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Edited by: Boyin Ding, University of Adelaide, Australia

Reviewed by: Nataliia Sergiienko, University of Adelaide, Australia; John Ringwood, Maynooth University, Ireland

This article was submitted to Sustainable Energy Systems and Policies, a section of the journal Frontiers in Energy Research

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A commercial wave energy system will typically consist of many interacting wave energy converters installed in a park. The performance of the park depends on many parameters such as array layout and number of devices, and may be evaluated based on different measures such as energy absorption, electricity quality, or cost of the produced electricity. As wave energy is currently at the stage where several large-scale installations are being planned, optimizing the park performance is an active research area, with many important contributions in the past few years. Here, this research is reviewed, with a focus on identifying the current state of the art, analyzing how realistic, reliable, and relevant the methods and the results are, and outlining directions for future research.

Wave energy has the potential to contribute significantly to the world's electricity consumption. To produce electricity in the range above a few MW, most wave energy concepts require that wave energy converters (WECs) are deployed together in arrays, or parks.

Interaction between the devices will affect the full performance, reliability, cost, and life-time of the park. The interaction can be hydrodynamical (scattered and radiated waves), mechanical (shared mooring and foundations), electrical (sea cables, substations, grid connection), and economic (shared capital and operational costs). As a result, many parameters will affect the interaction and the park performance, reliability, and costs. The number of devices and their separation distance, the park layout, mooring configurations, electrical and power take-off (PTO) systems, rated power of individual devices, constraints of subsystems, and so on—all parameters should be tuned to obtain the optimal design of the wave energy park before installation. In addition, environmental parameters have a large impact on the park. Wave climate, wave direction and variability, water depth, currents, and distance from shore are some factors that all affect the wave energy system, and different sites and environmental conditions will require different optimal solutions.

Many papers on wave energy parks have claimed to carry out optimization, whereas, in reality, they have only compared a few distinct configurations or tuned one parameter to obtain a minimum or maximum point. In recent years, there has been a vast increase in the research field of wave energy park optimization, and several global optimization algorithms have been developed and applied. Ideally, optimization of a wave energy park should find the best available solution to an objective function that considers all aspects of the park, including all costs and total revenue over the lifetime, reliability, constraints regarding available ocean area, deployment, and maintenance, allowed power fluctuations, water depth, etc. In short, the goal of the research should be to provide wave energy developers with clear, unbiased and reliable answers on how they should best design their park, given the constraints they are facing.

The aim of the current paper is to ask how far we are from providing such answers. The paper poses questions on how

The objective of this review paper is to review the state of the art of wave energy park optimization and to analyze how realistic, reliable, and relevant the current methods and results of the research area are. To arrive at this, first, the modeling methods and the optimization algorithms are reviewed. Modeling methods used in wave energy park optimization are reviewed in section 2, with focus on hydrodynamic modeling and WEC dynamics. Optimization methods are reviewed in section 3, and both simple “optimization” procedures such as comparing distinctive configurations or tuning single parameters are included, as well as global optimization algorithms such as evolutionary strategies. In section 4, we then return to the technical aspects reviewed in sections 2-3 and analyze how realistic (section 4.1), reliable (section 4.2), and relevant (section 4.3) the different methods and output are. Different approaches and systems are also compared to find general trends. Challenges and constraints for the research field are discussed in section 5, and future routes are outlined. Finally, some general conclusions are presented in section 6.

As mentioned in the introduction, a large wave energy system consists of hydrodynamical, mechanical, and electrical subsystems, all with their own complexities and requiring different modeling techniques. In the current section, different methods used to model hydrodynamics, wave-structure interaction, and WEC dynamics, including power take-off models, are reviewed.

The vast majority of works on hydrodynamical interactions in wave energy parks have been carried out based on linear potential flow theory. In other words, the fluid is assumed to be non-viscous, non-rotational, and incompressible, such that the governing equation of the fluid velocity reduces to the Laplace equation ΔΦ = 0, where

Hydrodynamic modeling of wave energy parks by means of analytical methods was initiated already in the early works on wave energy. Budal (

By applying an acoustic multi-body diffraction theory to water waves, the

Hydrodynamic modeling in offshore engineering is most often performed using numerical methods. The most commonly used approach is the boundary element method (BEM). At the EWTEC conference in 2015, over 37% of the papers in the wave energy tracks explicitly stated that a BEM software had been used (Penalba et al.,

In the above-mentioned methods, the underlying theory is based on potential flow, to linear or second non-linear order. When steep waves are considered, the assumptions of linearization are no longer valid, and in addition, viscosity may have a large impact on the dynamics of floating bodies. Computational fluid dynamics (CFD) methods solve the full Navier-Stokes equations using numerical methods. Different commercial and open-source CFD software packages are used increasingly to study the hydrodynamic properties of WECs, but the computational cost is still too high to allow for wave park optimization studies, even if Devolder et al. (

The dynamics of WECs differ fundamentally from those of traditional offshore structures, mainly due to the small scale of the WEC structures, the existence of a PTO, and the aim to absorb energy from the waves, which often requires large-amplitude motion at resonance with the waves. Many different PTO systems and methods of modeling them exist in the literature (see Folley et al., _{PTO}(

The equations of motion for a system of floating bodies can be described by Cummins' equation (Cummins,

The WEC dynamics can also be affected by the mooring system. For floating WECs, the main purpose of the mooring system is station-keeping. For several wave energy concepts, however, the mooring system is integrated into the PTO. This is the case, for example, when the WEC consists of a buoy connected to a direct-driven linear generator or when the buoy is tethered to pumps driving the PTO. The mooring system of WEC arrays is most commonly modeled as linear springs, as in the works by Vicente et al. (

By solving the equations of motion and obtaining the dynamics of the system, the absorbed power can be computed. Once the absorbed power has been computed, the hydrodynamical interaction can be evaluated using the interaction factor, or _{tot}, and the power that would have been absorbed by the same number

An interaction factor

Both analytical and numerical modeling are always connected with some approximations and uncertainty, and for an accurate understanding of the systems and for reliable results, physical experiments are required. However, experiments with wave energy arrays are both expensive and complex to carry out, and it is only in recent years that several large-scale physical experiments have been conducted and their results published.

Numerical predictions of the response of an array of five heaving floats in regular waves were compared to experimental measurements by Thomas et al. (

Recently, experiments with more units and increased complexity in terms of multiple degrees of freedom dynamics or with advanced control algorithms have been conducted. An experimental campaign of large arrays of up to 25 heaving point-absorber WECs was carried out in the shallow water wave basin of DHI in Denmark in the WECwakes project and has been presented by Troch et al. (

Arrays of up to six point-absorber WECs, each moving in six degrees of freedom were carried out both at the Australian Maritime College by Nader et al. (

The experimental works discussed above were all carried out in wave tanks. Experimental results from real sea tests of arrays of WECs are even more rare. Three fullscale WECs were deployed by Uppsala University at the Lysekil offshore site in 2009, and array interactions were studied (Rahm et al.,

Most studies on designing optimal wave energy parks have been based on comparisons of a few different configurations (Babarit,

Several layouts and parameter settings of arrays of heaving cylinders and surging barges were studied by Borgarino et al. (

Average absorbed power for aligned, staggered, and arrow arrays with three different separation distances (López-Ruiz et al.,

The next step after comparing distinctive configurations is to vary one parameter to find the optimal value, keeping all other parameters constant. Tissandier et al. (

As has been discussed above, many parameters may affect the performance of a wave energy park: the layout of the park, the separation distance between devices, the number and size of the devices in the park, the individual PTO settings of each unit, mooring configurations, control strategies, wave climate and wave direction, etc. Simple sweeps over single parameters are not sufficient to find optima in the broad and complex solution space of wave energy parks, and more advanced optimization algorithms are needed.

Non-linear programming optimization is useful for handling optimization problems that cannot be handled by simple parameter sweeps but which require optimization across a moderately large parameter space. These methods can also be adapted to handle non-linear constraints on the WEC motion, which was applied already in early works by Evans (

Optimization of the array layout for point-absorber, spherical WECs operating in heave only was performed by McGuinness and Thomas (

Optimization in terms of maximizing the energy absorption under WEC motion constraints was considered by Bacelli and Ringwood (

More examples of array optimizations coupled to WEC control algorithms exist (Garcia-Rosa et al.,

Metaheuristic optimization methods search the solution space for sufficiently good solutions, given specified constraints and convergence criteria. They are useful when the optimization problem is too large for all solutions to be evaluated and when the solution space is multi-peaked, and several different algorithms have been developed and applied to wave energy park optimization.

GAs were first applied for the layout of wave energy arrays in Child and Venugopal (

To evaluate the reliability and efficiency of different metaheuristic optimization strategies for wave energy applications, several recent works have carried out array optimizations using different algorithms and compared their results. A particle swarm optimization was compared to a GA by Faraggiana et al. (

Neshat et al. (

The research area of wave energy park optimization is still young, with many important developments in the past few years. But as the wave energy industry is currently moving toward ocean deployments and fullscale systems, it is important to identify the state of the art and where to focus future work. In this section, we return to the technical aspects of wave energy park modeling and optimization reviewed in sections 2, 3 and analyze how realistic, reliable, and relevant the methods and results are. These aspects are discussed in sections 4.1–4.3, respectively.

In this section, we analyze how well the WECs and environmental systems are modeled, i.e., whether the models are realistic representations of the studied physical systems.

As discussed before, most results on wave energy parks have been obtained under the assumption of potential flow theory, where the viscosity and the rotational behavior of the fluid have been neglected. Whereas some advances have recently been presented to model the hydrodynamical interaction between WECs with CFD methods (see section 2.1), it is still unthinkable to carry out optimization of large wave energy systems using high-fidelity viscous and rotational fluid models. Instead, physical experiments will have to be used to validate the analytical/numerical methods used for wave-structure modeling, which will be further discussed in section 4.2. But even within the assumptions of potential flow theory, further simplifications on the waves have often been assumed, and most arrays have been studied in a few sea states with long-crested waves. To study the performance in more realistic wave conditions, wave directionality and long-term wave spectra have been incorporated in recent works on park optimization.

The impact of the wave directionality on wave energy park performance has been studied in a number of works. As could be expected, the park performance of array layouts with rotational symmetry is less affected by wave directions than that of corresponding parks with rectangular or linear layouts (De Andrés et al.,

In reality, ocean waves are usually not long-crested but short-crested, i.e., they consist of many waves traveling in different directions simultaneously. Recently, some works have considered the performance and optimization of wave energy arrays in short-crested, or omni-directional, waves. Wave run-up and trapped wave modes were studied both experimentally and analytically in an array of bottom-mounted cylinders by Ji et al. (

Most array optimization studies have focused on optimization in one or a few wave conditions. The relevant objective should rather be the full life-cycle performance of the arrays, which requires a long-term assessment. However, the wave propagation models that are typically used to model waves over long time period or large domains cannot be used directly to model the dynamics and instant power of the WECs, and simplified estimations of the absorbed energy have previously been used to assess the potential of arrays over large domains (Defne et al.,

The wave climate over 25 years, comparable to the life-time of the devices, was used by López-Ruiz et al. (^{2} grid over the Swedish exclusive zone. A ten-year hindcast model was used by Gorr-Pozzi et al. (

Efforts to couple accurate hydrodynamic modeling close to the WEC with wave propagation models a distance away from the WEC have been made both for potential flow (McNatt et al.,

Comparison of the wave height obtained experimentally and numerically with a coupled MILDwave-NEMOH model (Verao Fernández et al., _{d} coefficient is defined as the ratio between the numerically calculated and the target significant wave height, and the wave height has been recorded by wave gauges (WG) in an array of nine heaving damped WECs.

A wave energy system consists of many parts, often including multi-body floats, moorings, power take-off systems, foundations, and electrical systems in generators, substations, sea cables, and grid connection points. Each of the subsystems is complex, and it is impossible to include all of the complexity of each subsystem in the optimization of a full wave energy park. But as the obtained results can be misleading if important aspects of the full system are neglected, proper attention must be given to the approximations made and how well they reflect the realistic wave energy system.

As discussed in section 2.2, the standard approach when optimizing wave energy parks is to model the PTO as a linear spring-damper system, where the power absorption is proportional to the Coulomb damping. In addition, the dynamical degrees of freedom are usually restricted; for example, point-absorbers are considered to move in heave only. Comparison of numerical PTO models to experimental data of scaled models (with simplified PTO systems) has been carried out in a number of works. Linear and non-linear PTO models for the Wavepiston WEC were compared in frequency and time domain by Read and Bingham (

Comparisons of numerical PTO models to fullscale, realistic WECs in offshore operations are rarer. Almost all tests with fullscale WECs have been conducted by companies and are kept within the company, but a few exceptions exist. A linear time-domain model for a single point-absorber WEC was validated against experimental data from a fullscale WEC by Eriksson et al. (

Mooring lines of floating wave energy systems may exhibit non-linear dynamics and snap loads, which will affect both the absorbed power and the life-time of the energy system. Whereas numerical models exist to study this dynamical behavior for isolated WECs (Bhinder et al.,

As for all sound science, the methods and the results must be reliable; the methods should produce stable and consistent results. Methods should be

With the physical experiments of wave energy parks carried out in the past few years, validation of analytical or numerical modeling methods has become possible. An analytical multiple scattering method was compared to experimental data for an array of bottom-mounted cylinders in short-crested waves by Ji et al. (

Predicted and measured capture factor for the power absorption of an array of five WaveStar models at a scale of 1:20 in irregular waves (Mercadé Ruiz et al.,

Physical tank tests and numerical modeling using WAMIT for OWCs were compared by Sharp et al. (

Experimental and numerical results for an array of five fixed OWCs.

Most of the experiments on arrays have been carried out using heaving buoys (Nader et al.,

Sinha et al. (

Arrays with WECs of different dimensions were studied by Göteman (

Many wave energy park optimization algorithms have started from prescribed layouts and optimized parameters such as separation distance between devices. A more unbiased approach was taken by McGuinness and Thomas (

Optimal configurations through single- or multi-objective optimization routines such as evolutionary algorithms were studied by Child and Venugopal (

Wave energy park optimization with metaheuristic optimization methods has produced similar results for small arrays at sites where the waves propagate within a narrow interval of predominant wave directions. The waves are propagating from left to right in all figure parts except

Corresponding optimization as in

It is of little value that the methods and results are realistic and reliable if the results are not relevant for the wave energy industry. In other words—are we optimizing the correct quantities? If the optimization algorithm maximizes the power output of a park, but this produces power peaks that are too large for any realistic electrical system or grid connection, the results are of little use for the development of large-scale wave energy systems. Or, if a relevant objective function is being optimized but at the expense of immensely increased costs, this again will not be a relevant option for the industry. In order to compete with other energy technologies, the objective function should maximize the total revenue over the lifetime of the devices within given constraints, such as maximal ocean area that can be used or maximal allowed power fluctuations. Many of the constraints can also be connected to economic values, for example, that increased separation distance between WECs requires more use of sea cable in the park, which increases the costs. Although most array optimization works have mainly focused on maximizing the absorbed power or the interaction factor, recent works have started to include more comprehensive objective functions.

Rapidly varying voltage magnitudes, known as flicker, is a major problem when integrating the produced electricity into the grid. Different voltage levels define different flicker severity over the short term (10 min) and long term (over 2 h) (Penalba and Ringwood,

Simplified economic cost functions have been implemented in wave energy park optimizations in several recent works. Sharp and DuPont (

where

Sea cables and electrical subsystems contribute significantly to the costs of the park, both in terms of installed capital costs and costs due to energy losses (Henfridsson et al.,

Wave energy park optimization from the perspective of minimizing cable costs.

The levelized cost of electricity (LCOE) computes the cost to generate electricity over the life-time of an energy system and is a useful measurement to compare different energy sources (IEA and NEA, _{out} delivered to the grid throughout the device's lifetime (Short et al.,

where

However, identifying economic parameters and values for a technology that has not yet reached full maturity is difficult, both because very little data is available and because it is expected that the costs and uncertainties will be reduced as the technology matures (Astariz and Iglesias,

Despite these challenges, some authors have attempted to include economic parameters in their optimization algorithms. Teillant et al. (

Optimization of wave energy parks based on economic considerations. The economic measures change with number of iterations and with number of devices in the park.

In the works above, the economic optimization of the full system with the LCOE as the objective function is still performed as a single-objective optimization. To the authors' knowledge, the only attempt to study the park optimization problem in a pure multi-objective way was presented by Arbonès et al. (

The state of the art of wave energy array modeling was reviewed by Babarit (

As the wave energy systems increase in size, so do the computational costs. Both with an increased number of WECs and an increased number of parameters to optimize, the time required for the optimization algorithms to converge quickly grows out of hand. Evaluation of each configuration is expensive, and the search space is non-convex and multi-modal (Neshat et al.,

Much work has been carried out to speed up the optimization process. Ruiz et al. (

Neshat et al. (

It can be expected that future works will continue to investigate and develop methods based on machine learning or other advanced algorithms to enable faster optimization of large parks. To ensure that the models are still realistic representations of the wave energy systems, the same systems should be modeled with several methods and the impact of approximations or constraints thoroughly evaluated.

Lack of available realistic data is a problem for the full wave energy sector, not only in the optimization of wave energy parks (Michelez et al.,

For completeness, in addition to the papers discussed above, economic values regarding wave energy systems can be found in Teillant et al. (

To reduce the uncertainty in numerical simulations of wave energy systems using CFD, several blind test workshops and paper series have recently been presented under the name CCP-WSI Blind Test Series (CCP-WSI,

In this paper, we have reviewed the state of the art of wave energy park optimization, with a focus on developments in the last five years. After discussing the different modeling and optimization methods that are used, we have analyzed whether the methods and results are realistic, reliable, and relevant.

It is not feasible to include all aspects of the waves when modeling wave energy parks. Whereas viscosity and rotation are important fluid properties when considering the detailed response and loads of single WECs, potential flow theory is the dominant assumption for large arrays of WECs. However, wave directionality, waves propagating in several directions simultaneously, and waves propagating over a long period of time should be considered when designing parks, and important steps have been taken in the last five years on realistic representation of the waves. Also, coupling high-fidelity fluid models close to the WECs to low-fidelity wave models further away from the bodies is a promising direction for a more realistic representation of the fluid-structure interactions. Optimization of large wave energy parks is still often carried out in the frequency domain with linear spring-damper systems as PTO. Some validation of this assumption has been carried out, but several works have also shown that linear frequency domain analysis cannot capture the non-linearities that are inherent in the system. Also, mooring dynamics have been shown to significantly affect the performance of parks and should not be neglected, and several important contributions have been published lately.

Straightforward comparison of different research groups' results is challenging, since the wave energy parks studied by the groups usually differ in terms of significant characteristics. However, some tendencies can be seen when comparing similar works.

Optimal layouts will position the WECs along lines perpendicular to wave direction at sites with a narrow interval of incident wave directions, but less obvious optimal layouts will emerge at sites where waves are coming from multiple directions. If there is not sufficient ocean area for perpendicular lines, the WECs will align along several lines, where the separation distance will depend on the predominant wave conditions.

However, these “perpendicular” layouts were in general obtained using single-objective functions, optimizing only the power output or interaction factor. Other optimal solutions might emerge when economic or multi-objective functions are considered, for example, when also minimizing the cable costs and losses.

Several works have shown that the total performance might be improved if devices of different individual dimensions are installed in the park.

The performance of large parks might be slightly lower in short-crested waves as compared to long-crested but with desired lower power fluctuations.

Some results have been published in recent years that have provided experimental data on large arrays and/or arrays with complex dynamics or PTO systems, but there are still very few publications with validation of park modeling and optimization. Although reasonable agreement has been shown, some works have also reported non-linear dynamics or deviations between numerical and experimental results.

As discussed above, the large, non-convex, and multimodal optimization problem of wave energy parks might be too large for conventional population-based optimization algorithms, and new approximate or machine learning methods might be useful complements for the evaluation of park performance, in particular for large parks. Simultaneously, careful consideration must be taken when making simplifying assumptions, as it has been shown that non-linear dynamics, mooring systems, and other complexities cannot be neglected. Validation of wave energy park modeling and optimization is still only in its initial phases. To increase reliability, more systems with increasing complexity should be validated, and the modeling uncertainties should be adequately quantified. In addition, biases are reduced and the reliability improved when different methods or systems are compared and shown to give similar results.

The relevance of the modeling results increases when the correct quantities are optimized. Multi-objective and/or economic cost functions should be used to provide useful guidelines on how wave energy parks should be designed, given a site and other constraints. If the research area is to move from developing

Figures 1 (left), 3, 4 (left), 5 (left), 6 (left), 7 (left), 11 (a,b,e,f,g), 12 (a,b,d), 13 (b), and 15 (upper right) have been reproduced from the references given in the figure captions, with permission from Elsevier Ltd.

Figures 4 (right) and 15 (left) have been reproduced from the references given in the figure captions, with permission from CRC Press/Balkema – Taylor & Francis Group.

Figures 11 (c), 12 (f), and 14 (left) have been reproduced from the references given in the figure captions, with permission from Springer Nature.

Figures 1 (right), 2, 8, 9, 12 (e), 13 (a), and 16 (left) have been reproduced from the references given in the figure captions under the Creative Commons CC-BY license.

The work in this review paper was initiated and lead by MGö. MGö wrote the manuscript with contributions from MGi, JE, and JI. All authors have proofread the final paper and are accountable for the content of the work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.