Dr Danny Coles
To date, pre-commercial tidal stream turbine arrays have been installed in the UK [1,2], America  and Canada . For the technology to contribute significantly to future electricity demand, arrays consisting of hundreds of turbines are needed. At this scale, the added drag of the turbines has a significant impact on the surrounding ambient flow field . This effect is described as array-scale blockage. Impacts include a reduction in flow speed through, and downstream of, the array, increases in flow speed around the array, and an increase in free surface elevation upstream of the array. Research has shown that by optimising the locations of turbines within an array, the array scale blockage impacts can be manipulated to increase energy production by up to 30% in some cases, relative to a regular array layout . Such significant increases in energy yield would clearly also reduce the cost of tidal stream energy, so there is much to be gained.
To estimate large-scale array energy yield, regional scale hydrodynamic models that typically span hundreds of kilometres are needed to accurately simulate the resource, and the impacts of energy extraction. These models take considerable time to run, prohibiting manual approaches to array optimisation. It has also been shown that manual approaches to array optimisation often result in a sub-optimal solution, because the optimal array is often not intuitive, due to complex interactions between the turbines, the bathymetry, and local and global blockage effects [5,6]. To overcome this, computational tools have been developed to improve array performance through gradient based optimisation . This approach has been implemented in the past to optimise arrays for power performance. However, optimising for array power alone does not take into account the diminishing returns in yield per turbine as the number of turbines in the array increases. This decrease in the average power per turbine, due to blockage effects, leads to additional turbines being placed in lower flow areas. This necessitates an approach that balances costs associated with adding turbines to an array against the additional yield gained from them, to decide on both an optimal number of turbines and their suitable locations.
In this blog, we present research that for the first time, sets out a method for optimising large arrays for cost of energy, using gradient based optimisation. This work was first presented in the journal article ‘Efficient economic optimisation of large-scale tidal stream arrays’, published in the Applied Energy journal . The research has been led by Zoe Goss, and is a collaboration between Imperial College London and the University of Plymouth.
Optimisation method and results
To design an array that minimises cost of energy, this research has developed the following steps:
1. Build and validate a hydrodynamic model of the region of interest in Thetis .
2. Define the array footprint (i.e. the allowable area for turbines to be installed). In Figure 1, an idealised channel domain is shown, with the array footprint marked by the white dashed rectangle, close to a circular island. The domain is discretised to include relatively small elements along the closed boundary and at the array.
Figure 1. Multi-scale triangular computational mesh across the idealised domain. This is overlaid on a map of depths which increase from 40 m to 3000 m at the open boundaries. An enlarged view of the regular isosceles triangular mesh used within the farm area is shown in white on the right.
3. Perform adjoint optimisation of the array design (i.e. the number of turbines and their positions) for power, using Thetis’ inbuilt gradient based optimisation functionality. Repeat this for a range of break-even power, which constrains the minimum average power per turbine. If a turbine’s performance falls below the break-even power, the optimisation process either moves it to a higher energy position, or removes it from the array. Figure 2 shows the optimal array layout for a range of break-even powers. Figure 3 shows the relationship between the break-even power and the array net average power, power per device and optimal number of turbines.
Figure 2. Array designs optimised with a break even power of (a) 0 MW, with 577 turbines, (b) 0.1 MW with 561 turbines, (c) 0.2 MW with 381 turbines, (d) 0.3 MW with 214 turbines, and (e) 0.4 MW with 89 turbines. The array boundary is shown in white. Areas with maximum turbine density are shown in yellow and no turbines shown in blue.
Figure 3. Variations in the total array power generated and the number of turbines for the optimal design as break-even power is increased. The average power per device (red dashed line) always stays higher than the break-even power (black dashed line line).
4. Take the mean power from each of the optimal array solutions, and interpolate to build an ‘emulator’ that predicts the optimal power that can be achieved over all possible numbers of turbines.
5. Use the emulator to provide average array power and number of turbines as inputs into an economic model that estimates the financial performance indicator of choosing. This could be LCOE, Net Present Value, Internal Rate of Return or Payback Period, for example. Establish the array size (i.e. number of turbines) that optimises the metric of interest.
Figure 4. (a) The emulator prediction for LCOE across the range of number of turbines, and (b) a snapshot around the optimal values. The optimal LCOE for each set of parameter values is shown as a black dot, and the input parameters are chosen to match the pessimistic, typical and optimistic values of LCOE.
6. Perform a full adjoint optimisation of the final, optimal array size chosen to produce a map of the spatial distribution of turbines and validate the predictions of the emulator.
The main benefit of this method is that once the emulator has been generated, the economic metric can be rapidly estimated across all possible numbers of turbines, with the corresponding array power found from the emulator. This negates the need to re-run an expensive optimisation loop in Thetis. The number of turbines that optimises the economic performance can then be obtained.
Additional steps can be added to the method to consider uncertainty in inputs to the economic model. For example, to estimate levelised cost of energy, CAPEX and OPEX figures are needed. In literature, CAPEX and OPEX figures vary widely, as indicated in Table 1. The range of LCOE outputs are shown in Figure 4 for the optimistic, typical and pessimistic cost input cases. Figure 5 shows the reduced range in LCOE estimates (P90-P10 range vs. pessimistic-optimistic range), achieved by performing a Monte Carlo analysis on the cost model.
Table 1. Estimates for the parameters used in the economic models, and the amount they vary .
Figure 5. Predictions for the optimal LCOE that can be achieved for each number of turbines, and the resultant array design that minimises it. Evaluated at the pessimistic, typical, and optimistic scenarios outlined in Table 1, as well as the 𝑃10, 𝑃50 and 𝑃90 values obtained through a Monte Carlo simulation, assuming uniform distributions of uncertain parameters.
Real world applications
A more complex application of the methodology presented here appears in . There it is used to predict the LCOE that can be achieved in the Alderney Race as the installed capacity is increased; this demonstrates the process applied to more complex, realistic sites. Figure 7 shows initial results, with the optimal array design over a range of break-even power in the Alderney Race. Figure 7 shows the relationship between the number of turbines and the estimated LCOE of the array, with consideration for the pessimistic to optimistic cost inputs, and P10 to P90 Monte Carlo analysis.
Figure 6. Optimal array design for break-even power of (a) 500 kW, (b) 400 kW, (c) 300 kW, (d) 200 kW, (e) 100 kW and (f) 0 kW. The number of turbines and net average power for the optimal array design can be seen for each break-even power.
Results show that LCOE is minimised by using approximately 40 turbines. Additional turbines must be located in lower flow regions that increases LCOE. However, once the number of turbines exceeds the optimum number, the rise in LCOE is relatively small, potentially validating the adoption of more turbines. Furthermore, as the industry develops, input costs will reduce from the pessimistic to optimistic range, resulting in significant LCOE reduction, as demonstrated here.
Figure 7. Prediction of optimal LCOE that can be achieved as a function of number of turbines, for the optimistic, pessimistic and typical scenarios outlined in table 7, and the P10, P50 and P90 values obtained through a Monte Carlo simulation assuming uniform errors.
Results in Figure 7 highlight that uncertainty in LCOE modelling remains high as a result of uncertainty in the cost inputs to model, even after implementing a Monte-Carlo analysis. Work is now ongoing to more accurately quantify the CAPEX and OPEX inputs to the LCOE model to further reduce the range in pessimistic to optimistic (or P10 to P90) LCOE estimates. Work is also ongoing to apply the array optimisation method to other real world sites.
For further details, please contact Dr Danny Coles or Zoe Goss:
Dr Danny Coles
 Black & Veatch, 2020, Lessons learnt from MeyGen Phase 1A. Final Summary Report
 Nova Innovation, 2021, Europe Case Study – Shetland Tidal Array, accessed online – 02/06/2021, https://www.novainnovation.com/markets/scotland-shetland-tidal-array/
 Verdant Power, 2021, The RITE project, accessed online – 02/06/2021, https://www.verdantpower.com/projects
 Sustainable Marine Power, 2021, Grand passage Nova Scotia Canada, accessed online – 02/06/2021, https://www.sustainablemarine.com/grand-passage
 Coles DS et al., 2020, The energy yield potential of a large tidal stream turbine array in the Alderney Race, Phil. Trans. R. Soc. A, 378: 20190502
 Funke SW et al., 2014, Tidal turbine array optimisation using the adjoint approach, Renewable Energy, 63:658-673
 Goss ZL et al., 2021, Efficient economic optimisation of large-scale tidal stream arrays, Applied Energy, 295:116975
 The Thetis Project, available online: https://thetisproject.org/
 Goss ZL et al., 2021, Economic analysis of tidal stream turbine arrays: a review, available online: https://arxiv.org/abs/2105.04718
 Goss ZL et al., 2020, Identifying economically viable tidal sites within the Alderney Race through optimization of levelized cost of energy, Phil. Trans. R. Soc. A, 378: 20190500