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In this work, we assess global offshore wind energy resources, wake-induced losses, array layout optimisation potential and climate change impacts. We first map global offshore ambient wind resource from reanalysis data. We estimate wake-induced losses using an engineering wake model, revealing that locations with low (high) resource typically experience larger (smaller) percentage losses. However, we further find that the specific wind speed distribution is important, with narrower distributions generally leading to greater losses. This is due to the overlap between the wind speed distribution and the high-sensitivity region of the turbine thrust and power curves. Broadly, this leads to much stronger wake-induced losses in the tropics (which experience the trade winds) than mid-latitudes. However, the tropics also experience a narrower wind direction distribution; our results demonstrate that this leads to greater potential for mitigation of wake effects via layout optimisation. Finally, we assess projected changes in wind resource and wake losses due to climate change under a high-emission scenario. Many regions are projected to decrease in ambient wind resources, and furthermore these regions will typically experience greater wake-induced losses, exacerbating the climate impact. These results highlight the different challenges and opportunities associated with exploiting offshore wind resources across the globe.

Rapid deployment of offshore wind is expected within the coming decades to help meet climate goals. With offshore wind turbine lifetimes of 25-30 years, and new offshore leases spanning 60 years, it is vital to consider long-term changes in potential wind power resource at the farm planning stage. Such changes may arise from multiple sources, including climate change, and increasing wake-induced power losses. In this work, we investigate and compare these two sources of long-term change in wind power, for a case study consisting of 21 wind farms within the German Bight. Consistent with previous studies, we find a small but significant reduction in wind resource due to climate change by the end of the 21st century under the high-emission RCP8.5 scenario, compared with a historical period, with a mean power reduction (over an ensemble of seven climate models) of 2.1%. To assess the impact of wake-induced losses due to increasingly dense farm build-out, we model wakes within the German Bight region using an engineering wake model, under various stages of (planned) build-out corresponding to the years 2010-2027. By identifying clusters of wind farms, we decompose wake effects into long-range (inter-cluster), medium-range (intra-cluster) and short-range (intra-farm) effects. Inter-cluster wake-induced losses increase from 0 for the 2010 scenario to 2.5% for the 2027 scenario, with intra-cluster losses also increasing from 0 to 4.3%. Intra-farm losses are relatively constant, at around 13%. While the evolution of wake effects therefore outweighs the climate effect, and impacts over a shorter timescale, both factors are significant. We also find evidence of non-linear interactions between the climate and wake effects. Both climate change and evolving wake effects must therefore be considered within resource assessment and wind farm planning.

Solving complex Partial Differential Equations (PDEs) accurately and efficiently is an essential and challenging problem in all scientific and engineering disciplines. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without increasing the overall mesh degree of freedom count. Conventional sophisticated mesh movement methods are extremely expensive and struggle to handle scenarios with complex boundary geometries. However, existing learning-based methods require re-training from scratch given a different PDE type or boundary geometry, which limits their applicability, and also often suffer from robustness issues in the form of inverted elements. In this paper, we introduce the Universal Mesh Movement Network (UM2N), which -- once trained -- can be applied in a non-intrusive, zero-shot manner to move meshes with different size distributions and structures, for solvers applicable to different PDE types and boundary geometries. UM2N consists of a Graph Transformer (GT) encoder for extracting features and a Graph Attention Network (GAT) based decoder for moving the mesh. We evaluate our method on advection and Navier-Stokes based examples, as well as a real-world tsunami simulation case. Our method outperforms existing learning-based mesh movement methods in terms of the benchmarks described above. In comparison to the conventional sophisticated Monge-Amp\`ere PDE-solver based method, our approach not only significantly accelerates mesh movement, but also proves effective in scenarios where the conventional method fails. Our project page is at https://erizmr.github.io/UM2N/.

This work proposes a novel data-driven model capable of providing accurate predictions for the power generation of all wind turbines in wind farms of arbitrary layout, yaw angle configurations and wind conditions. The proposed model functions by encoding a wind farm into a fully-connected graph and processing the graph representation through a graph transformer. The graph transformer surrogate is shown to generalise well and is able to uncover latent structural patterns within the graph representation of wind farms. It is demonstrated how the resulting surrogate model can be used to optimise yaw angle configurations using genetic algorithms, achieving similar levels of accuracy to industrially-standard wind farm simulation tools while only taking a fraction of the computational cost.

Wind farm modelling has been an area of rapidly increasing interest with numerous analytical as well as computational-based approaches developed to extend the margins of wind farm efficiency and maximise power production. In this work, we present the novel ML framework WakeNet, which can reproduce generalised 2D turbine wake velocity fields at hub-height over a wide range of yaw angles, wind speeds and turbulence intensities (TIs), with a mean accuracy of 99.8% compared to the solution calculated using the state-of-the-art wind farm modelling software FLORIS. As the generation of sufficient high-fidelity data for network training purposes can be cost-prohibitive, the utility of multi-fidelity transfer learning has also been investigated. Specifically, a network pre-trained on the low-fidelity Gaussian wake model is fine-tuned in order to obtain accurate wake results for the mid-fidelity Curl wake model. The robustness and overall performance of WakeNet on various wake steering control and layout optimisation scenarios has been validated through power-gain heatmaps, obtaining at least 90% of the power gained through optimisation performed with FLORIS directly. We also demonstrate that when utilising the Curl model, WakeNet is able to provide similar power gains to FLORIS, two orders of magnitude faster (e.g. 10 minutes vs 36 hours per optimisation case). The wake evaluation time of wakeNet when trained on a high-fidelity CFD dataset is expected to be similar, thus further increasing computational time gains. These promising results show that generalised wake modelling with ML tools can be accurate enough to contribute towards active yaw and layout optimisation, while producing realistic optimised configurations at a fraction of the computational cost, hence making it feasible to perform real-time active yaw control as well as robust optimisation under uncertainty., Comment: 16 Pages, 18 Figures, 3 Tables

Wind energy resource assessment typically requires numerical models, but such models are too computationally intensive to consider multi-year timescales. Increasingly, unsupervised machine learning techniques are used to identify a small number of representative weather patterns to simulate long-term behaviour. Here we develop a novel wind energy workflow that for the first time combines weather patterns derived from unsupervised clustering techniques with numerical weather prediction models (here WRF) to obtain efficient and accurate long-term predictions of power and downstream wakes from an entire wind farm. We use ERA5 reanalysis data clustering not only on low altitude pressure but also, for the first time, on the more relevant variable of wind velocity. We also compare the use of large-scale and local-scale domains for clustering. A WRF simulation is run at each of the cluster centres and the results are aggregated using a novel post-processing technique. By applying our workflow to two different regions, we show that our long-term predictions agree with those from a year of WRF simulations but require less than 2% of the computational time. The most accurate results are obtained when clustering on wind velocity. Moreover, clustering over the Europe-wide domain is sufficient for predicting wind farm power output, but downstream wake predictions benefit from the use of smaller domains. Finally, we show that these downstream wakes can affect the local weather patterns. Our approach facilitates multi-year predictions of power output and downstream farm wakes, by providing a fast, accurate and flexible methodology that is applicable to any global region. Moreover, these accurate long-term predictions of downstream wakes provide the first tool to help mitigate the effects of wind energy loss downstream of wind farms, since they can be used to determine optimum wind farm locations., Comment: 18 pages, 18 figures

Wind turbine wake modelling is of crucial importance to accurate resource assessment, to layout optimisation, and to the operational control of wind farms. This work proposes a surrogate model for the representation of wind turbine wakes based on a state-of-the-art graph representation learning method termed a graph neural network. The proposed end-to-end deep learning model operates directly on unstructured meshes and has been validated against high-fidelity data, demonstrating its ability to rapidly make accurate 3D flow field predictions for various inlet conditions and turbine yaw angles. The specific graph neural network model employed here is shown to generalise well to unseen data and is less sensitive to over-smoothing compared to common graph neural networks. A case study based upon a real world wind farm further demonstrates the capability of the proposed approach to predict farm scale power generation. Moreover, the proposed graph neural network framework is flexible and highly generic and as formulated here can be applied to any steady state computational fluid dynamics simulations on unstructured meshes., Comment: 20 pages, 12 figures

Given a partial differential equation (PDE), goal-oriented error estimation allows us to understand how errors in a diagnostic quantity of interest (QoI), or goal, occur and accumulate in a numerical approximation, for example using the finite element method. By decomposing the error estimates into contributions from individual elements, it is possible to formulate adaptation methods, which modify the mesh with the objective of minimising the resulting QoI error. However, the standard error estimate formulation involves the true adjoint solution, which is unknown in practice. As such, it is common practice to approximate it with an 'enriched' approximation (e.g. in a higher order space or on a refined mesh). Doing so generally results in a significant increase in computational cost, which can be a bottleneck compromising the competitiveness of (goal-oriented) adaptive simulations. The central idea of this paper is to develop a "data-driven" goal-oriented mesh adaptation approach through the selective replacement of the expensive error estimation step with an appropriately configured and trained neural network. In doing so, the error estimator may be obtained without even constructing the enriched spaces. An element-by-element construction is employed here, whereby local values of various parameters related to the mesh geometry and underlying problem physics are taken as inputs, and the corresponding contribution to the error estimator is taken as output. We demonstrate that this approach is able to obtain the same accuracy with a reduced computational cost, for adaptive mesh test cases related to flow around tidal turbines, which interact via their downstream wakes, and where the overall power output of the farm is taken as the QoI. Moreover, we demonstrate that the element-by-element approach implies reasonably low training costs., Comment: 27 pages, 14 figures

Extracting information on fluid motion directly from images is challenging. Fluid flow represents a complex dynamic system governed by the Navier-Stokes equations. General optical flow methods are typically designed for rigid body motion, and thus struggle if applied to fluid motion estimation directly. Further, optical flow methods only focus on two consecutive frames without utilising historical temporal information, while the fluid motion (velocity field) can be considered a continuous trajectory constrained by time-dependent partial differential equations (PDEs). This discrepancy has the potential to induce physically inconsistent estimations. Here we propose an unsupervised learning based prediction-correction scheme for fluid flow estimation. An estimate is first given by a PDE-constrained optical flow predictor, which is then refined by a physical based corrector. The proposed approach outperforms optical flow methods and shows competitive results compared to existing supervised learning based methods on a benchmark dataset. Furthermore, the proposed approach can generalize to complex real-world fluid scenarios where ground truth information is effectively unknowable. Finally, experiments demonstrate that the physical corrector can refine flow estimates by mimicking the operator splitting method commonly utilised in fluid dynamical simulation., Comment: published at ICML 2022

Neural networks are increasingly being used in a variety of settings to predict wind direction and speed, two of the most important factors for estimating the potential power output of a wind farm. However, these predictions are arguably of limited value because classical neural networks lack the ability to express uncertainty. Here we instead consider the use of Bayesian Neural Networks (BNNs), for which the weights, biases and outputs are distributions rather than deterministic point values. This allows for the evaluation of both epistemic and aleatoric uncertainty and leads to well-calibrated uncertainty predictions of both wind speed and power. Here we consider the application of BNNs to the problem of offshore wind resource prediction for renewable energy applications. For our dataset, we use observations recorded at the FINO1 research platform in the North Sea and our predictors are ocean data such as water temperature and current direction. The probabilistic forecast predicted by the BNN adds considerable value to the results and, in particular, informs the user of the network's ability to make predictions of out-of-sample datapoints. We use this property of BNNs to conclude that the accuracy and uncertainty of the wind speed and direction predictions made by our network are unaffected by the construction of the nearby Alpha Ventus wind farm. Hence, at this site, networks trained on pre-farm ocean data can be used to accurately predict wind field information from ocean data after the wind farm has been constructed., Comment: 8 pages, 6 figures

Mainstream numerical Partial Differential Equation (PDE) solvers require discretizing the physical domain using a mesh. Mesh movement methods aim to improve the accuracy of the numerical solution by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, mesh movement methods, such as the Monge-Ampere method, require the solution of auxiliary equations, which can be extremely expensive especially when the mesh is adapted frequently. In this paper, we propose to our best knowledge the first learning-based end-to-end mesh movement framework for PDE solvers. Key requirements of learning-based mesh movement methods are alleviating mesh tangling, boundary consistency, and generalization to mesh with different resolutions. To achieve these goals, we introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline based model provides more flexibility for large deformation, the GAT based model can handle domains with more complicated shapes and is better at performing delicate local deformation. We validate our methods on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampere method, our approach can greatly accelerate the mesh adaptation process, whilst achieving comparable numerical error reduction.

This research note documents the integration of the MPI-parallel metric-based mesh adaptation toolkit ParMmg into the solver library PETSc. This coupling brings robust, scalable anisotropic mesh adaptation to a wide community of PETSc users, as well as users of downstream packages. We demonstrate the new functionality via the solution of Poisson problems in three dimensions, with both uniform and spatially-varying right-hand sides., Comment: 5 pages, 2 figures. Appeared as a research note in the 30th International Meshing Roundtable

Particle Image Velocimetry (PIV) is a classical flow estimation problem which is widely considered and utilised, especially as a diagnostic tool in experimental fluid dynamics and the remote sensing of environmental flows. Recently, the development of deep learning based methods has inspired new approaches to tackle the PIV problem. These supervised learning based methods are driven by large volumes of data with ground truth training information. However, it is difficult to collect reliable ground truth data in large-scale, real-world scenarios. Although synthetic datasets can be used as alternatives, the gap between the training set-ups and real-world scenarios limits applicability. We present here what we believe to be the first work which takes an unsupervised learning based approach to tackle PIV problems. The proposed approach is inspired by classic optical flow methods. Instead of using ground truth data, we make use of photometric loss between two consecutive image frames, consistency loss in bidirectional flow estimates and spatial smoothness loss to construct the total unsupervised loss function. The approach shows significant potential and advantages for fluid flow estimation. Results presented here demonstrate that our method outputs competitive results compared with classical PIV methods as well as supervised learning based methods for a broad PIV dataset, and even outperforms these existing approaches in some difficult flow cases. Codes and trained models are available at https://github.com/erizmr/UnLiteFlowNet-PIV.

Depth-averaged shallow water models are widely used for the large-scale simulation of tidal turbine arrays. The relatively low computational complexity of this approach allows for layout optimisations aimed at improving the total array power output as well as an assessment of large-scale environmental impacts. In order to assess the suitability of using depth-averaged models to optimise array configurations, a comprehensive comparison between the wake profiles and power outputs predicted by a 2D shallow water model and a 3D actuator disc momentum (ADM) model is presented. Initially, a viscosity sensitivity analysis is presented to outline the limitations associated with using a constant eddy viscosity in the depth-averaged model and to outline the importance of correctly calibrating this value in line with the freestream velocity magnitude. Thereafter, the depth-averaged OpenTidalFarm (OTF) tool is used to optimise the positions of an array of 32 turbines in an ideal channel and the 3D Fluidity ADM-RANS model is used to assess the accuracy of the OTF predictions for the first time. It is shown that with the help of corrected power calculations a good agreement between the two models can be achieved, thus demonstrating the value of the eddy viscosity calibration implemented in the depth-averaged model.

Turbulence-resolving simulations of wind turbine wakes are presented using a high--order flow solver combined with both a standard and a novel dynamic implicit spectral vanishing viscosity (iSVV and dynamic iSVV) model to account for subgrid-scale (SGS) stresses. The numerical solutions are compared against wind tunnel measurements, which include mean velocity and turbulent intensity profiles, as well as integral rotor quantities such as power and thrust coefficients. For the standard (also termed static) case the magnitude of the spectral vanishing viscosity is selected via a heuristic analysis of the wake statistics, while in the case of the dynamic model the magnitude is adjusted both in space and time at each time step. The study focuses on examining the ability of the two approaches, standard (static) and dynamic, to accurately capture the wake features, both qualitatively and quantitatively. The results suggest that the static method can become over-dissipative when the magnitude of the spectral viscosity is increased, while the dynamic approach which adjusts the magnitude of dissipation locally is shown to be more appropriate for a non-homogeneous flow such that of a wind turbine wake.

Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models., Comment: Submitted to Geoscientific Model Development

Numerical models of the flow and wakes due to turbines operating within a real-scale offshore wind farm can lead to a prohibitively large computational cost, particularly when considering blade-resolved simulations. With the introduction of turbine parametrizations such as the actuator disk (AD) or the actuator line (AL) models, this problem has been partially addressed, yet the computational cost associated with these simulations remains high. In this work, we present an implementation and validation of an AL model within the mesh-adaptive three-dimensional fluid dynamics solver, Fluidity, under a unsteady Reynolds-averaged Navier-Stokes based turbulence modelling approach. A key feature of this implementation is the use of mesh optimization techniques, which allow for the automatic refinement or coarsening of the mesh locally according to the resolution needed by the fluid flow solver. The model is first validated against experimental data from wind tunnel tests. Finally, we demonstrate the benefits of mesh-adaptivity by considering flow past the Lillgrund offshore wind farm.

Despite decades of research in this area, mesh adaptation capabilities are still rarely found in numerical simulation software. We postulate that the primary reason for this is lack of usability. Integrating mesh adaptation into existing software is difficult as non-trivial operators, such as error metrics and interpolation operators, are required, and integrating available adaptive remeshers is not straightforward. Our approach presented here is to first integrate Pragmatic, an anisotropic mesh adaptation library, into DMPlex, a PETSc object that manages unstructured meshes and their interactions with PETSc's solvers and I/O routines. As PETSc is already widely used, this will make anisotropic mesh adaptation available to a much larger community. As a demonstration of this we describe the integration of anisotropic mesh adaptation into Firedrake, an automated Finite Element based system for the portable solution of partial differential equations which already uses PETSc solvers and I/O via DMPlex. We present a proof of concept of this integration with a three-dimensional advection test case., Comment: 5 page, 2 figures, Proceedings of the 25th International Meshing Roundtable, ed. Steve Owen and Hang Si, 2016

Extracting the optimal amount of power from an array of tidal turbines requires an intricate understanding of tidal dynamics and the effects of turbine placement on the local and regional scale flow. Numerical models have contributed significantly towards this understanding, and more recently, adjoint-based modelling has been employed to optimise the positioning of the turbines in an array in an automated way and improve on simple, regular man-made configurations. Adjoint-based optimisation of high-resolution and ideally 3D transient models is generally a very computationally expensive problem. As a result, existing work on the adjoint optimisation of tidal turbine placement has been mostly limited to steady-state simulations in which very high, non-physical values of the background viscosity are required to ensure that a steady-state solution exists. However, such compromises may affect the reliability of the modelled turbines, their wakes and interactions, and thus bring into question the validity of the computed optimal turbine positions. This work considers a suite of idealised simulations of flow past tidal turbine arrays in a 2D channel. It compares four regular array configurations, detailed by Divett et al. (2013), with the configuration found through adjoint optimisation in a steady-state, high-viscosity setup. The optimised configuration produces considerably more power. The same configurations are then used to produce a suite of transient simulations that do not use constant high-viscosity, and instead use large eddy simulation (LES) to parameterise the resulting turbulent structures. It is shown that the LES simulations produce less power than that predicted by the constant high-viscosity runs. Nevertheless, they still follow the same trends in the power curve throughout time, with optimised layouts continuing to perform significantly better than simplified configurations., Comment: Conference paper comprising 15 pages and 13 figures. Submitted to the Proceedings of the ECCOMAS Congress 2016 (VII European Congress on Computational Methods in Applied Sciences and Engineering), held in Crete, Greece on 5-10 June 2016

Due to the fractal nature of the domain geometry in geophysical flow simulations, a completely accurate description of the domain in terms of a computational mesh is frequently deemed infeasible. Shoreline and bathymetry simplification methods are used to remove small scale details in the geometry, particularly in areas away from the region of interest. To that end, a novel method for shoreline and bathymetry simplification is presented. Existing shoreline simplification methods typically remove points if the resultant geometry satisfies particular geometric criteria. Bathymetry is usually simplified using traditional filtering techniques, that remove unwanted Fourier modes. Principal Component Analysis (PCA) has been used in other fields to isolate small-scale structures from larger scale coherent features in a robust way, underpinned by a rigorous but simple mathematical framework. Here we present a method based on principal component analysis aimed towards simplification of shorelines and bathymetry. We present the algorithm in detail and show simplified shorelines and bathymetry in the wider region around the North Sea. Finally, the methods are used in the context of unstructured mesh generation aimed at tidal resource assessment simulations in the coastal regions around the UK., Comment: Pre-print of conference publication accepted in the Proceedings of 11th European Wave & Tidal Energy Conference (EWTEC 2015, http://www.ewtec.org/ewtec2015/ ). This paper was presented at the EWTEC 2015 conference on Tuesday 8 September 2015 in Nantes, France. Number of pages: 7. Number of figures: 6

Ocean modelling requires the production of high-fidelity computational meshes upon which to solve the equations of motion. The production of such meshes by hand is often infeasible, considering the complexity of the bathymetry and coastlines. The use of Geographical Information Systems (GIS) is therefore a key component to discretising the region of interest and producing a mesh appropriate to resolve the dynamics. However, all data associated with the production of a mesh must be provided in order to contribute to the overall recomputability of the subsequent simulation. This work presents the integration of research data management in QMesh, a tool for generating meshes using GIS. The tool uses the PyRDM library to provide a quick and easy way for scientists to publish meshes, and all data required to regenerate them, to persistent online repositories. These repositories are assigned unique identifiers to enable proper citation of the meshes in journal articles., Comment: Accepted, camera-ready version. To appear in the Proceedings of the 5th International Workshop on Semantic Digital Archives (http://sda2015.dke-research.de/), held in Pozna\'n, Poland on 18 September 2015 as part of the 19th International Conference on Theory and Practice of Digital Libraries (http://tpdl2015.info/)

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.

Hydrodynamic modelling is an important tool for the development of tidal stream energy projects. Many hydrodynamic models incorporate the effect of tidal turbines through an enhanced bottom drag. In this paper we show that although for coarse grid resolutions (kilometre scale) the resulting force exerted on the flow agrees well with the theoretical value, the force starts decreasing with decreasing grid sizes when these become smaller than the length scale of the wake recovery. This is because the assumption that the upstream velocity can be approximated by the local model velocity, is no longer valid. Using linear momentum actuator disc theory however, we derive a relationship between these two velocities and formulate a correction to the enhanced bottom drag formulation that consistently applies a force that remains closed to the theoretical value, for all grid sizes down to the turbine scale. In addition, a better understanding of the relation between the model, upstream, and actual turbine velocity, as predicted by actuator disc theory, leads to an improved estimate of the usefully extractable energy. We show how the corrections can be applied (demonstrated here for the models MIKE 21 and Fluidity) by a simple modification of the drag coefficient.