Tidal turbine array optimisation using the adjoint approach

• Published in: Renewable energy Vol. 63; pp. 658 - 673
• Main Authors: Funke, S.W., Farrell, P.E., Piggott, M.D.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01-03-2014; Elsevier
• Subjects: Adjoint method; Applied sciences; Array layout; Arrays; Energy; Exact sciences and technology; Gradient-based optimisation; Marine renewable energy; Mathematical analysis; Mathematical models; Natural energy; Optimization; Shallow water equations; Tidal energy; Tidal power; Tidal turbines; Tuning; Turbines; British Isles, Scotland, Highland, Caithness, Pentland Firth; British Isles, Scotland; Tidal turbines; Shallow water equations; Gradient-based optimisation; Adjoint method; Marine renewable energy; Array layout; Renewable energy
• ISSN: 0960-1481; 1879-0682
• DOI: 10.1016/j.renene.2013.09.031

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.
•The layout optimisation of tidal farms was formulated as a PDE-constrained optimisation problem.•Gradient-based optimisation and the adjoint approach allow the optimisation of many turbines with realistic flow models.•A shallow water model predicts the flow and the farm performance in each optimisation iteration.•The capability of this approach is demonstrated by optimising 256 turbines in a farm in the Inner Sound of the Pentland Firth, Scotland.•The software framework is open-source and available at http://opentidalfarm.org.