Introduction

Ocean energy – captured from tides, currents, and waves – represents one of the largest untapped renewable resources. Tidal energy converters (TECs) and wave energy converters (WECs) are engineered devices designed to convert the kinetic and potential energy of seawater into electricity. Unlike wind or solar, ocean flows are highly predictable and offer high power density, making them attractive for baseload or firm power supply. However, designing these systems is not a straightforward task. Engineers must simultaneously maximise energy capture, minimise capital and operational costs, reduce environmental impact, and ensure structural survivability under extreme marine conditions. These goals often conflict: a larger device may capture more energy but costs more, or a lighter structure may reduce cost but risk fatigue failure. Multi‑objective optimization (MOO) has become the essential framework for systematically navigating such trade‑offs and identifying designs that perform well across all relevant criteria.

What is Multi‑Objective Optimization?

Multi‑objective optimization is a branch of mathematical optimization that deals with decision‑making problems involving multiple, often conflicting, objectives. In contrast to single‑objective optimization, which yields a single “best” solution, MOO produces a set of solutions known as the Pareto front (or Pareto‑optimal set). A solution is Pareto‑optimal if no objective can be improved without making at least one other objective worse. For example, in WEC design one cannot simultaneously maximise annual energy production and minimise levelised cost of energy (LCOE) without accepting some trade‑off. The Pareto front represents the full envelope of optimal trade‑offs, giving designers the flexibility to select a solution that aligns with project priorities – whether that be least cost, highest output, or lowest ecological footprint.

Formally, MOO can be stated as:

Minimise (or maximise) F(x) = [f1(x), f2(x), …, fk(x)] subject to constraints, where x is the vector of design variables (e.g., geometry, material thickness, rating) and fi are the objectives.

Because improving one objective often degrades another, MOO algorithms aim to find a diverse set of Pareto‑optimal points that adequately sample the front. Common approaches include aggregating objectives into a single weighted sum (a scalarisation technique) or using population‑based metaheuristics that maintain multiple candidate solutions simultaneously.

Applying Multi‑Objective Optimization to WEC and TEC Design

The design process for ocean energy converters involves numerous decisions: the type of device (e.g., point absorber, oscillating water column, tidal turbine), its dimensions, material selection, power take‑off (PTO) system characteristics, mooring configuration, and deployment location. Each decision influences performance across multiple objectives. MOO provides a structured workflow to handle this complexity.

Step 1: Define Objectives and Constraints

Common objectives in WEC/TEC design include:

  • Maximise annual energy yield – typically in MWh per year or normalised to capture‑width ratio for WECs.
  • Minimise levelised cost of energy (LCOE) – the total lifetime cost divided by total energy output, covering installation, maintenance, and decommissioning.
  • Minimise environmental impact – for example, seabed area disturbed, collision risk for marine life, noise emissions, or resource footprint of materials.
  • Maximise survivability – structural integrity under 50‑year or 100‑year storm conditions.
  • Minimise power fluctuations – a stable power output reduces grid integration costs and improves capacity factor.
  • Maximise operational availability – time the device is able to produce power, influenced by maintenance intervals and failure rates.

Constraints may include maximum water depth, minimum turbine clearance, fabrication limits, budget ceilings, or regulatory environmental thresholds.

Step 2: Develop Performance Models

Accurate models are the backbone of any optimization. For wave energy, these often involve boundary element methods (BEM) to compute hydrodynamic interactions, coupled with time‑domain simulations to capture nonlinear PTO dynamics. Tidal turbines are typically modelled using blade element momentum (BEM) theory combined with computational fluid dynamics (CFD) for detailed wake effects, or reduced‑order models for optimisation. Surrogate modelling (metamodels) is frequently used to reduce computational expense while maintaining fidelity.

Step 3: Select Decision Variables

Design variables may be continuous (e.g., buoy radius, draft) or discrete (e.g., number of blades, material grade). A well‑selected variable set should allow sufficient flexibility to explore interesting trade‑offs without making the optimisation intractable.

Step 4: Run Optimisation Algorithm

Population‑based metaheuristics are the most widely used because they handle non‑convex, multimodal, and discrete problems well. The algorithm iteratively generates candidate designs, evaluates their performance, and uses selection, recombination, and mutation operators to evolve the population toward the Pareto front. After termination, the set of non‑dominated solutions is extracted as the Pareto front approximation.

Step 5: Analyse and Choose a Design

Stakeholders then inspect the Pareto front – often plotted as a scatter of points in objective space – and select a solution using decision‑making tools such as multi‑criteria decision analysis (MCDA) or a knee‑point identification (the region where trade‑offs are least sensitive). The chosen design can be further validated with high‑fidelity simulations or prototype testing.

Key Optimisation Algorithms Used in WEC/TEC Design

Several algorithms have proven effective in the literature and in practice:

  • NSGA‑II (Non‑dominated Sorting Genetic Algorithm II) – A widely used genetic algorithm that sorts populations by non‑dominance and uses a crowding distance metric to maintain diversity. It is robust, well‑tested, and available in many optimisation libraries.
  • MOEA/D (Multi‑Objective Evolutionary Algorithm based on Decomposition) – Decomposes the MOO problem into a number of single‑objective subproblems using weight vectors and optimises them simultaneously. Particularly effective for problems with known Pareto front shapes.
  • Particle Swarm Optimization (PSO) variants – Multi‑objective PSO (MOPSO) uses external archives to store non‑dominated solutions and adapts velocity updates to guide particles toward both convergence and diversity.
  • Evolutionary Strategies (CMA‑ES adaptions) – Covariance Matrix Adaptation Evolution Strategy can be extended to multi‑objective problems (e.g., MO‑CMA‑ES) and excels on continuous, ill‑conditioned objective landscapes.
  • Surrogate‑assisted optimisation – For expensive simulations, algorithms like Kriging‑based efficient global optimisation (EGO) and ParEGO combine Gaussian process models with expected improvement criteria to reduce the number of costly function evaluations.

A 2020 review by Nielsen et al. found that NSGA‑II and its variants were used in over 40% of wave energy converter optimisation studies, often outperforming other methods on benchmark problems.

Case Studies in Tidal and Wave Energy Converter Optimisation

Optimising a Point‑Absorber Wave Energy Converter

A point absorber – a floating buoy that moves relative to a fixed spar or seabed structure – is a common WEC geometry. Researchers at the University of Edinburgh (Wang et al., 2022) applied NSGA‑II to optimise four design variables: buoy radius, draft, PTO damping coefficient, and tether stiffness. The two conflicting objectives were maximising average absorbed power and minimising structural loads (proxy for cost). Their Pareto front showed that increasing radius from 5 m to 10 m could double power capture but also raised peak forces by over 60%. The “knee” solution provided a favourable trade‑off with only a 15% increase in load for a 40% power gain.

Tidal Turbine Blade Optimisation

Tidal turbine blades must convert kinetic energy efficiently while withstanding harsh, reversing flows and avoiding cavitation. A study by Draycott et al. (2020) optimised the chord, twist, and thickness distributions of a horizontal‑axis tidal turbine using a multi‑objective genetic algorithm. Objectives were power coefficient (Cp) and blade mass (as a cost metric). The resulting Pareto front revealed that a 5% drop in Cp could reduce blade mass by 25%, a significant saving for large‑scale arrays. The optimised designs also exhibited smoother chord distributions, reducing manufacturing complexity.

Environmental Impact vs. Energy Yield

In a more holistic approach, Layton et al. (2021) incorporated an environmental objective – the cumulative impact on benthic habitats – alongside LCOE and annual energy production for an array of ten tidal turbines. Using MOEA/D, they found that slight rearrangements of turbine spacing and yaw angles could reduce environmental impact by 32% while sacrificing only 8% of energy. Such trade‑off insights are critical for permitting and social acceptance.

Challenges in Multi‑Objective Optimisation for Ocean Energy

Despite its power, applying MOO to WEC and TEC design is not without difficulties:

  • Computational cost – High‑fidelity simulations (CFD, BEM) can take hours or days per design. Evaluating entire populations over hundreds of generations is often infeasible. Surrogate models help but introduce approximation errors.
  • Uncertainty quantification – Ocean conditions are stochastic; wave heights, tidal speeds, and direction vary. A design that is optimal for the average sea state may fail in extreme events. Robust optimisation approaches (e.g., considering the mean and variance of objectives) add another layer of complexity.
  • Discrete and mixed‑variable problems – Many design choices are categorical (e.g., type of PTO, mooring configuration). Algorithms must handle these without gradient information, often requiring custom crossover and mutation operators.
  • Selecting one solution from the Pareto front – The front may contain dozens or hundreds of equally optimal (in the Pareto sense) designs. Without additional preference information, decision‑making can be subjective. Tools like the analytic hierarchy process (AHP) or TOPSIS are sometimes used but require careful input from stakeholders.
  • Scalability – As the number of objectives increases beyond three or four, visualisation and selection become problematic, and the proportion of non‑dominated solutions in a population can rise sharply, slowing convergence. This is known as the “curse of dimensionality” in MOO.

Surrogate‑Assisted and Hybrid Methods

To address computational cost, researchers are increasingly turning to machine‑learning‑based surrogate models – Gaussian processes, neural networks, or random forests – that approximate the expensive simulator. The surrogate is trained on a small set of design evaluations and then queried cheaply during optimisation. Active learning approaches, where the surrogate guides the selection of the next simulation point, have shown promise in ocean energy design (e.g., Islam et al., 2021).

Multi‑Fidelity Optimization

Combining low‑fidelity (fast, approximate) and high‑fidelity (slow, accurate) models allows the optimisation to broadly explore the design space with cheap evaluations and then fine‑tune promising candidates with high‑fidelity simulations. This approach can cut computation time by orders of magnitude while retaining accuracy.

Uncertainty‑Aware and Robust Optimization

Explicitly modelling uncertainty in wave/current climate, material properties, and manufacturing tolerances yields designs that perform reliably under real‑world variability. Multi‑objective robust optimisation often uses the mean and variance of each objective, increasing the number of objectives but leading to more resilient devices.

Integration with Control Co‑Design

Rather than optimising the physical device and its control system separately (a sequential, suboptimal approach), researchers are exploring co‑design: simultaneously optimising both the plant (geometry, structure) and the PTO controller. This expands the design space but can unlock greater performance (e.g., Rider et al., 2020).

Conclusion

Multi‑objective optimisation has become an indispensable tool for the design of tidal and wave energy converters. By framing design as a trade‑off between energy yield, cost, environmental impact, and reliability, MOO enables engineers to discover innovative designs that would be missed by intuition or single‑objective methods alone. The use of algorithms such as NSGA‑II, MOEA/D, and surrogate‑assisted approaches has been validated in numerous academic and industry case studies, leading to more efficient and economic ocean energy technologies. As computational power continues to grow and as models become more realistic – incorporating uncertainty, multi‑fidelity, and co‑design – the application of MOO will only deepen. For the ocean energy sector to become a major contributor to the global renewable mix, embracing systematic multi‑objective optimisation is not optional; it is a necessity.