Source-linked AI summary
Tidal turbine array optimisation using the adjoint approach
S. W. Funke, P. E. Farrell, M. D. Piggott
TL;DR
The paper addresses how to optimise the positions and tuning of large tidal-turbine arrays despite nonlinear flow interactions and expensive physically accurate models. It combines a shallow-water flow model with gradient-based optimisation and adjoint equations, demonstrating successful optimisation from idealised cases to an industrially relevant 256-turbine site. The authors identify realistic forcing, bathymetry, validation, turbulence closure, and power-curve modelling as extensions needed for industrial application.
Problem
Large tidal arrays require optimised turbine positions and tuning, but nonlinear flow interactions and expensive accurate models make configuration search difficult.
Method
The paper formulates array design as PDE-constrained optimisation, using a two-dimensional nonlinear shallow-water model and adjoint equations to compute gradients efficiently.
Results
The method successfully increased power extraction in four idealised scenarios and a realistic Inner Sound case involving up to 256 turbines.
Takeaways & Limitations
Adjoint-based gradient optimisation enables physically realistic, automated tidal-farm optimisation for large turbine arrays.
Takeaways & Limitations
Industrial application requires realistic tidal forcing, bathymetry, validation against measurements, improved wake modelling, and a realistic power curve.
Abstract
from arXiv · showhide
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.
1. Introduction
Tidal turbine array optimisation must balance physically accurate flow modelling with computational feasibility. The paper formulates the problem as PDE-constrained optimisation and uses adjoint gradients to optimise large arrays automatically.
- Motivation: Hundreds of turbines are typically required for economically useful tidal power, making their positions and individual tuning important optimisation variables.The optimal configuration could substantially change captured energy and affect project economic viability.
- Motivation: Complex turbine-flow interactions and sensitive dependence of power on velocity make manual determination of the optimal configuration difficult.
- Existing approaches: Simplified flow models enable rapid exploration but cannot accurately capture complex flow behaviour, while physically accurate PDE models are too expensive for broad parameter searches.Prior work therefore often investigated only a handful of manually selected configurations in realistic settings.
- Proposed approach: The paper combines a two-dimensional nonlinear shallow water model with gradient-based optimisation and adjoint techniques to account for geometry, turbines, and flow interactions.The adjoint system efficiently computes gradients of extracted power with respect to turbine positions and tuning parameters.
- Scalability: The framework is designed to scale to realistic domains and large arrays, including a 256-turbine optimisation at an approximate cost of 200 flow solutions.
- Proposed approach: Adjoint gradients compute derivatives of one output with respect to many turbine inputs in a number of equation solves independent of turbine count.This makes the approach suitable for large-scale array optimisation compared with finite differences and gradient-free methods.
- Novelty: The method was presented as the first application of the adjoint method to turbine-array optimisation, with expected applicability beyond marine turbines.The paper contrasts this with prior work using simplified models or gradient-free methods.
2. Problem formulation
The turbine-array problem is formulated as a PDE-constrained optimisation over turbine positions and optional tuning parameters, subject to physical, site, and separation constraints. The shallow-water model represents turbines through spatially varying friction, while the objective measures kinetic power extraction.
- 2. Problem formulation: The abstract optimisation maximises J(z, m) subject to the PDE constraint F(z, m)=0, parameter bounds, and additional restrictions g(m).Here, z is the physical state and m contains the design parameters.
- 2.1. The design parameters: The state z=(u, η) solves the shallow-water equations, while m specifies turbine positions and optionally individual tuning parameters.The model uses depth-averaged velocity and free-surface displacement as state variables.
- 2.5. Box and inequality constraints: Turbines may be constrained to a prescribed deployment area and separated by a minimum distance to prevent overlap.Rectangular sites use box constraints, while pairwise distance restrictions enforce turbine separation.
- 2. Problem formulation: The reduced formulation treats z as an implicit function z(m), replacing the coupled problem with optimisation of the objective after solving the PDE for each m.The reduced form is chosen because it is usually preferable for time-dependent governing equations.
- 2.2. The PDE constraint: The physical constraint is the nonlinear shallow-water model with bottom friction, turbine friction, viscosity, gravity, and stationary or non-stationary dynamics.The boundary conditions prescribe inflow velocity, zero outflow free-surface displacement, and no-slip or free-slip conditions elsewhere.
- 2.3. The turbine parameterisation: Each turbine is represented by increased bottom friction over a small area, with a smooth bump function increasing toward its centre and multiplied across x and y.The array friction is the sum of individual turbine friction functions, using one support radius for equal-sized turbines.
- 2.3. The turbine parameterisation: The friction coefficient K can tune individual turbine energy extraction, although velocity cut-in and cut-out behaviour is not included.The paper notes that more sophisticated turbine parameterisations could be added.
- 2.4. The functional of interest: The objective is time-averaged power in the non-stationary case or power from increased friction in the stationary case, representing kinetic rather than electrical power.The objective must be differentiable for gradient-based optimisation; installation, service, and environmental impacts are excluded.
3. Numerical setup
The numerical setup uses a PDE-constrained, gradient-based optimisation loop with SQP and adjoint equations to compute turbine-parameter gradients efficiently. The shallow-water model is discretised with finite elements, with automatic differentiation implementing the discrete adjoint.
- Optimisation algorithm: Each optimisation iteration solves the forward problem, evaluates the functional, computes its gradient, checks termination, and updates the design parameters.
- Optimisation algorithm: The turbine configuration problem is solved using sequential quadratic programming through the SLSQP implementation in SciPy.
- Adjoint gradient computation: The adjoint shallow-water equations propagate information backwards from final time to initial time through adjoint velocity and free-surface variables.
- Implementation: Automatic differentiation derives and implements the discrete adjoint from the forward-model implementation without user intervention.
- Adjoint gradient computation: The adjoint approach computes the functional gradient using one adjoint system followed by a matrix-vector product whose cost is negligible in practice.
- Adjoint gradient computation: The gradient computation is effectively independent of turbine count because the adjoint PDE solve dominates the matrix-vector product cost.
- Forward discretisation: The forward equations use a Taylor-Hood finite element discretisation, with piecewise quadratic velocity and piecewise linear free-surface displacement.
- Forward discretisation: The non-stationary equations use implicit Euler time discretisation, while functionals use the quadrature rules of the finite element formulation.
4. Verification
The forward model is verified with manufactured solutions and convergence tests, while the adjoint gradient is checked using Taylor remainder convergence. The tests recover the expected discretisation orders and second-order Taylor remainder convergence.
- Forward-model verification: The manufactured-solution test adds PDE remainders as source terms so the selected sinusoidal solution becomes exact for the modified system.
- Forward-model verification: The spatial convergence test fixes a small time step so spatial discretisation error dominates the overall discretisation error.
- Forward-model verification: Second-order convergence is observed for the spatial discretisation, matching the order expected from the Taylor-Hood finite element pair.
- Forward-model verification: The temporal convergence test uses a fine mesh so temporal discretisation error dominates the overall discretisation error.
- Forward-model verification: First-order convergence is observed for temporal discretisation, matching the order expected from the implicit Euler method.
- Gradient verification: Second-order Taylor remainder convergence is obtained across random perturbations and numerical examples, supporting correct adjoint and gradient implementation.
5. Examples
The examples optimise tidal turbine layouts, and in one scenario also individual turbine tuning, across idealised stationary and non-stationary flow settings. Results show substantial power gains, with constraints and tuning choices materially affecting the optimised configurations and outputs.
- Experimental setup: 32 turbines are optimised in four idealised scenarios, using 320 m×160 m rectangular sites; scenario 3 also optimises individual turbine tuning.Scenarios include constant or sinusoidal inflow, depending on the case.
- Scenario 1: 76%: farm power increased from 54.5 MW to 95.7 MW in scenario 1 without minimum-distance constraints.The optimised turbines form two ⊐-shaped structures with their open ends facing the inflow.
- Scenario 1: 38%: farm power increased from 54.5 MW to 75.0 MW when turbines were constrained to remain at least 30 m apart.The constrained layout uses a >-shaped inflow alignment, an outflow wall, and staggered turbines to avoid direct wakes.
- Scenario 2: 96%: average power over one tidal cycle increased from 10.5 MW to 20.6 MW in non-stationary scenario 2.The optimised layout resembles scenario 1 but faces the closed basin side.
- Scenario 3: 39%: scenario 3 power increased from 30.9 MW to 42.9 MW when turbine friction coefficients were optimised alongside positions.This was 6% more energy than the optimal position-only solution, with most friction coefficients significantly reduced.
- Additional scenario: 22%: average power over one cycle increased from 48.4 MW to 59.0 MW in the symmetric-domain example.The optimised alignment consists of two distorted ∨ shapes whose open ends face the in-/outflow boundaries.
6. Farm optimisation in the Inner Sound of the Pentland Firth
The Inner Sound optimisation applies the framework to 128- and 256-turbine arrays in a semi-idealised Pentland Firth domain, producing higher extracted power and structured layouts. The comparison is subject to a non-realistic power-curve limitation.
- Computational setup: The optimisation used stationary shallow water equations for a semi-idealised Inner Sound domain with a 2 m turbine-site mesh and 1.5–200 m elements elsewhere.The domain contained 1.25 × 10^6 elements; simulations used increased viscosity and reduced turbine friction to ensure a steady-state solution.
- Optimisation results: 197 iterations increased the 128-turbine array’s power extraction from 600 MW to 741 MW, a 24% increase.The run used 197 gradient evaluations and 349 functional evaluations.
- Optimisation results: 133 iterations increased the 256-turbine array’s power extraction from 607 MW to 804 MW, a 33% increase.The run used 133 gradient evaluations and 185 functional evaluations.
- Optimised structures: Both optimised layouts formed boundary walls, dense eastern and western barrages, and arcing spurs aligned with the flow.Additional barrage columns in the 256-turbine case were staggered to minimise wake shadowing, while spurs were sized so most streamlines crossed two turbine rows.
- Physical interpretation: The southern and northern walls were interpreted as retaining and funnelling flow, while denser southern placement reflected southeastward incoming flow.The proposed mechanisms also motivate the western spurs and the arcing of barrages near the site boundaries.
- Caveat: Power extractions for the 128- and 256-turbine cases are not physically meaningful to compare because the model lacked a realistic power curve.The 128-turbine case reached 3.7 m s^-1 maximum velocity versus 3.0 m s^-1 for 256 turbines; a rated-speed limit could change the comparison.
7. Conclusions
The paper formulates tidal-farm layout as a PDE-constrained optimisation problem and uses adjoint gradients to optimise physically realistic flow models efficiently. It succeeds in idealised scenarios and a realistic-scale Pentland Firth application, while identifying extensions needed for industrial use.
- Contribution: Tidal-farm configuration is formulated as a PDE-constrained optimisation problem, enabling adjoint techniques to evaluate gradients rapidly.The formulation addresses the prohibitive cost of gradient-based optimisation without adjoint methods.
- Contribution: The method accounts for nonlinear geometry–turbine–flow interactions and supports physically realistic flow models with many turbines.Once model inputs are specified, the optimisation is fully automated.
- Results: The algorithm significantly increased extracted power in four idealised scenarios and was successfully applied to a major industrial-interest site with an industrially relevant turbine count.The realistic application used accurate shoreline geometry and a large number of turbines.
- Scope boundary: Industrial application requires realistic tidal forcing, wider-domain head-loss forcing, bathymetry, validation against measurements, improved wake modelling, and a realistic power curve.The authors identify these extensions as ongoing work.
- Availability: The optimisation software and examples are open-source at opentidalfarm.org.The paper states that the source code and all examples are publicly available.