experimental (PE) simulations. Again, it is possible to see the short duration test effect on the irregular wave errors that are averaging 60%.

5. Model Development

Statistical analyses were performed to better understand the influence of the three parameters (Hs, Tp, and direction) in the dataset that was obtained after running the simulations. The software used was JMP^® (v17.2.0), and the validated data for both breakwaters configurations during the numerical model were used in order to perform the analyses.

The power output and the CWR values were considered as response values, e.g., values that are depending on other parameters, and Hs, Tp, and direction were considered as the factors determining the response. By performing the least square fit, it is not only possible to assess the parameters’ influence (Table 12) but also to find a model that can represent the dataset.

A low p-value indicates that the observed data are unlikely to have occurred if the null hypothesis were true. Therefore, when a p-value is small, it suggests that there is strong evidence to reject the null hypothesis in favor of the alternative hypothesis [28]. Consequently, a low p-value suggests that the variable likely does have an influence on the outcome being studied. Thus, as can be observed in Table 12, the significant wave height showed itself to be the most influencing parameter in the power output and in the CWR overall, followed by the period and finally by the direction.

Table 12. Influence of parameters on the power output.

Factor	Logworth	p Value
Hs (1,3)	22.56	0.00025
Tp (6,12)	16.48	0.00089
Direction (80,115)	5.47	0.00102

Table 12. Influence of parameters on the power output.

Factor	Logworth	p Value
Hs (1,3)	22.56	0.00025
Tp (6,12)	16.48	0.00089
Direction (80,115)	5.47	0.00102

The difference in the logworth value between those two parameters is later explained by the influence under different response. Figure 11 displays the multivariate analysis for the three factors and the four responses. Hs values show good correlation with the power output as discussed before, with Pearson coefficients of 0.89 (Figure 11g) and 0.95 (Figure 11j) for vertical and sloped configurations, respectively. However, T_S values dominate good correlation with CWR values, rating 0.70 (Figure 11b) and 0.83 (Figure 11e) for vertical and sloped, respectively.

Lastly, even though the direction of incident waves seems to have a significantly lower effect on the response of the device, it can be seen in Figure 11 that the peak of the chart is always reached for perpendicular directions (e.g., 90°) (Figure 11c,f,i,l). Lower variations of wave directions are desired not only for the device’s lifetime (less stress in structure) but also for the energy production. The directions of waves near shore are not varying much (in contrast with offshore locations) and can be largely determined in advance owing to the natural phenomena of refraction and reflection [8], thus allowing the structural protection mode to be programmed according to the incident waves.

Figure 12 displays the model adjustments using the data obtained during the numerical simulation versus the data predicted by the models. All of the four models achieved higher values of correlation (higher than 0.80), proving the capacity of using mathematical models as good tools while assessing hinged WEC feasibility for a given wave resource in certain location. However, it is important to mention that these mathematical models were produced with a limited range of values and configurations, a fact that restricts their application only to extrapolations to the model itself. Nevertheless, they do not fail to fulfill the role of proving the possibility of using models to predict efficiencies and productions based on wave resources.

Once again, the relationship between Hs and the device’s power and Tp with efficiency became clear. From the model equations shown together with the straight lines, it is possible to verify that the models related to the CWR do not even present Hs or even the direction in their terms.

Finally, the maximum desirability of the model is reached by varying the factors until the four responses are the highest possible (considering that both CWR and power output are to be maximized). Within the dataset analyzed, such configuration happens to be with 3 m of Hs, 6 s of Tp, and 90° of incident wave direction. Crossing these data with the wave resource distribution, it is possible to quickly assess the potential of a hinged WEC in a particular location. The closer the wave resource to the perfect configuration layout, the higher will be the production and the efficiency of the device.

6. Conclusions

This study investigated the potential of a hinged WEC device integrated into breakwater structures for renewable energy production through numerical and physical modeling. The numerical model was validated with experimental results obtained with a 1:14 scale physical model. The findings demonstrate the viability of the approach adopted, with hinged devices offering a promising solution for possibly supplying significant power to ports and harbors worldwide, especially where a good wave resource exists.

Numerical modeling results indicate that the adopted model accurately forecasts the energy production of hinged devices, aiding in their design and optimization. The analysis shows that vertical breakwaters achieved an average power output of 18 kW, while sloped breakwaters averaged 25 kW. Both configurations performed best under wave conditions of 3 m Hs and 9 s Tp. The most frequent SST led to power outputs of 18 kW for vertical breakwaters and 25 kW for sloped breakwaters, demonstrating their efficiency and potential for wave energy harvesting. Further research should address the limitations of numerical simulation as a better real marine environment representation. The research also revealed that sloped breakwater geometries exhibit superior energy harvesting capabilities compared to vertical structures. This highlights the importance of the analysis of the structure available for deployment.

Furthermore, the statistical study identified key factors influencing power generation in hinged systems. Significant wave height was found to be directly linked to power production, while wave period plays a crucial role in determining the overall efficiency of the system. Additionally, mathematical models were developed for the relations between wave height, wave period, and direction to the power output.

Finally, this study emphasizes the strong role of mathematical modeling in the development of hinged WECs. Not only can these models predict power output, but they also serve as an instrumental tool for design optimization. This paves the way for the implementation of cost reduction and optimization in WECs that are currently still facing such problems, especially high testing costs.