[Paper Review] Tidal turbine array optimisation using the adjoint approach

Simon W. Funke, Patrick E. Farrell+1 more|arXiv (Cornell University)|Apr 5, 2013

Wind Energy Research and Development45 references16 citations

TL;DR

This paper presents a gradient-based optimisation framework for tidal turbine arrays using the adjoint method to efficiently compute sensitivity gradients of power output with respect to turbine positions and tuning parameters. By solving the adjoint equations, the method achieves near-linear scalability with turbine count, enabling high-fidelity optimisation of up to 256 turbines in realistic bathymetry, significantly increasing power extraction while maintaining computational feasibility.

ABSTRACT

Oceanic tides have the potential to yield a vast amount of renewable energy. Tidal stream generators are one of the key technologies for extracting and harnessing this potential. In order to extract an economically useful amount of power, hundreds of tidal turbines must typically be deployed in an array. This naturally leads to the question of how these turbines should be configured to extract the maximum possible power: the positioning and the individual tuning of the turbines could significantly influence the extracted power, and hence is of major economic interest. However, manual optimisation is difficult due to legal site constraints, nonlinear interactions of the turbine wakes, and the cubic dependence of the power on the flow speed. The novel contribution of this paper is the formulation of this problem as an optimisation problem constrained by a physical model, which is then solved using an efficient gradient-based optimisation algorithm. In each optimisation iteration, a two-dimensional finite element shallow water model predicts the flow and the performance of the current array configuration. The gradient of the power extracted with respect to the turbine positions and their tuning parameters is then computed in a fraction of the time taken for a flow solution by solving the associated adjoint equations. These equations propagate causality backwards through the computation, from the power extracted back to the turbine positions and the tuning parameters. This yields the gradient at a cost almost independent of the number of turbines, which is crucial for any practical application. The utility of the approach is demonstrated by optimising turbine arrays in four idealised scenarios and a more realistic case with up to 256 turbines in the Inner Sound of the Pentland Firth, Scotland.

Motivation & Objective

• To address the challenge of optimally configuring large tidal turbine arrays to maximise energy extraction under complex flow interactions.
• To overcome the prohibitive computational cost of brute-force or heuristic optimisation in high-dimensional parameter spaces.
• To develop a scalable, automated optimisation framework that integrates physically accurate shallow water models with efficient gradient computation.
• To demonstrate the method’s effectiveness on both idealised and realistic tidal sites, including the Inner Sound of the Pentland Firth.
• To enable practical deployment of large-scale tidal farms by providing a computationally efficient, mathematically rigorous optimisation pipeline.

Proposed method

• Formulates the turbine array optimisation problem as a PDE-constrained optimisation, with the shallow water equations governing the flow dynamics.
• Uses a two-dimensional finite element model to simulate the flow and power extraction for a given turbine configuration.
• Employs the adjoint method to compute the gradient of the total power output with respect to turbine positions and tuning parameters at a cost nearly independent of the number of turbines.
• Solves the adjoint equations backward through time, propagating sensitivity from the power output to the design parameters.
• Integrates the adjoint gradient into a gradient-based optimisation algorithm to iteratively improve array layout and tuning.
• Validates the approach on idealised geometries and a realistic site with complex bathymetry and up to 256 turbines.

Experimental results

Research questions

• RQ1 How can the power output of a tidal turbine array be maximised under complex, nonlinear flow interactions and site constraints?
• RQ2 Can adjoint-based optimisation enable efficient gradient computation for large-scale turbine arrays with hundreds of turbines?
• RQ3 What are the optimal turbine layouts and individual tuning strategies in complex tidal environments?
• RQ4 How does the adjoint method compare in computational efficiency to traditional trial-and-error or metaheuristic approaches?
• RQ5 To what extent can physically accurate shallow water models be used in automated array optimisation without prohibitive cost?

Key findings

• The adjoint method enables gradient computation for power output with respect to turbine positions and tuning parameters at a cost nearly independent of the number of turbines, making large-scale optimisation feasible.
• In idealised scenarios, the optimisation algorithm increased power extraction significantly, with improvements exceeding 50% in some configurations compared to regular layouts.
• For the realistic Inner Sound of the Pentland Firth case, the method successfully optimised arrays of up to 256 turbines, achieving high power output despite complex bathymetry and flow dynamics.
• The optimised layouts exhibited physical intuition, such as turbine density decay near walls to funnel flow and strategic placement on southern walls to retain flow in the domain.
• The total power output for 128 and 256 turbine cases was nearly equal due to the cubic dependence of power on flow speed, though this would diverge under realistic rated-speed constraints.
• The approach is fully automatic once model inputs are specified, and extends naturally to include environmental impact or economic profit as optimisation objectives.