# Layout Optimisation of Wave Energy Converter Arrays

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## Abstract

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## 1. Introduction

## 2. Array Power Absorption

- The main forces are due to the action of gravity, the power take-off (PTO) system, and the interaction between the waves and the WECs.
- The forces are computed within the context of linearised potential flow theory, a constant water depth, and small amplitude motions compared to the wavelength and the water depth.
- The PTO is represented separately for each mode of motion in the array as a linear damper and a spring in parallel.

## 3. Optimisation Problem

#### 3.1. Array Layout Description

- a: distance between rows.
- b: distance between columns.
- $\alpha $: row’s angle with respect to the x-axis.
- $\delta $: angle between rows and columns.

#### 3.2. Covariance Matrix Adaptation Evolution Strategy

- Initialise $\underline{m}$, $\sigma $ and $\Sigma $.
- Generate a population of $\lambda $ individuals using Equation (14).
- Evaluate the objective function $f\phantom{\rule{0.166667em}{0ex}}q\phantom{\rule{0.166667em}{0ex}}N$ and set fitness value to all individuals.
- If any termination condition is met, go to the next step; otherwise repeat steps 2–5.
- Retrieve the best encountered individual (optimal solution) and finish.

- The objective function has been evaluated more than ${N}_{f}=1000$ times.
- The standard deviation of a, b, $\alpha $ and $\delta $ are smaller than $\left(\right)open="("\; close=")">D-\tilde{R}$, $\left(\right)open="("\; close=")">D-\tilde{R}$, $\pi \phantom{\rule{0.166667em}{0ex}}{10}^{-3}$ and $\frac{\pi}{6}\phantom{\rule{0.166667em}{0ex}}{10}^{-3}$, respectively.
- From generation $2\phantom{\rule{0.166667em}{0ex}}{N}_{g}$ and for every ${N}_{g}$ generations, the best fitness value has not been improved in the last ${N}_{g}$ generations (stagnation), where ${N}_{g}$ is the closest integer to $\frac{{N}_{f}}{5\phantom{\rule{0.166667em}{0ex}}\lambda}$.

#### 3.3. Genetic Algorithm

- Select two individuals (parents) from $\left(\right)$ with associated probabilities $\left(\right)$.
- Create a new individual ${\underline{Z}}_{k}^{\left(\right)}$ as a copy of the parent with the largest fitness value.
- Change ${N}_{c}$ parameters of the new individual by those of the parent with the smallest fitness value (crossover); ${N}_{c}$ is a random choice between 1 and 2. Which parameter is swapped is a random choice among a, b, $\alpha $ and $\delta $. The random choice is performed assuming a uniform distribution over all possible options without replacement.
- Change a parameter of the new individual by a random value uniformly distributed within the box $0\le \underline{Z}\le 1$ (mutation), or change nothing. Whether or not mutation occurs is a random choice with associated probabilities of 0.7 or 0.3, respectively. Which parameter mutates is a random choice among a, b, $\alpha $ and $\delta $.

- Initialise population of $\lambda $ individuals.
- Evaluate objective function $f\phantom{\rule{0.166667em}{0ex}}q\phantom{\rule{0.166667em}{0ex}}N$ and set fitness value to all individuals.
- Select the $\mu $ best individuals.
- If any termination condition is met, go to the next step; otherwise repeat steps 2–5.
- Retrieve the best encountered individual (optimal solution) and finish.

- The objective function has been evaluated more than ${N}_{f}=1000$ times.
- From generation $2\phantom{\rule{0.166667em}{0ex}}{N}_{g}$ and for every ${N}_{g}$ generations, the best fitness value has not been improved in the last ${N}_{g}$ generations (stagnation), where ${N}_{g}$ is the closest integer to $\frac{{N}_{f}}{5\phantom{\rule{0.166667em}{0ex}}\lambda}$.

#### 3.4. Glowworm Swarm Optimisation

- Initialise population of $\lambda $ glowworms.
- Evaluate objective function $f\phantom{\rule{0.166667em}{0ex}}q\phantom{\rule{0.166667em}{0ex}}N$ and set the luciferin level of all glowworms using Equation (17).
- Identify neighbours and calculate associated probabilities for all glowworms using Equation (18).
- Randomly select leading glowworms according to the previously calculated probabilities.
- Update glowworms’ positions using Equation (19).
- Update glowworms’ neighbourhood range using Equation (20).
- If any termination condition is met, go to the next step; otherwise repeat steps 2–7.
- Retrieve the best encountered glowworm position (optimal solution) and finish.

- The objective function has been evaluated more than ${N}_{f}=1000$ times.
- From generation $2\phantom{\rule{0.166667em}{0ex}}{N}_{g}$ and for every ${N}_{g}$ generations, the best fitness value has not been improved in the last ${N}_{g}$ generations (stagnation), where ${N}_{g}$ is the closest integer to $\frac{{N}_{f}}{5\phantom{\rule{0.166667em}{0ex}}\lambda}$.

## 4. Case Study

## 5. Results and Discussion

#### 5.1. Performance of Optimisation Algorithms

#### 5.2. Optimal Array Layout Solutions

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Falnes, J. A review of wave-energy extraction. Mar. Struct.
**2007**, 20, 185–201. [Google Scholar] [CrossRef] - Barstow, S.; Mørk, G.; Mollison, D.; Cruz, J. The wave energy resource. In Ocean Wave Energy; Springer: Berlin, Germany, 2008; pp. 93–132. [Google Scholar]
- World energy demand and economic outlook. In International Energy Outlook 2016; U.S. Energy Information Administration: Washington, DC, USA, 2016; pp. 7–17.
- Clément, A.; McCullen, P.; Falcão, A.F.O.; Fiorentino, A.; Gardner, F.; Hammarlund, K.; Lemonis, G.; Lewis, T.; Nielsen, K.; Petroncini, S.; et al. Wave energy in Europe: Current status and perspectives. Renew. Sustain. Energy Rev.
**2002**, 6, 405–431. [Google Scholar] [CrossRef] - Falcão, A.F.O. Wave energy utilization: A review of the technologies. Renew. Sustain. Energy Rev.
**2010**, 14, 899–918. [Google Scholar] [CrossRef] - Magagna, D.; Uihlein, A. Wave energy. In JRC Ocean Energy Status Report; Publications Office of the European Union: Luxembourg, 2014; pp. 31–43. [Google Scholar]
- Uihlein, A.; Magagna, D. Wave and tidal current energy—A review of the current state of research beyond technology. Renew. Sustain. Energy Rev.
**2016**, 58, 1070–1081. [Google Scholar] [CrossRef] - Kallesøe, B.S.; Dixen, F.H.; Hansen, H.F.; Køhler, A. Prototype test and modeling of a combined wave and wind energy conversion system. In Proceedings of the 8th European Wave and Tidal Energy Conference, Uppsala, Sweden, 7–10 September 2009; pp. 453–459. [Google Scholar]
- Iuppa, C.; Contestabile, P.; Cavallaro, L.; Foti, E.; Vicinanza, D. Hydraulic performance of an innovative breakwater for overtopping wave energy conversion. Sustainability
**2016**, 8, 1226. [Google Scholar] [CrossRef] - Falcão, A.F.; Henriques, J.C. Oscillating-water-column wave energy converters and air turbines: A review. Renew. Energy
**2016**, 85, 1391–1424. [Google Scholar] [CrossRef] - Contestabile, P.; Iuppa, C.; Di Lauro, E.; Cavallaro, L.; Andersen, T.L.; Vicinanza, D. Wave loadings acting on innovative rubble mound breakwater for overtopping wave energy conversion. Coast. Eng.
**2017**, 122, 60–74. [Google Scholar] [CrossRef] - Babarit, A. Impact of long separating distances on the energy production of two interacting wave energy converters. Ocean Eng.
**2010**, 37, 718–729. [Google Scholar] [CrossRef] - Borgarino, B.; Babarit, A.; Ferrant, P. Impact of wave interactions effects on energy absorption in large arrays of wave energy converters. Ocean Eng.
**2012**, 41, 79–88. [Google Scholar] [CrossRef] - Penalba, M.; Touzón, I.; López-Mendia, J.; Nava, V. A numerical study on the hydrodynamic impact of device slenderness and array size in wave energy farms in realistic wave climates. Ocean Eng.
**2017**, 142, 224–232. [Google Scholar] [CrossRef] - Child, B.F.M.; Venugopal, V. Optimal configurations of wave energy device arrays. Ocean Eng.
**2010**, 37, 1402–1417. [Google Scholar] [CrossRef] - Moarefdoost, M.M.; Snyder, L.V.; Alnajjab, B. Layouts for ocean wave energy farms: Models, properties, and heuristic. Omega
**2017**, 66, 185–194. [Google Scholar] [CrossRef] - Budal, K. Theory for absorption of wave power by a system of interacting bodies. J. Ship Res.
**1977**, 21, 4. [Google Scholar] - Sharp, C.; DuPont, B. Wave energy converter array optimization—A review of current work and preliminary results of a genetic algorithm approach introducing cost factors. In Proceedings of the ASME 2015 International Design Engineering Technical Conference and Computers and Information in Engineering Conference, Boston, MA, USA, 2–5 August 2015. [Google Scholar]
- McNatt, J.C.; Venugopal, V.; Forehand, D. A novel method for deriving the diffraction transfer matrix and its application to multi-body interactions in water waves. Ocean Eng.
**2015**, 94, 173–185. [Google Scholar] [CrossRef] - Sandia National Laboratory’s Reference Model Project (RMP). Available online: http://energy.sandia.gov/energy/renewable-energy/water-power/technology-development/reference-model-project-rmp/ (accessed on 10 June 2017).
- Nava, V.; Topper, M.B.R.; Ruiz-Minguela, P.; de Andrés, A.; Jeffrey, H. A critical discussion about optimisation approaches for ocean energy array design. In Proceedings of the 2nd International Conference on Renewable Energies Offshore (RENEW2016), Lisbon, Portugal, 24–26 October 2016; pp. 383–391. [Google Scholar]
- Optimal Design Tools for Ocean Energy Arrays (DTOcean). Available online: http://www.dtocean.eu (accessed on 10 June 2017).
- Hansen, N. The CMA Evolution Strategy: A Tutorial. arXiv
**2016**, 1604, 1–39. [Google Scholar] - Krishnanand, K.N.; Ghose, D. Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions. Swarm Intell.
**2009**, 3, 87–124. [Google Scholar] [CrossRef] - Mercadé Ruiz, P.; Ferri, F.; Kofoed, J.P. Experimental validation of a wave energy converter array hydrodynamics tool. Sustainability
**2017**, 9, 115–134. [Google Scholar] [CrossRef] - Kagemoto, H.; Yue, D.K.P. Interactions among multiple three-dimensional bodies in water waves: An exact algebraic method. J. Fluid Mech.
**1986**, 166, 189–209. [Google Scholar] [CrossRef] - Babarit, A.; Delhommeau, G. Theoretical and numerical aspects of the open source BEM solver NEMOH. In Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6–11 September 2015. [Google Scholar]
- Wilhelm, S.; Manjunath, B.G. Tmvtnorm: A package for the truncated multivariate normal distribution. Sigma
**2010**, 2, 2. [Google Scholar] - Hasselmann, K. Measurements of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dtsch. Hydrogr. Z.
**1973**, 8, 95. [Google Scholar]

**Figure 1.**Array layout description. The black dots represent wave energy converter (WEC) positions. The continuous red line is the bounds of the array deployment, the blue line is a row, and the green line is a column. The x-axis is aligned with the main direction of propagation of the ambient waves.

**Figure 4.**Optimisers’ performance and required number of function evaluations prior to termination. The results are shown for the 100 runs performed by each optimiser.

**Figure 5.**Covariance matrix adaptation evolution strategy’s (CMA’s) performance and required number of function evaluations prior to termination. The results are shown for 100 runs performed using population sizes of 8 and 40.

**Figure 7.**Search-space discretisation (SSD), best layout parameters found by the optimisers, and the reference layout (REF) parameters. Grid points resulting from the SSD are shown as the intersections between black lines.

**Figure 8.**Optimal layout parameters found by the optimisers over the 100 runs and the reference layout parameters (REF).

**Table 2.**Distances between best layout solutions in the parameter space, their q-factor and their yearly power production.

Parameter | CMA | GA | GSO | REF |
---|---|---|---|---|

$\parallel \underline{Z}-{\underline{Z}}^{\left(\mathrm{REF}\right)}\parallel $ | 0.63 | 0.03 | 0.14 | 0.00 |

q | 0.93 | 0.90 | 0.90 | 0.90 |

P (MW) | 3.22 | 3.23 | 3.21 | 3.21 |

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## Share and Cite

**MDPI and ACS Style**

Ruiz, P.M.; Nava, V.; Topper, M.B.R.; Minguela, P.R.; Ferri, F.; Kofoed, J.P.
Layout Optimisation of Wave Energy Converter Arrays. *Energies* **2017**, *10*, 1262.
https://doi.org/10.3390/en10091262

**AMA Style**

Ruiz PM, Nava V, Topper MBR, Minguela PR, Ferri F, Kofoed JP.
Layout Optimisation of Wave Energy Converter Arrays. *Energies*. 2017; 10(9):1262.
https://doi.org/10.3390/en10091262

**Chicago/Turabian Style**

Ruiz, Pau Mercadé, Vincenzo Nava, Mathew B. R. Topper, Pablo Ruiz Minguela, Francesco Ferri, and Jens Peter Kofoed.
2017. "Layout Optimisation of Wave Energy Converter Arrays" *Energies* 10, no. 9: 1262.
https://doi.org/10.3390/en10091262