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Abstract

The behaviour of arrays of 12 heaving point absorbers in concentric arrangements is numerically assessed in a frequency domain model. The floaters are attached to a central cylindrical bottom-mounted structure. Each point absorber is restricted to the heave mode and is assumed to have its own linear power take-off system consisting of an external damping coefficient enabling power extraction and a supplementary mass coefficient tuning the point absorber to the incoming waves. The external damping and supplementary mass coefficients are optimized to maximize the power absorption by each floater in the array, with a restriction on the total control force that can be applied on the floaters. Various concentric arrangements with different radii and number of concentric circles are analysed to determine the most efficient among them. Moreover, the influence of the presence of a central bottom-mounted pillar and the effect of change in its dimension and shape on the power absorption are also studied.

Keywords

Concentric arrays force restriction optimal control point absorbers wave energy

Introduction

Wave energy conversion technologies can be classified in many different ways based on the working principle and location. Recent reviews on the existing wave energy converter (WEC) technologies can be found in Falcão (2010) and Guedes Soares et al. (2012, 2014). If the horizontal dimension of the device is much smaller than the typical wavelength of the ocean waves, the WEC is called a point absorber; otherwise, it is called a terminator or attenuator depending on if it is aligned normally or parallel to the prevailing wave direction. The point absorbers – floating or submerged – convert the vertical motion of ocean waves into linear or rotational motion for driving electrical generators by means of a power take-off (PTO) system. Point absorbers are very attractive because it can harness energy from a wavefront much larger than its own horizontal width (Budal and Falnes, 1975). However, the point absorbers must operate in arrays to produce considerable amounts of power from ocean waves. For multiple bodies, optimal control parameters are not only dependant on the incoming waves but also on the position and the behaviour of other floaters. Furthermore, optimal control theory has shown that multiple converters can exploit constructive interference to enhance power absorption by the WEC.

Several theoretical models have been developed by researchers in order to deal with waves and interacting bodies. Budal (1977) and Falnes (1980) adopted the point absorber approximation theory to derive the expressions for the maximum wave power which can be extracted by an array of point absorbers. The approximation relies on the assumption that the bodies are small compared to the incident wavelengths so that the wave scattering within the array can be neglected while calculating the interactions. A theory accounting more accurately for the wave-body interactions is the plane-wave approximation which is based on the assumption that bodies are widely spaced relative to the incident wavelengths, so that the radiated and circular scattered waves can be locally approximated by plane waves (McIver and Evans, 1984).

In this study, the performance of 12 heaving point absorbers arranged in concentric arrays is evaluated. The floaters are attached to a central cylindrical bottom-mounted platform by means of an arm. The concentric arrays differ in their radii and the number of circles into which the floaters are arranged. The effects of the dimension and the shape of the central pillar are also examined.

Concentric arrays

In order to analyse the performance of concentric array of point absorbers, the study is carried out for three concentric arrangements c₁, c₂ and c₃, with floaters placed at different radii. The configuration c₁ has an outer radius of 18 m and an inner radius of 9 m, the configuration c₂ has an outer radius of 24 m and an inner radius of 12 m and the configuration c₃ has an outer radius of 30 m and inner radius of 15 m. The configuration c₂ is shown in Figure 1, for which most of the calculations are carried out in this article.

Figure 1.

Concentric array of floaters for configuration c₂.

The floaters in each configuration are placed in the same position relative to the central cylindrical platform, as shown in Figure 1. The floaters have been numbered 1–12 in all the arrays, which makes it easier to compare the performance of each floater in different arrangements. The floaters numbered 1–6 are placed on the outer circle, and the other six numbered 7–12 are placed on the inner circle. The floaters placed in the inner and outer radii are 60° apart from each other. Moreover, the floaters in the inner circle are placed 30° from the corresponding floaters on the outer circle. The results of these concentric arrays are compared with linear and grid arrangements as in Sinha et al. (2015) and Backer et al. (2009), respectively. From the previous study (Sinha et al., 2015), a cone-cylinder floater shape, with an apex angle of 90° and a diameter of 5 m extended by a cylindrical part of 0.5 m, is chosen to be used in all the concentric arrays.

The floaters are attached, by means of an arm, to a central cylindrical bottom-mounted platform as shown in Figure 2. This platform, with a radius of 5 m and height of 50 m, is mounted on the seabed. Furthermore, the radius of this cylindrical platform is varied to study its effect on the performance and power absorption by each floater. Moreover, the shape of the platform is also changed. The incoming waves propagate in the direction of positive x-axis as shown next to Figure 2. In Figure 1, the floaters numbered 1, 2, 7, 12 and 6 are the ones facing the incoming waves and will be referred like that in the further sections. The floaters numbered 3, 9, 10, 5 and 4 will be referred to as placed on the right side of the configuration.

Figure 2.

Bottom-mounted cylindrical platform connected with point absorbers.

Numerical modelling

Frequency domain modelling

The equation of motion of $N$ multiple bodies, oscillating in heave in a regular wave with angular frequency $ω$ can be expressed as

- ω^{2} (M + M_{a} + M_{s u p}) \vec{Z} + j ω (B_{e x t} + B_{h y d}) \vec{Z} + K \vec{Z} = {\vec{F}}_{e x}

(1)

where $\vec{Z}$ is the complex amplitude of the floater positions, $M$ is the mass matrix of the floaters and $K$ is the matrix of hydrostatic restoring coefficients or stiffness matrix. The added mass and hydrodynamic damping coefficients are denoted by $M_{a}$ and $B_{h y d}$ , respectively. The added mass and hydrodynamic damping coefficients are both symmetric $N \times N$ matrices with the hydrodynamic interaction coefficients on the non-diagonal positions. The vector ${\vec{F}}_{e x}$ contains the complex amplitudes of the heave exciting forces, and the hydrodynamic parameters $M_{a}$ , $B_{h y d}$ and ${\vec{F}}_{e x}$ are calculated based on the boundary element method (BEM) using WAMIT. Since the natural frequencies of the floaters are generally smaller than the incident wave frequencies, the floaters are often tuned towards the characteristics of the incident waves to augment power extraction. This can be achieved by latching techniques (Budal and Falnes, 1980) by introducing a supplementary spring term (Ricci et al., 2007) or a supplementary mass term (Vantorre et al., 2004). In this approach, a tuning force proportional to the acceleration has been implemented by means of a supplementary mass matrix, $M_{s u p}$ , which is a diagonal matrix containing the supplementary mass coefficients of each floater on the diagonal positions and is given by

M_{s u p} = m_{s u p}^{j} I

(2)

where $m_{s u p}^{j}$ is the supplementary mass coefficient for the $j th$ floater, and I is the $N \times N$ identity matrix. A linear external damping matrix $B_{e x t}$ has been applied to enable power extraction. It is also a diagonal matrix containing the external damping coefficients of each floater on the diagonal positions and is given by

B_{e x t} = b_{e x t}^{(j)} I

(3)

where $b_{e x t}^{(j)}$ is the external damping coefficient for the $j th$ floater, and I is the $N \times N$ identity matrix.

The average power absorption in regular waves is given by

P_{a b s} = \frac{1}{2} b_{e x t} ω^{2} z_{A}^{2}

(4)

where $z_{A}$ is the amplitude of the floater heave. In the case of irregular waves, the power absorption is obtained using the linear superposition of the floater responses given by

P_{a b s} = \sum_{i = 1}^{n_{f}} ω_{i}^{2} Z_{i}^{*} B_{e x t} Z_{i}

(5)

where $n_{f}$ is the number of frequencies, and $Z_{i}^{*}$ is the complex conjugate transpose of the heave amplitude $Z_{i}$ . The numerical computations for the determination of the hydrodynamic coefficients are performed for 40 frequencies, ranging between 0.035 and 0.3 Hz with $Δ f = 0.0068 Hz$ .

The two control parameters $b_{e x t}$ and $m_{s u p}$ are optimized in order to maximize the absorbed power. This optimization is carried out using sequential quadratic programming (SQP) method, and a force restriction is introduced in order to avoid unrealistic solutions, such as extremely large floater motions. This restriction is described in the next section. The optimization is carried out individually for each of the floater in the array. All the floaters are assumed to be equipped with their own identical PTO mechanism.

Force restriction

In some cases, the optimal control parameters for the maximum power absorption result in very large control forces. The tuning force, in particular, might become very large and can even be a multiple of the damping force. If this tuning force is to be delivered by the PTO, it might result in a very uneconomic design of the PTO system. For this reason, it is important to study the response of each floater in the case when the total control force is restricted. If the force spectrum is expressed as

S_{F, i}^{j} = \frac{{F_{A, i}^{j}}^{2}}{2 Δ ω}

(6)

and the significant amplitude of the force is defined as

F_{A, s i g n}^{j} = 2 \sqrt{\int_{0}^{\infty} S_{F, i}^{j} (ω) d ω}

(7)

Therefore, the significant amplitude of the damping and tuning forces for the $j th$ floater is given by

F_{b_{e x t}, A, s i g n}^{j} = 2 \sqrt{\int_{0}^{\infty} S_{F_{b_{e x t}}, A}^{j} (ω) d ω}

(8)

and

F_{m_{\sup}, A, s i g n}^{j} = 2 \sqrt{\int_{0}^{\infty} S_{F_{m_{\sup}}, A}^{j} (ω) d ω}

(9)

respectively.

Hence, the significant amplitude of the total force is given by

F_{t o t, A, s i g n}^{j} = \sqrt{{F_{b_{e x t}, A, s i g n}^{j}}^{2} + {F_{m_{s u p}, A, s i g n}^{j}}^{2}}

(10)

For the scope of this article, the total force which can be exerted on each floater is limited to a maximum significant value of 200 kN.

Wave climate

The calculations are performed for eight reference sea states of floater measurements as shown in Table 1. The majority of the calculations will be carried out for the sea state SS-5. The generated spectrum is based on the parameterized JONSWAP spectrum (Guedes Soares and Caires, 1995).

Table 1.

Reference sea states.

Sea state	$H_{s} (m)$	$T_{p} (s)$
SS-1	0.25	5.24
SS-2	0.75	5.45
SS-3	1.25	5.98
SS-4	1.75	6.59
SS-5	2.25	7.22
SS-6	2.75	7.78
SS-7	3.25	8.29
SS-8	3.75	8.85

Results and discussion

The results for the average power extraction $P_{e x t}$ , external damping coefficients $b_{e x t}^{j}$ and supplementary mass coefficients $m_{s u p}^{j}$ for each floater arranged in concentric arrays c₁, c₂ and c₃ are presented in this section. All the results are calculated for sea state SS-5. The various concentric arrangements are evaluated and their results are compared with those of other array configurations. Furthermore, the effect of the dimensions of the central bottom-mounted platform on the performance of the floaters is also evaluated.

Concentric arrays

The concentric arrangements of c₁, c₂ and c₃ are analysed and the results are presented for each floater. It can be seen from Figure 3(a) that power absorbed by each floater varies for different arrangements. The floaters facing the incoming waves extract more power, performing the best for configuration c₁ and the worst for configuration c₃, that is, these floaters absorb less power $P_{e x t}$ as the radius of the concentric circles increases. However, for the floaters on the right side of the array, the power absorption does not vary much with change in radius, with configuration c₃ performing the best and closely followed by arrays c₂ and c₁.

Figure 3.

(a) Power extracted by each floater, (b) external damping coefficients for each floater, (c) supplementary mass coefficients for each floater and (d) absorption efficiency of each floater.

Figure 3(b) and (c) give the external damping and supplementary mass coefficients for each floater for configurations c₁, c₂ and c₃. The control parameters do not vary much among the floaters, with configuration c₃ having the most uniform values. The uniformity of control parameters among the floaters in an array is a good sign as it will be easier for the PTO mechanism to control the floater movements and in turn maximize the power absorption. The external damping coefficients $b_{e x t}^{j}$ are the highest for the floaters facing the incoming waves and their corresponding supplementary mass coefficients $m_{s u p}^{j}$ are the lowest. Both these coefficients have been optimized to maximize the power absorption by each floater in the array, with constraints on the total force that can be exerted on each floater by the PTO system.

In linear and grid arrangements, the external damping and the supplementary mass coefficients vary a lot for different floaters (see Backer et al., 2009), unlike the uniform values for these concentric arrays of floaters. Therefore, it will be easier to control the floaters in concentric array and maximize the power absorption.

Figure 3(d) shows the efficiency of each floater for different concentric arrangements. As mentioned earlier, the floaters facing the incoming waves extract the most power and this value is the maximum for configuration c₁. It is observed that these floaters are the most efficient. The floaters on the right side of the arrays are the most efficient for configuration c₃. This graph helps us to understand which floater positions are the most efficient for different concentric arrangements and will be useful for effective floater positioning while designing the WEC.

In Table 2, the total power extracted by all the floaters in various concentric arrays is compared with that of linear and grid arrays as presented in Sinha et al. (2015) for SS-5. The results for the other sea states in the wave climate are presented later in this section. The concentric arrangement c₂ performs the best, closely followed by c₁ and then c₃.

Table 2.

Total power absorption for different concentric arrangements.

Power	c₁	c₂	c₃	Linear	Grid
$P_{e x t} (kW)$	501.21	504.63	491.75	429.3	464.22

It is observed that with increase in the radius of the concentric configuration from c₁ to c₂, the power absorption increases, however, increasing the radius of the configuration again from c₂ to c₃, there is a decrease in power absorption. The configuration c₃ has the least value among the three arrangements. Hence, the choice of radius for arranging the floaters into concentric circles is important while designing the WEC. The concentric array of floaters in configuration c₂ extracts around 15% and 8% more power than linear and grid arrays, respectively, which is a notable improvement.

Table 3 shows the difference in percentage in the total amount of power extracted by 12 heaving point absorbers arranged in concentric arrays as compared to that in linear and grid arrays, for all the sea states in the wave climate. It can be seen that the concentric array c₂ performs significantly better than the linear and grid arrays for all the sea states, absorbing on an average more than 15% and 7% power, respectively. Hence, we can say that the concentric array of floaters is much more efficient in terms of power extraction. Moreover, for a concentric array, it is easier to control the movement of the floaters, in addition to better accessibility and easier maintainability of WEC device.

Table 3.

Difference in total power absorption in linear and grid arrangements compared to concentric array c₂.

Sea state	Linear (%)	Grid (%)
SS-1	20.80	3.39
SS-2	20.94	4.58
SS-3	20.72	6.84
SS-4	16.73	7.69
SS-5	14.91	8.01
SS-6	13.59	7.83
SS-7	12.49	7.44
SS-8	11.43	7.03

Effect of platform dimensions

In this section, the radius of the cylindrical platform is changed to analyse its effect on the performance of the floaters for configuration c₂. The calculations are performed for three values of platform radius – r = 3, 5 and 7 m.

It can be seen from Figure 4 that the floaters facing the incoming waves extract higher power $P_{e x t}$ when the platform radius is the largest, whereas the floaters on the right side of the configuration absorb higher power when the platform radius is the lowest.

Figure 4.

Power extracted by each floater with change in the pillar dimensions.

Table 4 shows the total power absorption with change in platform radius, for SS-5. The total power absorption increases with increase in the radius of the platform. The platform with radius 7 m extracts around 5% more power as compared to the one with a radius of 3 m. However, the difference is not large.

Table 4.

Total power absorption for different platform dimensions and shapes.

Power	r = 3 m	r = 5 m	r = 7 m	Square platform
$P_{e x t} (kW)$	491.32	504.63	517.65	511.49

The calculations are also carried out for a bottom-mounted square platform of dimensions $10 m \times 10 m$ , and it is seen that the total power absorption is close to the one obtained with a cylindrical platform. Therefore, the difference is not much even with the change in shape. Hence, in order to find the best possible dimension and shape of the platform, both structural and cost-effective analyses must be carried out together with this study on the hydrodynamic performance of the floaters.

Effect of change in the number of concentric circles

The effect of change in the number of concentric circles into which the 12 heaving point absorbers are arranged is presented in this section. The new array arrangement is denoted by c₄ with floaters placed into three concentric circles as shown in Figure 5. The floaters in the outer circle are placed at a radius of 24 m as in the configurations before; however, there are only four floaters in each of the circle. The floaters in the second circle are placed at a radius of 17 m and at an angle of 30° from the floaters on the outer circle. Finally, the floaters on the inner circle are placed at a radius of 10 m and at an angle of 60° from the floaters on the outer circle. The results for the new concentric configuration c₄ are compared with that of c₂.

Figure 5.

Concentric array of floaters arranged into three concentric circles.

Figure 6 shows the power extracted by each floater in both the configurations c₂ and c₄ for sea state SS-5. There is a variation in the performance of each floater as they are placed in different positions relative to the central platform. For configuration c₄, the floaters at positions 5, 8, 9 and 12 absorb the most power with the floater numbered 12, being the closest to the platform, performing the best. Moreover, the external damping coefficients and the supplementary mass coefficients for the floaters are more uniform in the case of configuration c₂ as compared to c₄.

Figure 6.

Power extracted by each floater for two different concentric arrays.

Even though there is a difference in the performance of each floater, the total power absorption is almost the same (around 504 kW) in both the arrays. Thus, changing the array arrangement from two to three concentric circles, that is, from c₂ to c₄ is not beneficial and will lead to higher costs. The study was also extended to four-circle concentric arrangement, and it is observed that there is a slight increase in the total extracted power. However, the increase in the power values is not big enough to make it economically viable. Moreover, a concentric arrangement with more circles would be more difficult to maintain and more complex to control.

Conclusion

The behaviour of arrays of 12 heaving point absorbers arranged in various concentric arrays is analysed in unidirectional irregular waves. The concentric arrays vary in terms of the outer and inner radii where the floaters are arranged. Six floaters are placed on the outer circle and six of them are placed on the inner circle, equidistant from each other – 60° apart. The external damping and supplementary mass coefficients have been optimized to maximize the power absorption by each floater in the array. The concentric array with an outer radius of 24 m and an inner radius of 12 m extracts the most power, and the total power absorption initially increases with increase in the array radius and then decreases. Hence, it is important to consider the radius of concentric array while designing the WEC for optimal performance.

The floaters facing the incoming waves absorb the most power, as expected, but the results are the best for the array c₁, the one with the least radius. The study carried out is important to know the efficiency of each position of the floater in the array, relative to the fixed platform. The effect of changing the platform radius is analysed, and it is seen that the platform with the largest radius, r = 7 m, absorbs 5% more power as compared to the one with a radius of 3 m. Therefore, a structural and cost benefit analysis, together with the results obtained in this article, must be performed to determine the best possible dimension and shape of the bottom-mounted platform for optimal power extraction.

Furthermore, the floaters are arranged into concentric arrangements with more circles and from the results, it is observed that the increase in power absorption is not high enough to overcome the increase in the costs and the loads to build and maintain a concentric arrangement with more than two circles. This preliminary hydrodynamic study performed in this article will be useful to design an efficient WEC with concentric array of floaters.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study has been supported by the Portuguese Foundation for Science and Technology (FCT) through its annual funding to the Centre for Marine Technology and Ocean Engineering (CENTEC).

Author biographies

Ashank Sinha received B. Tech (Hons.) degree in mechanical engineering from Indian Institute of Technology Bhubaneswar, India in 2012 and M.Sc. degree in Naval Architecture and Marine Engineering from Instituto Superior Técnico, University of Lisbon in 2015. He is currently an assistant researcher with Centre for Marine Technology and Ocean Engineering (CENTEC), IST, Lisbon.

Debabrata Karmakar received his Ph.D. degree in Ocean Engineering and Naval Architecture from Indian Institute of Technology Kharagpur, India in 2009. He is currently working as an assistant professor in the Department of Applied Mechanics and Hydraulics at National Institute of Technology Suratkhal, India.

Prof. C. Guedes Soares received the M.S. and Ocean Engineer degrees from the Massachusetts Institute of Technology, Cambridge, in 1976, the Ph.D. degree from the Norwegian Institute of Technology, Trondheim, Norway in 1984, and the Doctor of Science degree from the Technical University of Lisbon in 1991. He is a Professor of Naval Architecture and Marine Engineering and the President of the Centre for Marine Technology and Ocean Engineering (CENTEC), which is a research center of the University of Lisbon that is recognized and funded by the Portuguese Foundation for Science and Technology.

References

Backer

Vantorre

Beels

. (2009) Performance of closely spaced point absorbers with constrained floater motion. In: Proceedings of the 8th European wave and tidal energy conference, Uppsala, 7–10 September, pp. 806–817.

Budal

(1977) Theory for absorption of wave power by a system of interacting bodies. Journal of Ship Research 21: 248–253.

Budal

Falnes

(1975) A resonant point absorber of ocean waves. Nature 256: 478–479.

Budal

Falnes

(1980) Interacting point absorbers with controlled motion. In: Count

(ed.) Power from Sea Waves. London: Academic Press, pp. 129–142.

Falcão

(2010) Wave energy utilization: A review of the technologies. Renewable and Sustainable Energy Reviews 14: 899–918.

Falnes

(1980) Radiation impedance matrix and optimum power absorption for interacting oscillators in surface waves. Applied Ocean Research 2: 75–80.

Guedes Soares

Caires

(1995) Changes in spectral shape due to the effect of finite water depth. In: Proceedings of the 14th international conference on offshore mechanics and arctic engineering (ed Guedes Soares

), vol. II, pp. 547–556. New York: ASME, Denmark, 18–22 June.

Guedes Soares

Bhattacharjee

Karmakar

(2014) Overview and prospects for development wave and of offshore wind energy. Brodogradnja 65(2): 87–109.

Guedes Soares

Bhattacharjee

Tello

. (2012) Review and classification of wave energy converters. In: Guedes Soares

Garbatov

Sutulo

. (eds) Maritime Engineering and Technology. London: Taylor & Francis Group, pp. 585–594.

10.

McIver

Evans

(1984) Approximation of wave forces on cylinder arrays. Applied Ocean Research 6: 101–107.

11.

Ricci

Saulnier

Falcao

(2007) Point absorber arrays: A configuration study off the Portuguese west-coast. In: Proceedings of the 7th European wave and tidal energy conference, Porto, 11–13 September.

12.

Sinha

Karmakar

Guedes Soares

(2015) Numerical modelling of an array of heaving point absorbers. In: Guedes Soares

(eds) Renewable Energies Offshore. London: Taylor & Francis Group, pp. 383–391.

13.

Vantorre

Banasiak

Verhoeven

(2004) Modelling of hydraulic performance and wave energy extraction by a point absorber in heave. Applied Ocean Research 26: 61–72.

Hydrodynamic performance of concentric arrays of point absorbers

Abstract

Keywords

Introduction

Concentric arrays

Numerical modelling

Frequency domain modelling

Force restriction

Wave climate

Results and discussion

Concentric arrays

Effect of platform dimensions

Effect of change in the number of concentric circles

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

Author biographies

References