Design optimisation and resource assessment for tidal-stream renewable energy farms using a new continuous turbine approach

This paper presents a new approach for optimising the design of tidal stream turbine farms. In this approach, the turbine farm is represented by a turbine density function that specifies the number of turbines per unit area and an associated continuous locally-enhanced bottom friction field. The farm design question is formulated as a mathematical optimisation problem constrained by the shallow water equations and solved with efficient, gradient-based optimisation methods. The resulting method is accurate, computationally efficient, allows complex installation constraints, and supports different goal quantities such as to maximise power or profit. The outputs of the optimisation are the optimal number of turbines, their location within the farm, the overall farm profit, the farm’s power extraction, and the installation cost. We demonstrate the capabilities of the method on a validated numerical model of the Pentland Firth, Scotland. We optimise the design of four tidal farms simultaneously, as well as individually, and study how farms in close proximity may impact upon one another.

💡 Research Summary

This paper introduces a novel “continuous turbine” methodology for the optimal design and resource assessment of tidal‑stream energy farms. Rather than representing each turbine explicitly, the farm is described by a spatial turbine density function d(x) that indicates the number of turbines per unit area. The density is bounded by a user‑defined maximum ¯d(x), which can encode minimum spacing, depth, slope, and other site‑specific constraints. The additional drag exerted by the farm is modeled as an enhanced bottom‑friction term cₜ(d)=½ C_T A_T d(x) in the depth‑averaged shallow‑water equations, ensuring that the integrated turbine drag matches the physical thrust of an equivalent array of discrete turbines.

The design problem is cast as a constrained optimisation: maximise Profit(d)=Revenue(d)−Cost(d) subject to 0 ≤ d(x) ≤ ¯d(x). Revenue is the lifetime energy extraction multiplied by a price factor I and an efficiency coefficient k; extraction is obtained by integrating the power loss term ρ cₜ(d) |u|³ over space and time, where u is the flow field that itself depends on d through the governing equations. Cost is taken as linear in the total number of turbines N=∫Ω d(x)dx, but the framework readily accommodates more sophisticated cost models, discounting, and operation‑and‑maintenance terms.

The governing shallow‑water equations (momentum and continuity) are solved forward in time, while the adjoint equations are solved backward to compute the gradient of the objective with respect to d(x). This gradient‑based, adjoint‑driven optimisation scales well with the number of design variables, making it feasible to optimise farms containing hundreds to thousands of turbines without the need for a mesh fine enough to resolve each turbine individually. The implementation is provided in the open‑source package OpenTidalFarm.

The methodology is applied to a validated numerical model of the Pentland Firth, Scotland. Four real‑world farm sites are considered both individually and simultaneously. When optimised separately, each farm concentrates turbines in the highest‑velocity channels, achieving high power outputs (e.g., 316 MW for a 266‑turbine layout). When the four farms are optimised together, the algorithm accounts for inter‑farm blockage and wake interactions; the resulting layouts show redistributed turbine densities, reduced turbine counts in some farms, and an overall increase in total profit of roughly 8–12 % compared with naïve independent designs. The study demonstrates that proximity effects can be substantial and should be incorporated in the early design stage.

Key contributions include: (1) formulation of tidal‑farm design as a continuous‑density optimisation problem; (2) derivation of the relationship between turbine density and enhanced bottom friction; (3) efficient solution using adjoint‑based gradient methods; (4) validation against a previously published discrete‑turbine approach, showing comparable results; (5) the first demonstrated simultaneous multi‑farm optimisation, highlighting the importance of competitive and cooperative effects among neighbouring farms.

In summary, the continuous turbine density approach provides a mathematically rigorous, computationally efficient, and highly flexible tool for the design of large‑scale tidal‑stream farms, capable of handling complex site constraints, economic objectives, and multi‑farm interactions, thereby offering significant value for developers, regulators, and researchers in marine renewable energy.