Development and Performance Analysis of a Novel Wave Energy Converter Based on Roll Movement: A Case Study in the BiMEP

Abstract

With the growing concern for environmental issues, progress has been made recently in the promotion of new technologies in the field of renewable energies. This article studies a new wave energy converter that uses the heel generated by the mechanical energy of the waves to transform it into electrical energy by means of a mobile mass, coupled to an electrical generator, which moves from port to starboard and vice versa. The advantage of this converter is that it is capable of incorporating the energy conversion unit inside the converter, as well as allowing the placement of a set of several devices within the same collector, and of modifying the roll period to adapt it to the wave conditions of the installation area. To do this, on one side, two models of wave energy converters were compared by varying the beam to check whether it is better to have a smaller or larger beam by carrying out roll decay tests and simulations for different waves. Moreover, the maximum power available in the moving mass of the power take-off was calculated theoretically for two situations of different transverse metacentric height to check which is more efficient, reaching 2 MW for some waves.

1. Introduction

Nowadays, concerned about the planet’s environmental future, countries around the world are seeking to minimise gas emissions from fossil fuels as much as possible by promoting the use of alternative energies such as renewables. In relation to this, agreements have been reached such as the 2015 Paris Agreement [1] and the Green Deal initiative introduced by the European Commission, which seeks to promote innovative technologies in the field of renewable energies [2]. The European Commission is committed to a range of targets for 2030. These include reducing greenhouse gas emissions by 55%, increasing the share of renewables to 32.5% of electricity consumption, and improving energy efficiency by 21.5%. In addition to the more established sources such as solar, wind, and hydro, this will require the use of other renewables such as wave energy [3].

According to a study carried out by Greenpeace in 2005, there is wave energy available on the Spanish coasts of between 20 GW and 30 GW, with a power ceiling of 84.4 GW and an electricity generation ceiling of 296 TWh per year. This last figure represents 105.7% of the peninsular electricity demand estimated for the year 2050 [4]. Aware of this potential, Spain has taken important steps in this field by creating two sites dedicated to the design and evaluation of wave energy converters (WECs). These sites are the Plataforma Oceánica de las Islas Canarias (PLOCAN) [5,6], where devices such as Wavepiston [7] and Tveter Power [8] have been installed, and the Biscay Maritime Energy Platform (BiMEP) [9], where devices such as MARMOK-A-5 [10] and Penguin [11] have been installed.

Before WECs can be installed at these sites, the design must be validated by extensive numerical simulations and protypes must go through wave flume testing and other experimental trials. Prior to the construction of a scale model, numerical simulations can help us understand the performance of the device and optimise the performance of the converter [12]. Subsequently, experimental tests can show very closely what performance and efficiency the designed WEC might achieve. These tests are usually carried out in a flume, which is a controlled environment that allows the simulation and analysis of wave performance [13]. Designs for which flume testing has been reported include an oscillating water column [14], point absorbers [15], and even a pendulum WEC [16,17], as is the case of the AMOG WEC, where tank testing was carried out with a scale prototype of the proposed device.

Indeed, regarding pendulum WECs, Oceantec patented a device (patent reference WO2008/040822A1 [18]) which, due to the balancing movement of the collector, was able to generate electrical energy thanks to an internal gyroscope, similar to the concept underlying the device proposed herein. A 1:4 scale model of the Oceantec WEC was installed in Cala Murgita for some months in 2008 [19].

Another device capable of generating electrical energy by means of a rolling motion is the aforementioned AMOG WEC. This device generates energy thanks to the fact that it is able to use the rolling to generate a pendulum motion relative to the collector. The generator with the gearbox is positioned at the highest point of the WEC. A demonstration of this WEC was installed in Falmouth, UK in 2019 [20]. A similar case is that of the SEAREV which, like the Penguin, is a floating hull with the difference that electrical energy is generated inside this converter by means of a circular pendular movement. This relative pendular movement means that electrical energy is generated potentially through a direct connection to a generator or through the use of a hydraulic system. In this case, the estimated output is 400 kW. To achieve this, this device has undergone a process of geometric optimisation and maximisation of energy extraction which should be applied to all WECs [21,22]. Another example of a pendulum WEC is the PeWEC which is designed for the climate of the Mediterranean Sea. This device extracts energy by means of a pendulum installed inside the hull. It was tested in the INSEAN wave basin at a scale of 1:12 [23,24].

As can be seen, several WECs that take advantage of roll motion produce energy using a pendulum system. A special case is that of the Penguin, which is able to generate energy thanks to a combination of both pitch and roll movements. Thanks to these types of movement generated by the collector, an eccentric mass rotates inside it, capable of generating up to 1 MW [25].

Another important point to note is that most of the aforementioned designs are able to hold the entire power take-off (PTO) system inside the hull, which is important to avoid corrosion of the moving parts due to exposure to salt water. Nonetheless, none seem to have been able to contain more than one PTO inside their hull [26].

The WEC proposed in this study bases its energy conversion on a sensor body capable of transforming the movement of the waves into a balancing movement. Precisely, this balancing movement generates that, when the device is tilted, the decks are in an inclined plane situation. These inclined planes have rails on which a mass mounted on bearings circulates in an attempt to reduce the friction coefficient. Finally, the linear back-and-forth motion of the mass is transformed into a rotational motion that can be used by a conventional generator by means of a series of pulleys and belts.

Due to the proposed PTO, this WEC, in addition to taking into account the beneficial design features already existing in the field, can include multi-generation by installing different masses along the length of the device. Another advantage is the optimisation of space thanks to the simplification of the power take-off. Finally, this WEC allows to modify the transverse metacentric height (GM_t) by means of a series of interconnected ballast tanks and pumps feeding the tanks. Having ballast tanks at different heights, the centre of gravity of the collector can be modified, in turn modifying the GM_t so that the device maximises its performance.

This study makes a significant contribution to the development of a new wave energy converter, offering a novel solution to optimising energy extraction from wave motion. The main objectives of this study are to optimise the WEC and estimate the potential energy production of new wave energy converter based on roll movement. For these purposes, first, the performance of two innovative wave energy harvesting devices was compared under wave conditions specific to the BiMEP area. This comparison was carried out through decay simulations and tests with the most recurrent wave patterns in this area, the research aims to identify the most suitable model for efficient energy extraction, while advancing the practical applications of wave energy and contributing to global efforts in renewable energy solutions. Subsequently, to estimate the potential energy production, the maximum power available in the mass will be determined using an experimental design.

2. Material and Methods

2.1. Device Description and Situation

The WEC on which this article is based (WO 2024/209063 A1) [27] will be installed at BiMEP (43.37° N, 2.88° W). This WEC is able to extract energy from the waves by means of a barge-like shaped collector that is due to the waves managing to generate a roll movement. Inside the collector there are one or more PTOs arranged as shown in Figure 1.

The PTO is responsible for converting the mechanical energy of the collector (1) into electrical energy due to a roll (2) generated by the waves. For this purpose, a mass (3) is mounted on rails (4) which moves along the beam (B) from port to starboard and vice versa. At the end of the rails there are stoppers (5) that help to brake the mass. For the generation of energy, the mass has a belt or chain (6) incorporated in it, which due to the linear movement of the mass transmitting the movement to this belt or chain, which in turn moves two pulleys or gears (7) which are located on the port and starboard sides. This rotational movement of the pulleys or gears is transmitted to the electric generator (10) through a shaft (8b) which could have a multiplier and which joins the pulley or gear with the electric generator. Subsequently, at the output of the generator, there is an AC/DC converter (11) to be able to either store or send the generated electrical energy to the grid.

2.2. Analysis of Wave Characteristics

Before even starting to design the WEC, it is essential to study the waves in the BiMEP area. In addition, a statistical analysis of the waves in the BiMEP was carried out to make a prediction of the favorable waves in order to optimise the number of trials to be carried out in the test channel. To obtain an accurate sample of the waves, we analysed data obtained directly from a buoy installed in the BiMEP by the IHCantabria. Specifically, data were collected for a full year from 1 May 2020 to 30 April 2021. It should be noted that in order to better analyse the wave data obtained in the peak months, from November to April, both months are included.

Table 1. Wave types in the BiMEP between November 2020 and April 2021 inclusive.

		Peak Period(s)
		2	2.5	3	3.5	4	4.5	5	5.5	6	6.5	7	7.5	8	8.5	9	9.5	10	10.5	11	11.5	12	12.5	Total
Significant wave height (m)	0.5	0.11	0.29	1.09	1.09	0.17	0.20	0.17	0.11	0.03	0.03	0.06	-	-	-	-	-	-	-	-	-	-	-	3.36
	0.75	0.06	0.80	0.92	1.75	0.95	0.83	0.32	0.63	0.32	0.23	0.03	-	-	-	-	-	-	-	-	-	-	-	6.83
	1	-	0.06	0.63	2.30	0.89	1.38	0.89	0.86	0.92	0.92	0.43	0.17	0.26	0.29	0.14	0.03	-	-	-	-	-	-	10.16
	1.25	-	0.03	0.52	0.49	0.34	0.49	0.77	1.21	1.44	1.32	0.46	0.63	0.52	0.55	0.11	0.03	-	0.06	-	-	-	-	8.96
	1.5	-	-	0.17	0.34	0.49	0.49	0.69	1.03	0.83	0.75	0.37	0.80	0.83	1.18	0.60	0.11	0.06	0.03	0.06	0.03	-	-	8.87
	1.75	-	-	-	0.20	0.63	0.83	0.60	0.55	0.92	0.43	0.72	0.77	1.21	1.09	1.09	1.12	0.23	0.60	0.20	0.03	0.03	0.03	11.28
	2	-	-	-	0.03	0.37	0.66	0.75	0.80	0.43	0.95	0.46	0.63	1.18	1.41	1.72	1.81	0.63	0.52	0.37	0.03	-	-	12.74
	2.25	-	-	-	-	-	0.69	1.00	0.43	0.46	0.66	0.49	0.37	0.52	0.86	0.92	1.58	0.98	0.49	0.52	0.06	-	-	10.02
	2.5	-	-	-	-	-	0.37	0.63	0.80	0.26	0.72	0.49	0.43	0.20	0.49	0.57	0.89	0.63	0.40	0.26	0.63	0.55	-	8.32
	2.75	-	-	-	-	-	0.09	0.11	0.29	0.20	0.49	0.26	0.29	0.34	0.40	0.37	0.60	0.60	0.26	0.34	0.11	0.11	-	4.88
	3	-	-	-	-	-	0.11	0.11	0.20	0.40	0.37	0.26	0.06	0.09	0.14	0.17	0.40	0.34	0.06	0.26	0.09	-	-	3.07
	3.25	-	-	-	-	-	0.06	0.03	0.17	0.23	0.43	0.26	0.09	0.06	0.00	0.11	0.37	0.29	-	0.03	0.26	-	-	2.38
	3.5	-	-	-	-	-	-	0.03	0.03	0.03	0.09	0.34	0.06	0.03	0.17	0.06	0.26	0.17	-	-	-	-	-	1.26
	3.75	-	-	-	-	-	-	-	0.06	0.06	0.20	0.32	0.32	0.17	0.11	0.06	0.03	0.06	-	-	-	-	-	1.38
	4	-	-	-	-	-	-	-	-	0.17	0.23	0.29	0.09	0.03	0.09	-	-	0.09	-	-	-	-	-	0.98
	4.25	-	-	-	-	-	-	-	-	0.06	0.03	0.03	0.46	0.20	0.06	0.06	-	0.06	0.03	-	-	-	-	0.98
	4.5	-	-	-	-	-	-	-	-	0.03	0.17	0.55	0.34	0.23	0.03	-	0.06	0.03	0.03	-	-	-	-	1.46
	4.75	-	-	-	-	-	-	-	-	-	-	0.40	0.20	0.14	0.09	0.11	0.06	0.03	0.03	-	-	-	-	1.06
	5	-	-	-	-	-	-	-	-	-	-	0.14	0.03	-	-	0.06	0.03	0.03	0.03	-	-	-	-	0.32
	5.25	-	-	-	-	-	-	-	-	-	-	0.11	0.14	-	-	-	-	0.06	-	-	-	-	-	0.32
	5.5	-	-	-	-	-	-	-	-	-	-	0.11	0.14	-	-	0.03	-	0.03	-	-	-	-	-	0.32
	5.75	-	-	-	-	-	-	-	-	-	-	-	0.03	-	0.06	0.00	-	0.03	-	-	-	-	-	0.11
	6	-	-	-	-	-	-	-	-	-	-	-	-	-	0.06	0.00	-	-	-	-	-	-	-	0.06
	6.25	-	-	-	-	-	-	-	-	-	-	-	-	-	0.03	0.09	0.06	-	-	-	-	-	-	0.17
	6.5	-	-	-	-	-	-	-	-	-	-	-	-	-	-	0.11	0.06	-	-	-	-	-	-	0.17
	6.75	-	-	-	-	-	-	-	-	-	-	-	-	-	-	0.23	0.03	-	-	-	-	-	-	0.26
	7	-	-	-	-	-	-	-	-	-	-	-	-	-	-	0.29	-	-	-	-	-	-	-	0.29
	Total	0.17	1.18	3.33	6.20	3.85	6.20	6.11	7.18	6.77	8.01	6.57	6.06	6.00	7.09	6.92	7.52	4.33	2.53	2.04	1.23	0.69	0.03	100.00

shows the data of the significant height and the peak period of the waves occurred in BiMEP for the months with the highest wave activity.

Therefore, as can be seen in Table 1, the most frequent waves are found between 1.5 m and 2.5 m of significant height. Meanwhile, with respect to the peak period, waves between 5.5 and 9.5 s are the most common ones.

The colour coding used in Table 1 indicates the following frequencies of occurrence:

• ≥1%, green, representing 27.96% of all the waves of these types.
• 0.75% to 0.999%, blue, representing 20.47% of all the waves of these types.
• 0.5% to 0.749%, yellow, representing 18.17% of all the waves of these types.
• 0.25% to 0.499%, orange, representing 22.07% of all the waves of these types.
• <0.25%, red, representing 11.34% of all the waves of these types.

On the other hand, it should be noted that the WEC will use the slopes of the waves to extract energy. Therefore, it is essential to calculate the slopes of each type of wave. To do this, the following mathematical equation is used [28]:

$${\mathsf{\theta }}_0=\mathsf{\pi }·{\mathrm{H}}_{\mathrm{s}}/\mathsf{\lambda }$$

(1)

where H_s is the significant wave height and λ is the wave length defined by the following equation:

$$\mathsf{\lambda }=({\mathrm{gT}}_{\mathrm{p}}^2/2\mathsf{\pi })\mathrm{Tanh}\left(2\mathsf{\pi }\mathrm{h}/\mathsf{\lambda }\right)$$

(2)

where g is gravity, which in this particular case has a value of 9.80 m/s², T_p is the period between wave peaks, and h is the depth. When h/λ ≥ 1/2 the conditions are considered to correspond to deep water, when 1/20 < h/λ < 1/2, the conditions correspond to intermediate waters and in case h/λ ≤ 1/20, it is shallow water [29]. For the zone of interest for this study, the buoy is located at 43.4682° N and 2.8848° W in a depth of 80 m [30]; therefore, Equation (2) is used for periods less than 10 s and for wave peak periods equal or greater than 10 s, Equation (3) is used:

$$\mathsf{\lambda }={\mathrm{gT}}_{\mathrm{p}}^2/2\mathsf{\pi }$$

(3)

Figure 2 shows the wave slope values for each wave typology.

In Figure 2, it can be seen that the most frequent waves have a wide range of slopes from 2° to 18°, or, expressed in percentages, from 3.49% to 32.49%.

2.3. Model Design

Once we know the frequency of the waves in the area where the device is to be placed and the maximum slopes that the waves in the area can offer, the design of the floating device can begin. For the design of the collector, calculations are carried out before starting to shape the device. This requires a high block coefficient to be able to survive extreme wave conditions. On the other hand, it is necessary to take into account the hull roll period (T) which depends directly on the transverse metacentric height (GM_t) of the WEC [31] as follows:

$$T=2\pi K_{xx}^{\prime }/\sqrt{g·GMt}$$

(4)

where K′_xx is the transverse turning radius of the device and g is gravity. The transverse turning radius is defined between the values of 0.33 B to 0.45 B [32] where B is the beam, defined in Figure 1. For the calculations in this study, a K′_xx with a value of 0.4 B has been taken into account.

These two device dimensions were selected after a deep analysis in which several boats of different dimensions of length, beam, prop, block coefficient, etc., were designed using the MAXSURF Naval Architecture Software (2024), testing the stability and simulating different waves incident to the device. However, it was finally decided, on the one hand, to have an extensive length to be able to take advantage of a larger space in the wave front, having more wave power to take advantage of. Second, each device, due to the beam, as mentioned in Equation (4), has a different balance period linked, so we wanted to observe what happened by having two devices with the same transverse metacentric height, but different beam. Third, we wanted to observe how increasing the sleeve affects the decay tests and the effect it would have on the natural frequency and settling time.

Given this, we decided to build two different devices to check whether altering the beam of the device, and therefore the hydrostatic values of the converter, changed its performance, resulting in a greater or lesser amount of energy extraction from the waves. Both hulls were designed using MAXSURF Naval Architecture Software (2024), using the Modeler programs, to shape the hull of the collector and to be able to see the hydrostatic values before the distribution of masses. Specifically, in the Stability analysis module, the loads can be distributed inside the device; while in the Motions module, simulations can be made of the performance of the hull in certain types of waves and decay tests can be performed, the latter being the main reason why this program is used. Figure 3 shows the two devices modelled in MAXSURF Naval Architecture Software (2024) and

Table 2. Hydrostatic characteristics of model 1 and model 2 designed with the MAXSURF Modeler.

	Model 1	Model 2
Length (m)	30	30
Beam (m)	10	16
Draft (m)	2.48	1.53
Depth (m)	6.05	6.05
Displacement (tons)	707	707
Block coefficient	0.949	0.947

shows the hydrostatic characteristics of each of the models.

Table 3. Mass distribution of model 1 obtained with MAXSURF Stability.

	Total Mass (tons)	Centres of Gravity
		Longitudinal (m)	Transverse (m)	Vertical (m)
Lightship	276.4	0	0	3.18
Ballast	360	0	0	0.20
Sliding mass	20	0	−5 < x < 5	0.30
Electrical equipment	50.6	0	0	0.35
Total	707	0	−0.14 < x < 0.14	1.38

and

Table 4. Mass distribution of model 2 obtained with MAXSURF Stability.

	Total Mass (tons)	Centres of Gravity
		Longitudinal (m)	Transverse (m)	Vertical (m)
Lightship	463.4	0	0	1.98
Ballast	173	0	0	0.20
Sliding mass	20	0	−8 < x < 8	0.30
Electrical equipment	50.6	0	0	0.35
Total	707	0	−0.23 < x < 0.23	1.38

, respectively, show the mass distribution of the smaller and larger beam models.

It should be noted that the value of “x”, the centre of gravity of the sliding mass used to generate electrical energy, ranges between −5 and 5 m from port to starboard and vice versa in one model and between −8 and 8 m in the other due to the difference in the beam. This influences the transverse centre of gravity of the collectors and means that the centre of gravity of the device with the smaller beam varies between −0.14 and 0.14 m and that of the device with the larger beam between −0.23 and 0.23 m. This causes the device to heel due to the change of masses as the moving mass is on one side. If the mass were at the end of the beam, this would result in having a list angle of 2.45° with the smaller beam and 0.95° with the larger beam.

Righting arm (GZ) curves were plotted for each of the models to check the stability of the collectors and therefore their survival. In this way, it is possible to determine up to how far the devices can heel without tipping over. Figure 4 and Figure 5 show these GZ curves for heel angles between −30° and 130°, taking into account that the sliding mass will be located at the centreline (x = 0).

As can be seen, the two models have high survivability. According to the MAXSURF Naval Architecture Software (2024) simulations, capsize would occur at heel angles of over 100°.

Having observed the survivability of the two converters, it is possible to simulate roll decay tests in MAXSURF Motions (2024), according to ITTC 7.5-02-07-04.5 recommendations [33]. It is expected that values such as the damping coefficient of the device as well as its natural roll frequency can be obtained from such simulations. Figure 6 and Figure 7 show the roll decays of the two models and

Table 5. Coefficients, frequencies, and settling time extracted from decay tests.

	MAXSURF 10 Metre Beam		MAXSURF 16 Metre Beam
	Mean	Std. Desv.	Mean	Std. Desv.
Max. overflow (%)	79.160	-	79.161	-
Damping coefficient	0.074	-	0.074	-
Natural frequency (Hz)	1.035	0.008	1.578	0.008
Damping frequency (Hz)	1.038	0.008	1.582	0.009
Settling time (s)	52.226	-	34.255	-

the coefficients and frequencies of these models.

The values shown in Table 5 for the data in Figure 6 and Figure 7 can be calculated using the following equation [34]:

$${\mathrm{Percentage}\mathrm{C}}_{\mathrm{Max}}=100\exp \left(-\mathsf{\pi }\mathsf{\xi }/\sqrt{1-{\mathsf{\xi }}^2}\right)$$

(5)

In particular, the damping coefficient, ξ, can be extracted, since it is closely related to the maximum overshoot (C_Max). On the other hand, the natural frequency, ω₀, is calculated from the damping coefficient and the damping frequency, ω_d, which is obtained by measuring the times between peaks or valleys. These three parameters are related in the way shown in Equation (6) [34].

$${\mathsf{\omega }}_0=\sqrt{1-{\mathsf{\xi }}^2}/{\mathsf{\omega }}_{\mathrm{d}}$$

(6)

One of the most important characteristics obtained from roll decay tests is the settling time (t_s). For values of 0 < ξ < 0.9 using the 2% t_s criterion, it is approximately 4 times the time constant of the system (T₀). Equation (7) shows how this constant is related to the natural frequency and the damping coefficient [34].

$${\mathrm{t}}_{\mathrm{s}}=4{\mathrm{T}}_0=4/{\mathsf{\xi }\mathsf{\omega }}_0$$

(7)

2.4. Wave Tank Tests

Once all these data were available and shown to be compatible with the devices behaving in an ideal way, prototypes were built on a scale of 1:53 in polylactic acid using a 3D printer. Figure 8 shows model 1 at that 1:53 scale in flume in a wave test.

The completed hulls were brought to the wave test channel of the Bilbao School of Engineering at the University of the Basque Country, Bilbao, Spain. There, simulations were carried out under different wave conditions (see below). Heel angles were measured based on changes in the position of marks on the bow of each model, using a GoPro Hero 10 that can record video at 240 frames per second. The recordings obtained were run through Tracker, a point tracking program that, given a measurement and an axis of coordinates, shows the displacement of a selected point.

To test which device would best perform the function of extracting energy from the waves, they were subjected to six of the most frequent BiMEP wave types characterised by peak periods and significant heights respectively of the following:

(1)

4.50 s and 1.00 m;

(2)

5.00 s and 2.25 m;

(3)

6.00 s and 1.25 m;

(4)

8.00 s and 2.00 m;

(5)

9.50 s and 2.25 m;

(6)

9.50 s and 1.75 m.

For this test, the devices were anchored in the channel by four chains, each attached to the catcher hooks (one at the bow and one at the stern at the height of the waterline) at one end and to a magnet on the bottom of the channel at the other. Two mooring lines were attached to each hook, to prevent significant drift in any direction. It should be underlined that despite the hull being anchored, it was given considerable margin to move freely to allow the extraction of energy from the waves.

Notably, the two models were tested with the aforementioned waves without any moving masses inside them, and hence, both models had a total weight of 4 kg. These experimental tests were carried out to observe which of the two models could achieve higher heel heights in the most frequent waves in the BiMEP area. By having higher heels, it is estimated that a higher maximum available power could be achieved and, therefore, the extraction of energy from the waves could be carried out better and have a higher performance.

2.5. Maximum Theorical Power Calculation

Once it has been determined which device performs best for the waves in the BiMEP area, the theoretical maximum power available (MPA) must be calculated. This maximum power available is the maximum power that the moving mass has, and therefore, the maximum power available before it is transformed into electrical energy.

Due to the complexity of performing a large number of experimental tests and the cost of each of them, it was decided to perform experimental tests only on the values that limit the wave spectrum in the BiMEP. This limitation was carried out with 4 waves shown below, characterised by their peak period and significant height respectively:

(1)

3.5 s and 0.5 m;

(2)

9.5 s and 0.5 m;

(3)

3.5 s and 5 m;

(4)

9.5 s and 5 m.

These 4 waves are added to the 6 previously tested with the intention of having a representative value of the wave spectrum and, based on that, to carry out an experimental design (DOE). This experimental design was proposed to investigate all possible combinations of the levels of the experimental factors (significant wave height (Hs) and peak period (Tp)) that influence the result (MPA) across the response surface. This method not only provides us with the relationship between independent variables (factors) and a dependent response (output), but also optimises the response and simplifies the modelling using second-order (quadratic) polynomials.

In order to perform the experimental design properly, normalities were obtained for both the significant height and the peak period of the simulated waves in the wave trough. The significant height was a normal distribution because in the Kolmogorov–Smirnov test, the p value was 0.718. In addition, out of 53 samples, the mean was 2.703 m and the standard deviation was 1.180. On the other hand, in the case of the peak period, it was a normal distribution because the Kolmogorov–Smirnov test had a p value of 0.644. Likewise, a mean of 5.406 s and a standard deviation of 1.387 s were obtained for 53 samples.

In order to perform the DOE of two transverse metacentric heights, estimate the extractable potential energy, and facilitate data collection, LabVIEW software (2020) was used to simulate the performance of the device with 20 tonnes of mobile mass on it.

To calculate it, the formula for the balance motion of the device, which is determined by Equation (8), must first be entered [35]:

$${\mathrm{I}}_{\mathrm{xx}}^{\prime }\ddot{\mathsf{\varphi }}+\mathrm{b}\dot{\mathsf{\varphi }}+\mathrm{g}\Delta \mathrm{GZ}\left(\mathsf{\varphi }\right)={\mathrm{M}}_{\mathrm{wave}}$$

(8)

where Φ is the vessels’ roll, and therefore, $\ddot{\mathsf{\varphi }}$ and $\dot{\mathsf{\varphi }}$ are the vessel’s angular acceleration and velocity respectively, I′_xx is the longitudinal moment of inertia of the system, Δ is the displacement of the device, and GZ is the vessel’s righting lever. I′_xx can also be expressed as [32]:

$${\mathrm{I}}_{\mathrm{xx}}^{\prime }={\left({\mathrm{K}}_{\mathrm{xx}}^{\prime }\right)}^2·\left(\Delta -\mathrm{m}\right)+\mathrm{c}(\mathrm{m}({\mathrm{x}}^2+{\mathrm{z}\prime }^2)\left(1-\mathrm{m}/\Delta \right)$$

(9)

m is the mass of the sliding mass, x the horizontal position of the mass, z′ the distance from the centre of gravity without adding the mass to its plane of motion, and c the effect of the added mass.

Considering the terms in Equation (8), b $\dot{\mathsf{\varphi }}$ is the vessel’s damping moment and gΔGZ(Φ)is its righting moment. The vessel’s viscous coefficient, b, can be defined as follows [32]:

$$\mathrm{b}={\mathsf{\omega }}_0·2\mathsf{\xi }·{\mathrm{I}}_{\mathrm{xx}}^{\prime }$$

(10)

and the natural frequency can be written as [31]

$${\mathsf{\omega }}_0=\sqrt{\mathrm{g}\Delta ·{\mathrm{GM}}_{\mathrm{t}}/{\mathrm{I}}_{\mathrm{xx}}^{\prime }}$$

(11)

On the other hand, the vessel’s righting moment can be expressed as follows [32]:

$$\mathrm{GZ}={\mathrm{GZ}}_0-\left(\mathrm{m}/\Delta \right)·\mathrm{xcos}\mathsf{\varphi }$$

(12)

The first term of the equation is the original righting lever, GZ₀, without taking into account the sliding mass. In our case, however, as the mass moves in the plane of the vessel, we also have to consider its transverse motion by adding the moving mass effect.

Finally, we have the momentum of the wave, which can be calculated as follows [32]:

$${\mathrm{M}}_{\mathrm{wave}}={\mathrm{c}\mathsf{\theta }}_0\sin {\mathsf{\omega }}_{\mathrm{e}}\mathrm{t}$$

(13)

where c is the restoring moment coefficient and, ω_e is the wave encounter frequency.

It should be noted that if you would have a different wave incidence, the value of the maximum wave slope would differ from what you have on the basis of the following formula [32]:

$${\mathrm{M}}_{\mathrm{wave}}={\mathrm{c}\mathsf{\theta }}_0^{\prime }\sin {\mathsf{\omega }}_{\mathrm{e}}\mathrm{t}$$

(14)

Being θ₀′, the maximum angle for an incident wave direction other than 90°. This parameter is related to the maximum slope of the wave by the following formula [32]:

$${\mathsf{\theta }}_0={\mathsf{\theta }}_0^{\prime }\sin \mathsf{\varsigma }$$

(15)

where ς is the angle of incidence of the wave with respect to the WEC.

On the other hand, to calculate the theoretical maximum energy of the mass, the velocity of the mass must be calculated. To do this, the instantaneous acceleration of the mass is calculated according to Newton’s laws of motion:

$$\sum \mathrm{F}=\mathrm{m}·\mathrm{a}=\mathrm{m}·\mathrm{g}·\sin \mathsf{\alpha }-\mu ·\mathrm{m}·\mathrm{g}\cos \mathsf{\alpha }$$

(16)

where ΣF is the sum of the forces to which the moving mass is subjected, a is the acceleration of the moving mass, m is the mass in kilograms of the moving mass, g is the gravitational constant at the earth’s surface, which for the location where the tests were carried out is 9.80 m/s² [36], μ is the coefficient of friction of the moving mass with the rails, which in this case is 0.059, and α is the slope of the inclined plane to which the moving mass is subjected, which in this case is the heel angle of which the device.

Once the acceleration of the moving mass and its force is obtained, it can be used to obtain the instantaneous velocity, v. The maximum available power of the moving mass, P, is then calculated as follows:

$$\mathrm{P}=\mathrm{F}·\mathrm{v}$$

(17)

In addition, the displacement of the moving mass can also be calculated from the velocity. This is vitally important as it is a variable in Equation (9), so it will have to be fed back as the position of the moving mass on the inclined plane affects the heel of the device.

All of this, as mentioned above, was then entered into LabVIEW to determine the maximum speeds of the mass as well as its force. Then, the aforementioned DOE was carried out predicting the rest of the values using Statgraphics to obtain the maximum power values for 2 different GM_t, one of 2.64 m, which corresponds to a value of 4.94 s of balance period, and another of 1.32 m, 6.98 s of roll period, in order to observe whether the change in the GM_t, all other conditions being identical, can alter the energy production.

3. Results

For comparison of the performance of the two models, both the 10 m beam (Figure 9) and the 16 m beam (Figure 10), they were subjected to experimental tests. These experimental tests consisted of simulating six most frequent wave conditions of the BiMEP. Subsequently, which of the two devices had the greater heel and thus the greater possibility of higher energy production was tested.

From the results shown in Figure 9 and Figure 10, it can be seen that the 10 m beam model has a greater heel in the six wave conditions studied and is therefore estimated to perform better in the BiMEP wave conditions. Therefore, we proceeded to carry out the tests with the mobile mass only for the 10 m beam arrangement. It is worth noting the particular case of wave 2, which presents a significant amplification, generating heeling angles of 35° for the 10 m beam device. Therefore, the experimental design of the maximum available power will be carried out only for the 10 m beam model.

In this experimental design, different transverse metacentric heights will be tested for the 10 m beam model. Figure 11 shows a transverse metacentric height of 2.64 m.

As can be seen, the maximum available power can reach more than 2 MW. However, taking into account Table 1, which corresponds to the most frequent waves in the BiMEP, the maximum available power in the mobile mass can reach 800 kW.

As for the results of the maximum available power for a GM_t of 1.32 m, the result is shown in Figure 12.

In Figure 12 it can be seen that the maximum available power is lower than in the case of 2.64 m GM_t. In this case, maximum available powers of 900 kW can be reached, and a maximum available power of 200 kW can be reached in the range of most frequent waves given in the BiMEP area following Table 1.

4. Conclusions

Based on the analysis of the GZ and roll decay curves, it can be concluded that devices with this design will self-right effectively and have high survivability in wave conditions that could cause significant heel, returning to equilibrium.

The analysis of heeling in devices without a moving mass inside shows that the device with the smaller beam will experience higher amplifications, meaning it will heel more than the device with the larger beam under identical wave conditions. This is because the larger beam device stabilises faster, which is less desirable for its intended purpose. Therefore, for the calculation of maximum available mass power with two different GM_t values, only the model with the smaller beam was used.

Simulations on the smaller beam model for different wave types at the BiMEP revealed the maximum power available for the PTO moving mass. It was observed that for waves with the same period, a greater significant wave height results in a steeper slope, causing the mass to move in the same or even better manner than with lower waves, which is beneficial for energy generation.

Considering the results in Figure 11 and Figure 12, the maximum available power is higher with a ballast rolling period of 4.94 s compared to 6.98 s. This may be due to several factors, such as higher wave slopes for a 5 s period compared to a 7 s peak wave period, as shown in Figure 4. To this, the effect of the list angle or the heel generated by having the mass of 20 tonnes on one side must be added, and it must also be added that, due to the friction coefficient, a minimum slope is required for the mass to start to move. Consequently, Figure 11 and Figure 12 show that the 0 kW line represents the point where the mass starts moving, with few 7 s peak period values reaching this threshold.

Although the results for a ballast with a 7 s balance period are not ideal, the maximum available power is still satisfactory, with more than 2 MW generated for BiMEP wave conditions. For the most frequent wave type, the available power reaches 800 kW. Additionally, as a ship-like WEC, small ballast tanks could be added to adjust the centre of gravity and GM_t. This allows the device to be adapted to different wave conditions by varying ballast heights, optimising performance for different locations and seasons. For instance, BiMEP wave conditions vary between summer and winter, and further studies could explore this adaptation.

In addition to the ability to adjust the centre of gravity for optimised energy extraction, this new device shares the advantages of conventional WECs, such as high energy output. Unlike many others, it houses the PTO inside, protecting it from seawater and reducing operational and maintenance costs. A multiple generator system could further increase energy production with a single WEC, avoiding the need for multiple conventional devices, which would increase environmental impact, visual pollution, and costs. Additionally, the device boasts excellent survivability and is able to withstand heel angles up to 100° and return to equilibrium without compromising structural integrity.