Identifying economically viable tidal sites within the Alderney Race through optimization of levelized cost of energy

Abstract

Costs of tidal stream energy generation are anticipated to fall considerably with array expansion and time. This is due to both economies of volume, where arrays comprising of large numbers of turbines can split fixed costs over a greater number of devices, and learning rates, where the industry matures and so arrays of the same size become cheaper due to lessons learned from previous installations. This paper investigates how tidal energy arrays can be designed to minimize the levelized cost of energy (LCOE), by optimizing not only the location but also the number of devices, to find a suitable balance between decreased costs due to economies of volume and diminishing returns due to global blockage effects. It focuses on the Alderney Race as a case study site due to the high velocities found there, making it a highly suitable site for large-scale arrays. It is demonstrated that between 1 and 2 GW could be feasibly extracted as costs in the tidal industry fall, with the LCOE depending greatly on the assumed costs. A Monte–Carlo analysis is undertaken to account for variability in capital and operational cost data used as inputs to the array optimization. Once optimized, the estimated P50 LCOE of an 80 MW array is £110/MWh. This estimate aligns closely with the level of subsidy considered for tidal stream projects in the Alderney Race in the past.

This article is part of the theme issue ‘New insights on tidal dynamics and tidal energy harvesting in the Alderney Race’.

1. Introduction

The Alderney Race contains a significant tidal energy resource, with a maximum average power potential of 5.1 GW and large regions of the Race exhibiting velocities of up to 5 m s⁻¹ [1]. In order to develop tidal stream energy projects in the Alderney Race, there is a need to understand how the resource can be harnessed most cost-effectively. Through phased array development, the cost of energy can be reduced as the industry matures and array deployments within the Alderney Race expand.

Newly developed methods for optimizing the placement of tidal stream turbines within arrays have demonstrated the potential to increase array efficiency (energy yield per turbine) by up to 100% in some cases, thereby providing an avenue for further cost of energy reduction [2]. This gradient-based optimization approach is implemented within a full hydrodynamics solver in order to link the changes to the hydrodynamics caused by the iterative movement of turbines within the optimization to the resulting power of the array. This is a critical requirement when optimizing large arrays in order to account for array scale blockage, as has been demonstrated in [3].

In this paper, we implement array optimization in the Alderney Race to minimize the levelized cost of energy (LCOE). We implement a proxy for learning rates and economies of volume based on information in the literature in order to propose optimal phased array development in the Alderney Race, where these cost reductions unlock ongoing array development. The array optimization is implemented within the Thetis hydrodynamic model, which is described in §2. The model is validated using four bed-mounted ADCP datasets and tidal gauge data from around the model domain (§3). The optimization approach is described in §4a. Results from the optimization are presented and discussed in §4b.

2. Hydrodynamic model

The nonlinear shallow water equations are discretized using the finite-element method, via the flexible coastal ocean modelling software Thetis [4,5], implemented within the Firedrake [6] finite-element code generation framework. The P_1DG − P_1DG velocity-pressure finite-element pair is used for spatial discretization. The semi-implicit Crank–Nicolson time-stepping method is used for temporal discretization, with a constant time step of Δt = 600 s. These discretization options are second-order accurate in space and time, as verified for the three-dimensional version of the Thetis model in [7] and for the two-dimensional depth-averaged version (which is the one used in this work) in [8]. The model domain, which covers the majority of the English Channel, is shown in figure 1. The model is forced by Q1, O1, P1, K1, M2, S2, N2, K2 and M4 constituent elevation forcing at the open boundaries, implemented via the Uptide¹ package using data extracted from the OSU Tidal Prediction Software (OTPS) [10]. Thetis solves the non-conservative form of the nonlinear shallow water equations;

$$\frac{\mathrm{\partial }\eta }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (H\mathbf{u})=0$$ 2.1

and

$$\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}-\nu {\mathrm{\nabla }}^2\mathbf{u}+f{\mathbf{u}}^{\perp }+g\mathrm{\nabla }\eta =-\frac{{\tau }_b}{\rho H},$$ 2.2

where η is the free surface perturbation, t is time, H is the total water depth, fu^⊥ is the Coriolis forcing described by equation (2.4), where u^⊥ is the velocity vector rotated counter-clockwise 90°, u = (u, v) is the depth-averaged velocity vector, ν is the kinematic viscosity of the fluid, g is acceleration due to gravity, τ_b is the bed friction and ρ is the water density. Over the majority of the domain the kinematic viscosity is set to a value of 10 m² s⁻¹. This artificially increased value acts primarily as a stabilization mechanism and accounts for scales of motion not resolved at the mesh resolutions used. At the open boundaries, the viscosity is increased to a value of 1000 m² s⁻¹, over a region extending 50km from the open boundaries. This acts as a further stabilization mechanism for any spurious flow behaviour that can be generated through minor inconsistencies (e.g. due to different resolution and bathymetry data employed) between our model and set-up, and the configuration used to generate the tidal harmonic forcing data. This has negligible impact on the flow speeds within areas of interest for energy extraction. The location of the open boundaries are also chosen, in part, to be far enough away from the array location that changes in hydrodynamics do not propagate significantly back to the open boundaries, which would invalidate the assumption of unchanged boundary forcing. Justification of the selection of the time step and kinematic viscosity described above is given through a sensitivity test in §3bi.

Figure 1.

The unstructured triangular mesh used in the shallow water English Channel model, with the locations of the four ADCPs (labelled pointers) and the 68 tide gauges (unlabelled pointers) used to validate the model [9]. The farm area considered is also shown within the Alderney Race. (Online version in colour.)

The bottom friction, τ_b, is calculated using the Manning formulation;

$$\frac{{\tau }_b}{\rho }=g{n}^2\frac{||\mathbf{u}||\mathbf{u}}{H^{1/3}},$$ 2.3

where n is the Manning coefficient, which is uniformly applied over the whole domain and tuned during model calibration in §3biii.

The Coriolis forcing, fu^⊥, is represented by the beta-plane approximation, due to the size of the domain, such that;

$$f=f_0+\beta y,$$ 2.4

where the Coriolis and Rossby parameters are given by

$$f_0=2\mathrm{\Omega }\sin (\zeta )\text{and}\beta =\frac{1}{R}2\mathrm{\Omega }\cos (\zeta ),$$ 2.5

respectively, where Ω is the angular frequency of the Earth’s rotation, ζ the latitude and R is the Earth’s radius.

Additional meteorological forcings such as wind and atmospheric pressure have not been included here. Some studies have found that the effects of wind-driven waves can have a notable impact on the tidal power extraction, especially in extreme and winter conditions [11,12]. However, tide-induced currents are the dominant forcing for tidal current estimation, especially at the depths concerned for tidal energy extraction [13]. For this reason, the exclusion of these additional meteorological forcings is common in regional scale hydrodynamic modelling work, such as [1,14–16].

Bathymetry data were obtained from the Marine Digimap database [17], with 1 arc-second resolution (≈30 m) over the northern half of the Channel and 6 arc-second resolution (≈180 m) over the southern half of the Channel. There is a small region around Normandy that is not covered by the Marine Digimap database. For this region bathymetry was obtained from the General Bathymetric Chart of the Oceans (GEBCO) 2014 dataset with 30 arc-second resolution (≈900 m) [18].

Coastlines were adapted from the global self-consistent, hierarchical, high-resolution geography database (GSHHG) [19]. Coastline geometry has been simplified to remove estuaries and islands (except for the islands closest to Alderney). This was done to reduce computational expense as it enables lower mesh resolution to be used around the coastlines of the domain while having minimal impact on the region of interest [20].

The model employs an unstructured triangular mesh, allowing variable mesh resolution across the domain, as shown in figure 1. A mesh independence study was carried out to establish the most suitable mesh resolution to achieve accurate ambient flow results at acceptable computational expense. Table 1 summarizes the resolution and number of nodes/elements in the four meshes considered in the mesh independence study.

Table 1.

Resolution and node/element count of meshes used in the mesh independence study.

	tidal plot	Alderney Race	shorelines	rest of domain	nodes	elements
Mesh 1	1000 m	2000 m	4000 m	10 000 m	2876	5760
Mesh 2	750 m	1500 m	3000 m	10 000 m	4086	8180
Mesh 3	500 m	1000 m	2000 m	10 000 m	7126	14 260
Mesh 4	375 m	750 m	1500 m	10 000 m	11 485	22 978

Results from the mesh independence study are presented in §3bii, demonstrating that mesh independence was achieved with Mesh 3. Mesh 3 has a resolution of 500 m within the tidal energy plots in the Alderney Race, 2000 m around coastlines and 10 000 m within the rest of the domain. The mesh comprises of 14 260 unstructured triangular elements and 7126 nodes.

3. Calibration and validation

(a). Methods

The Thetis model was calibrated by varying the bed drag to establish the value of the Manning coefficient that achieves the highest level of agreement with field measurements of flow speed and tidal elevation. In similar tidal calibration studies (e.g. [21]) non-dimensional quadratic drag coefficients, c_D, are typically in the range 0.0025—0.0075. The quadratic drag coefficient can be converted to a Manning coefficient using [22];

$$c_D=g{n}^2{h}^{-1/3},$$ 3.1

where g is acceleration due to gravity, n is the Manning coefficient and h = 35m is the representative depth in the Alderney Race. This results in a Manning coefficient range of approximately 0.03 to 0.05 sm^−1/3. In this study, five values of Manning coefficient were considered: 0.02, 0.025, 0.03, 0.035 and 0.04 sm^−1/3. Section 3a provides a description of the methods used to carry out the model calibration and validation. Results from the calibration and validation studies are presented in §3b and 3c.

(i). Flow speeds

An industry partner provided data from three bed-mounted ADCP campaigns within the Alderney Race, which were used to calibrate the model. The ADCP datasets provide 10 min averages of flow speed and direction within 1 m vertical bins spanning the majority of the water column. The datasets cover a one-month period. The flow speed and directions obtained from the ADCPs were depth averaged to compare against simulated results.

Five separate simulations were run using the aforementioned Manning coefficients. The model accuracy was quantified using three different metrics. The normalized root mean squared error (NRMSE) provides a comparison between simulated and measured flow speeds;

$$\text{NRMSE}=\frac{1}{max({\text{mes}}_i)}\sqrt{\frac{1}{N}\sum _{i=1}^{i=N}{({\text{mod}}_i-{\text{mes}}_i)}^2},$$ 3.2

where mes_i is the measured data obtained from the bed-mounted ADCPs at time step i and mod_i is the model prediction at time step i.

The index of agreement (IA), also known as the Willmott Index [23], is used as a relative covariability of the model predictions and ADCP observations about an estimate of the ‘true’ mean. Bias (B) and relative bias (RB) are also used to quantify the systematic error of the model in simulating the flow. This is consistent with the approach taken in [24,25]. These quantities are defined mathematically as

$$\text{IA}=1-\frac{\sum _{i=1}^{i=N}{({\text{mod}}_i-{\text{mes}}_i)}^2}{\sum _{i=1}^{i=N}{(|{\text{mod}}_i-\overline{{\text{mes}}_i}|+|{\text{mes}}_i-\overline{{\text{mes}}_i}|)}^2}$$ 3.3

and

$$B=\overline{{\text{mod}}_i}-\overline{{\text{mes}}_i},RB=100\frac{B}{\overline{{\text{mes}}_i}}.$$ 3.4

The model was validated against an additional sea bed-mounted ADCP dataset (ADCP 4) obtained in the Alderney Race [26]. This dataset covered a period of 14.6 days, providing flow speeds and directions at hourly intervals. The locations of the four ADCP deployments in the Alderney Race are shown in figure 1.

(ii). Tidal elevations

Free surface elevation data were extracted from 68 locations around the domain to compare against tide gauge data. The tide gauge locations are shown in figure 1 [9]. For each of the Manning coefficients used to tune the model, the Thetis simulation was run over a two-month period and the elevation predictions at each of the tide gauge and ADCP locations were extracted. Least squared regression was performed on the η time signal, via the Uptide python package, to calculate the model phase and amplitude for each of the harmonic constituents in the tidal forcing. These were then compared either directly to the tide gauge phase and amplitude readings or to phase and amplitudes calculated in the same manner from elevation time series readings at the location of each of the ADCPs. The model amplitude error, 100 · (amp_mod − amp_mes)/amp_mes, and phase error, (pha_mod − pha_mes)°, for the two most dominant constituents in the region, M2 and S2, are presented below.

The time period of the signal must be long enough to distinguish each pair of constituents, and this required period can be calculated in Uptide using the Rayleigh Criterion. Most pairs can be resolved within a month, however, S2 & K2 and P1 & K1 require much longer to resolve, and so, for the purpose of harmonic validation, the latter of each pair (the constituent which is smaller in magnitude in the region) was removed. All nine constituents were included in models for all other purposes except harmonic validation.

(b). Calibration results

(i). Model stability

The impact of varying both the kinematic viscosity and time step were investigated using a Manning coefficient value of 0.03 sm^−1/3. Increasing the time step through Δt = 30, 60, 300, 600, 1200 s resulted in an average velocity magnitude NRMSE for ADCPs 1–3 of 11.7%, 11.5%, 11.8%, 11.8% and 12.5%, respectively. A time step of 600 s was selected for future runs since (a) it was computationally prohibitive to use a smaller time step within the adjoint optimization loop, and (b) only minor changes in model results where exhibited for the time step range considered when compared with ADCP data.

ADCP 4 was used to test the sensitivity of the model to changes in the kinematic viscosity, using ν = 1, 10, 100, 1000 m² s⁻¹. The ν = 1 m² s⁻¹ case resulted in model instability. The remaining kinematic viscosity cases resulted in a velocity magnitude NRMSE of 10.2 %, 10.8 % and 12.3 %, respectively. A viscosity of ν = 10 m² s⁻¹ was selected since it provided the lowest NRMSE in velocity while also providing model stability.

(ii). Mesh sensitivity

Figure 2a compares the flow speed time series obtained from model simulations (using Meshes 1–4) and the ADCP measurements. The difference in peak flow speeds obtained by Meshes 1 and 2 is approximately 0.2 m s⁻¹. Further refinement of the mesh reduces the difference in peak flow speeds (obtained by Mesh 3 and 4) to 0.1 m s⁻¹.

Figure 2.

The sensitivity of the Thetis model velocity magnitude predictions to (a) changes in the mesh resolution for a Manning coefficient of 0.03 sm^−1/3, at ADCP 3, and to changes in the Manning coefficient, at (b) ADCP 1, (c) ADCP 2 and (d) ADCP 3, all while using Mesh 3. (Online version in colour.)

Table 2 shows the NRMSE for each combination of mesh and Manning coefficient tested. There was on average a 0.66% improvement in NRMSE between Meshes 3 and 4 across the range of Manning coefficients tested. Mesh 1 took under 2 h to model 60 days, Mesh 2 took under 4 hours, Mesh 3 took just over 5 h and Mesh 4 took just over 11 h. This is a significant increase in the computational expense for a relatively small difference between models. The computational expense for a single model run is of key importance as the array optimization aspect of the modelling is fully coupled to the hydrodynamics—for every iteration of the optimization algorithm, the forward model needs to be re-run to calculate the updated hydrodynamics. Therefore, Mesh 3 was chosen for the following work.

Table 2.

The Thetis model NRMSE, averaged over ADCPs 1–3.

	Manning coefficient (sm−1/3)
	0.02	0.025	0.03	0.035	0.04
Mesh 1	13.7%	11.8%	12.6%	15.2%	18.6%
Mesh 2	13.7%	11.8%	12.3%	14.9%	18.5%
Mesh 3	13.5%	11.2%	11.8%	14.2%	17.7%
Mesh 4	14.5%	11.0%	10.4%	12.9%	16.3%

(iii). Calibration: flow speed

Figure 2b–d compare the flow speed time series obtained using Mesh 3 for each of the Manning coefficients considered. These results are compared against field data obtained from the ADCPs. To protect the confidential nature of the ADCP data we do not specify a start date and instead show time in days from the start of the model. The time-series data demonstrate that the flow speeds are relatively sensitive to the Manning coefficient, as there is approximately a 0.5 m s⁻¹ difference in peak flow speeds across the range of Manning coefficients used.

Table 3 provides the error metrics achieved for each of the Manning coefficients considered at each ADCP location. Closest agreement between simulated and measured NRMSE velocities was achieved using a Manning coefficient of 0.025 sm^−1/3 and 0.03 sm^−1/3, depending on location within the Alderney Race. Closest agreement with ADCP 1 and 2 was achieved using n = 0.025 sm^−1/3, resulting in an average NRMSE across the three ADCPs of 12.4 and 9.7%, respectively. Reasonable agreement was also achieved using n = 0.03 sm^−1/3, which achieved an NRMSE of 15% and 11%, respectively. Closest agreement with ADCP 3 was achieved with n = 0.03 sm^−1/3 (9.4%), with n = 0.035 sm^−1/3 (11.1%) and n = 0.025 sm^−1/3 (11.6%) also having close levels of performance. The biases show that for Manning coefficients of 0.03 sm^−1/3 and above the model underestimates the flow speeds at all ADCP locations. For a Manning coefficient of n = 0.02 sm^−1/3, flow velocities at ADCP 3 are overestimated. For a Manning coefficient of n = 0.025 sm^−1/3, flow velocities at ADCP’s 2 and 3 are overestimated.

Table 3.

The Thetis velocity amplitude model error metrics at ADCPs 1–3 for different Manning coefficients.

	Manning coefficient (sm−1/3)
	0.02	0.025	0.03	0.035	0.04
NRMSE 1	11.7%	12.4%	15.0%	18.4%	21.9%
NRMSE 2	11.4%	9.7%	11.0%	13.2%	16.4%
NRMSE 3	17.4%	11.6%	9.4%	11.1%	14.7%
NRMSE (avg)	13.5%	11.2%	11.8%	14.2%	17.7%
IA 1	0.938	0.926	0.885	0.826	0.762
IA 2	0.944	0.954	0.936	0.902	0.843
IA 3	0.895	0.946	0.960	0.938	0.884
IA (Avg)	0.925	0.942	0.927	0.889	0.830
bias 1 (m s−1)	−0.06	−0.22	−0.36	−0.50	−0.62
bias 2 (m s−1)	0.10	−0.06	−0.21	−0.32	−0.44
bias 3 (m s−1)	0.27	0.09	−0.08	−0.24	−0.38
relative bias 1	−3.4%	−12.7%	−21.3%	−29.2%	−36.4%
relative bias 2	6.1%	−3.6%	−13.2%	−20.9%	−28.5%
relative bias 3	16.6%	5.7%	−5.1%	−14.8%	−23.6%

(iv). (Calibration: free surface elevation)

Table 4 provides a comparison of modelled versus actual M2 and S2 elevation amplitudes and phases at the ADCP locations. Closest agreement was achieved using a Manning coefficient of 0.03 sm^−1/3, where M2 and S2 phase were predicted very well ($0.1\text{–}{1.0}^{\circ }$ and $0.5\text{–}{2.1}^{\circ }$ error, respectively) and the M2 and S2 amplitudes had errors ranging between 9.7–17.7% and 1.3–10.2% respectively.

Table 4.

The Thetis model prediction and error in the M2 and S2 phase and amplitude, for each Manning coefficient tested. Harmonic analysis is performed on η readings taken at the locations of ADCPs 1–3.

	Manning coefficient (sm−1/3)
ADCP	0.02	0.025	0.03	0.035	0.04
M2 amplitude
1	1.82 (−13.3%)	1.86 (−11.3%)	1.90 (−9.7%)	1.90 (−9.3%)	1.90 (−9.6%)
2	1.78 (−14.0%)	1.82 (−11.9%)	1.85 (−10.5%)	1.86 (−10.0%)	1.86 (−10.2%)
3	1.90 (−20.0%)	1.94 (−18.5%)	1.96 (−17.7%)	1.96 (−17.5%)	1.96 (−17.9%)
S2 amplitude
1	0.75 (−3.9%)	0.76 (−3.3%)	0.76 (−3.3%)	0.75 (−4.3%)	0.74 (−5.9%)
2	0.74 (−2.0%)	0.74 (−1.2%)	0.74 (−1.3%)	0.74 (−2.2%)	0.72 (−3.7%)
3	0.79 (−9.5%)	0.79 (−9.5%)	0.78 (−10.2%)	0.77 (−11.4%)	0.76 (−13.1%)
M2 phase
1	197.23 ° (−1.2 °)	198.52 ° (0.1 °)	199.38 ° (1.0 °)	199.78 ° (1.4 °)	200.01 ° (1.6 °)
2	198.79 ° (−1.8 °)	199.98 ° (−0.6 °)	200.74 ° (0.1 °)	201.01 ° (0.4 °)	201.14 ° (0.5 °)
3	198.58 ° (−0.6 °)	199.56 ° (0.4 °)	200.20 ° (1.0 °)	200.47 ° (1.3 °)	200.62 ° (1.5 °)
S2 phase
1	254.47 ° (−3.7 °)	256.41 ° (−1.7 °)	257.60 ° (−0.5 °)	258.27 ° (0.1 °)	258.82 ° (0.7 °)
2	255.87 ° (−5.1 °)	257.70 ° (−3.2 °)	258.81 ° (−2.1 °)	259.34 ° (−1.6 °)	259.80 ° (−1.1 °)
3	255.75 ° (−3.5 °)	257.32 ° (−1.9 °)	258.30 ° (−0.9 °)	258.84 ° (−0.4 °)	259.33 ° (0.1 °)

Table 5 shows the M2 and S2 elevation amplitude errors from the Thetis model relative to the tide gauge readings. Closest agreement was achieved for n = 0.025 sm^−1/3, with n = 0.02 sm^−1/3 achieving very similar performance. For a Manning coefficient of 0.03 sm^−1/3, as shown in figure 3, the M2 and S2 normalized phase errors were 16.7% and 16.4%, respectively, whereas for a Manning coefficient of 0.025 sm^−1/3, they were 13.8% and 13.4%, respectively. This suggests that further work could use a variable bed friction throughout the channel, based on maps of bed type such as the ‘Service Hydrographique et Ocèanographique de la Marine Service’ (SHOM) [27], as used in [24], since a coefficient of 0.03 sm^−1/3 generally produced better results for ADCPs and gauges within the Race, whilst a coefficient of 0.025 sm^−1/3 produced better results for tide gauges spread throughout the wider Channel.

Table 5.

The Thetis model M2 and S2 amplitude errors over the 68 tide gauge locations.

	Manning coefficient (sm−1/3)
	0.02	0.025	0.03	0.035	0.04
M2 RMSE	0.219 m	0.233 m	0.281 m	0.347 m	0.416 m
M2 NRMSE	13.0%	13.8%	16.7%	20.6%	24.7%
S2 RMSE	0.081 m	0.078 m	0.096 m	0.123 m	0.152 m
S2 NRMSE	13.8%	13.4%	16.4%	21.2%	26.1%

Figure 3.

The M2 and S2 amplitude of the 68 tide gauges across the English Channel against the Thetis model predictions, for a Manning coefficient of n = 0.03 sm^−1/3.

Inspection of the ADCP 1–3 velocity time-series data against the Thetis model and harmonic analysis of the M2 and S2 elevation phase and amplitude have shown that closest agreement was achieved with a Manning coefficient of 0.03 sm^−1/3. The average NRMSE of ADCP 1–3 velocity time-series data and the harmonic analysis of the elevation at the 68 tide gauges showed slightly better results for 0.025 sm^−1/3 (11.2% versus 11.8%). On balance, for this work, a Manning coefficient of 0.03 sm^−1/3, applied uniformly over the domain, was established as the most suitable value to take forward to the array optimization study.

(c). Validation

The model was calibrated with ADCPs 1–3, leaving ADCP 4 from [26] for independent validation. Table 6 provides a comparison of the NRMSE, bias, relative bias and index of agreement at the location of ADCP 4, obtained from the five bed drag coefficient cases.

Table 6.

The Thetis velocity amplitude model error at ADCP 4 for different Manning coefficients.

	Manning coefficient (sm−1/3)
	0.02	0.025	0.03	0.035	0.04
NRMSE	21.8%	11.3%	10.2%	15.7%	21.5%
bias (m s−1)	0.19	−0.01	−0.16	−0.31	−0.44
relative bias	13.8%	−0.6%	−11.7%	−22.9%	−32.3%
IA	0.859	0.956	0.960	0.898	0.809

The NRMSE and index of agreement results suggest that the n = 0.03 sm^−1/3 case provides the best agreement with ADCP 4 data, reinforcing the choice of the Manning coefficient from the calibration tests. However, the difference between these results and those obtained with n = 0.025 sm^−1/3 is small and the bias is lowest at n = 0.025 sm^−1/3. For comparison, the RMSE at ADCP 4 with n = 0.03 sm^−1/3 is 0.243 m s⁻¹, whereas [24] achieved a RMSE of 0.15–0.26 m s⁻¹, a relative bias of between −1% and −8%, and an IA of 0.962–0.990 using a model with 100 m resolution and a time step of 10 s.

4. Economic resource assessment

(a). Method

Turbines were introduced into the Thetis model via the inclusion of a turbine drag coefficient, c_t, which appears within an additional sink term in the momentum equation of the form;

$$\frac{c_t}{\rho H}||\mathbf{u}||\mathbf{u}.$$ 4.1

The additional turbine drag coefficient is added to the existing sea-bed drag coefficient at the location of the turbines. The added turbine drag coefficient is applied continuously over the array area, and is allowed to vary spatially depending on the local density of turbines. This continuous approach for parameterizing array drag was favoured over the more computationally expensive discrete approach, where each individual turbine is allocated its own drag term [28]. The discrete approach requires a mesh with element sizes constrained to be below the turbine rotor diameter to allow individual turbines to be resolved. The continuous approach is particularly well suited to the problem at hand because it allows for the number of turbines as well as their positions to be simultaneously optimized for. In [2], flow modelling from the continuous and discrete turbine representations are compared, showing that the farm wake and bypass flow modelled by both methods are largely consistent. However, the continuous method does not resolve the individual wakes within the farm area. Inter-array flow effects such as the impingement of wakes on downstream turbines are mitigated to an extent in this work by constraining the minimum longitudinal spacing between turbines to ten rotor diameters. This turbine spacing constraint is informed by guidelines published by the European Marine Energy Centre (EMEC) [29]. In taking this approach, this work provides a valid approach to modelling large-scale arrays in order to model the impacts of array scale blockage on energy yield.

The turbine drag coefficient c_t can be found from a spatially varying turbine density field, d(x), via the relationship;

$$c_t(\text{d}(x))=\frac{1}{2}\hspace{0.17em}{C}_T\hspace{0.17em}{A}_T\hspace{0.17em}\text{d}(x),$$ 4.2

where C_T is the thrust coefficient, A_T is the swept area of a turbine and d(x) corresponds to the number of turbines per metre squared of seabed [2].

The swept area and rated power of the turbines used in this work are assumed to be 16 m and 2 MW, respectively. This is informed by (a) the scale of turbines in operation currently and (b) the depths within the Alderney Race. Typically, turbines in operation at present have rotor diameters and rated capacity ranging between 9 m–18 m and 100 kW–1.5 M, respectively, e.g. the Nova Innovation turbines at the Shetland Tidal Array [30] and the SIMEC Atlantis Energy and Andritz Hydro Hammerfest turbines at the MeyGen array [31]. Studies suggest that increasing the rotor diameter and rated power can be key drivers for reducing the cost of tidal stream energy in the future [32], and SIMEC Atlantis Energy have developed a 2MW turbine with 20–24 m rotors for the next phase of the MeyGen project [33]. However, the rotor diameters of the turbines are limited here in regions of the East Race since it is relatively shallow. On balance, 16 m diameter 2 MW turbines were chosen to be representative of future commercial turbines that may be deployed in this region. It is acknowledged that in reality, a variety of turbine sizes are needed given the spatial variation in depth across the Race, as demonstrated in [3], and this is an area identified for further work.

Below the rated speed the thrust and power coefficient are assumed equal to C_T = 0.8 [34] and C_P = 0.41 [31], respectively. Above rated speed, the coefficients are scaled by $u_{\text{rated}}^3/u^3$ to maintain constant drag and power. The turbine density is set to zero outside of the array area, Ω, shown in figure 1, and within the array it is allowed to vary continuously up to a density value which corresponds to a spacing of 2.25 diameters laterally (centre-to-centre) and 10 diameters longitudinally [29].

To optimize the array, a gradient-based optimization algorithm is employed which uses Thetis’s adjoint to obtain the required gradient, where c_t is the control parameter for evaluating the sensitivity of the optimization functional, and is the quantity that is updated at every optimization iteration. Further details may be found in [2]. The array design, including the number of turbines (obtained via the integral of the turbine density field) and their spatial distribution, is optimized with respect to a functional J which here takes the form;

$$J=P_{\text{avg}}-P_{\text{BE}}\times n_t,$$ 4.3

where P_avg is the average power generated by the whole array and n_t is the number of turbines in the array. P_BE is the break even power, which is the time-averaged power that a turbine must generate to be economically viable for the project. In this work, the break even power is varied between 0 and 700 kW. A 2 MW turbine that achieves a time-averaged power of 700 kW is performing with a capacity factor of 35%. An increase in break even power results in a decrease in the optimal number of turbines in the array. In this case, adding additional turbines has a detrimental impact on the power generated per turbine, because the added turbines increase array blockage, slowing the flow velocities and hence reducing the available power to the array, so that the break even power cannot be achieved. This is overcome within the optimization by reducing the number of turbines in the array.

The power generated by the array is calculated at each time step by integrating over the farm area, Ω, via the approximation

$$P=\frac{1}{2}\rho A_T{\int }_{\mathrm{\Omega }}{C}_P\hspace{0.17em}d(x)\hspace{0.17em}||\mathbf{u}|{|}^3\hspace{0.17em}\text{d}x.$$ 4.4

The power can be time averaged and combined with the number of turbines, n_t, in the array design at that iteration, which can be obtained from c_t via

$${\int }_{\mathrm{\Omega }}{c}_t(d(x))\hspace{0.17em}\text{d}x=\frac{1}{2}{C}_T{A}_T{n}_t.$$ 4.5

The optimization was run over a representative 14.5-day spring-neap cycle, ensuring a sufficient duration to capture the dominant tidal variations while also achieving acceptable computational time for the optimization to converge. The model was spun up for one day before each optimization iteration.

Once the optimization was completed for each of the break even powers considered, the LCOE of each optimal array was calculated. The LCOE describes the fixed price per MWh that needs to be received by the developer for an array to break even over its lifetime. It is calculated using

$$\text{LCOE}=\frac{{\text{CA}}_f+{\text{CA}}_t\times n_t+\sum _{i=1}^L(({\text{O}}_f+{\text{O}}_t\times n_t)/{(1+r)}^i)}{\sum _{i=1}^L(E_i/{(1+r)}^i)},$$ 4.6

where CA_f and CA_t are the fixed and turbine-dependent components of the capital expenditure (CAPEX), defined below. L is the operational lifetime of the array in years, where each year is denoted by i. O_f and O_t are the fixed and turbine-dependent components of the operational expenditure (OPEX) per year, also defined below. E_i is the energy generated in year i and r is discount rate applied annually.

LCOE uses discounted cash flow analysis to take into account the notion that money has a greater value at present than in the future, with r being the extent to which the value of revenues and expenditures depreciate each year. New forms of energy are considered risky to investors, so may have a high discount rate, however as tidal energy becomes more established, the discount rate is likely to fall [35]. In this model, the simplifying assumption is made that the annual energy generation, E_i, is constant year on year. In practice, the yield is likely to vary annually due to the 18.6-year lunar nodal cycle and degradation of the turbine performance for example; however, this is not considered at this stage of investigation.

The CAPEX, CA, are the costs incurred in the production and installation of a tidal project. The OPEX, O, are the costs incurred in the maintenance and operation of the array. Both CA and O are split into fixed and turbine-dependent components.

The fixed costs are spent regardless of the size of the array, so that the total shared fixed cost per turbine reduces as the number of turbines in the array increases. This economy of volume is a mechanism that has driven down the cost of energy in the wind industry [36]. The turbine costs increase linearly with the number of turbines. For example, there may be fixed costs for resource assessments of new sites, establishing a manufacturing centre for turbines, or array to grid connections, but the costs of materials will increase linearly with the number of turbines.

The inputs used for each of these fixed/turbine costs were calculated in previous work by drawing together inputs from multiple sources [35,37–39] and are summarized in table 7. We consider a range of values for each cost, ranging between pessimistic and optimistic values quoted in literature. We also carry out a Monte Carlo analysis to find the P10, P50 and P90 values of LCOE, by assuming a uniform distribution over each of the cost inputs in table 7 with the pessimistic and optimistic values as their upper limits to randomly vary between. P10, P50 and P90 are the values for which 10%, 50% and 90% of the LCOE estimates are better (lower) than, respectively. This approach of optimizing arrays to minimize LCOE was first presented in [40] for idealized array cases.

Table 7.

Estimates for the optimistic (opt), typical (typ) and pessimistic (pes) parameters used in the economic models, and the amount they vary.

		value range
symbol	description	pes	typ	opt	units
CAf	fixed CAPEX	14.4	9.2	5.6	£m per year
CAt	turbine-dependent CAPEX	4.4	3.3	2.4	£$\text{m per year per turbine}$
Of	fixed OPEX	0.87	0.32	0.27	£$\text{m per year}$
Ot	turbine-dependent OPEX	0.26	0.15	0.094	£$\text{m per year per turbine}$
r	discount rate	0.15	0.10	0.05	n.a.
L	lifetime of an array	20	25	30	years

(b). Results

Figure 4 shows the optimal array designs for the range of break even powers considered. In general, the optimal array layouts span the Alderney Race in fence-like structures orientated perpendicular to the flow direction. This is consistent with findings in [41]. Orientating the turbines in these fence-like structures maximizes power generation by preventing the flow from passing around the array as a result of the added turbine drag.

Figure 4.

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 P_BE. The array density is allowed to increase up to a maximum of $1/({\text{Ø}}^2\times 2.25\times 10)$, where $\text{Ø}$ is the turbine diameter of 16 m. (Online version in colour.)

At high levels of break even power, the optimal array contains a low number of turbines in order to mitigate against array blockage effects that detrimentally impact upon power generation. This is synonymous with the early stages of tidal energy development, where the relatively high CAPEX/OPEX and discount rate only permit turbines to be installed in the highest resource areas, otherwise the cost of energy becomes unacceptably high. As the break even power decreases, larger-scale arrays become more viable, as it becomes economically acceptable to install turbines in lower energy regions of the Race. It is only once the break even power reduces below £300/MWh that the array contains turbines in the West Race (i.e. in Alderney territorial waters). This result suggests that to minimise the cost of energy, early stage developments should take place in French territorial waters, however, this may not be the case once different turbine designs (i.e. rotor diameter and rated speed) are considered, as the deeper West Race (Alderney territorial waters) may be more suitable for turbines with a larger rotor diameter, for example.

Figure 5 shows the relationship between the break even power and (a) net average array power and corresponding average power per device and (b) number of turbines. Decreasing the break even power reduces the acceptable level of power generation per turbine, so that larger scale arrays become more viable. This has the same impact as reducing CAPEX and/or OPEX, where reductions in CAPEX/OPEX mean that turbines can generate less power to achieve the same LCOE. Mechanisms that enable these CAPEX/OPEX cost reductions include learning taken from previous projects as the industry develops, and cost reduction unlocked by development of the supply chain, for example [32,35]. Economies of volume are inherently modelled as a cost-reduction mechanism in this paper through the distinction of the fixed- and turbine-dependent components of both CAPEX and OPEX. Learning rates as a cost reduction mechanism can be investigated by comparing the impact of moving from the pessimistic to typical to optimal scenarios presented in table 7.

Figure 5.

Variation of the macro parameters of the optimal array design with the break even power used in the functional. (Online version in colour.)

Increasing the number of turbines increases the total array drag, slowing the flow in the region of the array and thus reducing the available power. When the reduction in CAPEX/OPEX costs achieved from economies of volume outweigh the negative impacts of array blockage, it becomes economically viable to install more turbines. However, the detrimental impacts of array blockage result in a diminishing return on array power as the array size increases. This is illustrated in figure 6, which shows the relationship between the number of turbines and (a) net average array power, and (b) net average power per device. As the number of turbines increases to very high levels (greater than 2000 turbines), the net average array power begins to plateau, levelling out to an upper bound of approximately 1 GW by 15 000 turbines.

Figure 6.

A prediction for the power generated in the optimal array design for all numbers of turbines, achieved by curve fitting between the results of each P_BE optimization. (Online version in colour.)

Figure 7 shows the relationship between the number of turbines in the array and the minimum estimated LCOE. The optimal relationship between number of turbines and LCOE is shown for the pessimistic, typical and optimistic scenarios outlined in table 7, as well as the P10, P50 and P90 estimates obtained from the Monte Carlo analysis. The optimal LCOE estimates are also presented in table 8. As the number of turbines increases from 0, there is a steep decrease in LCOE, which is enabled through economies of volume. As the number of turbines approaches the optimal value there is a turning point, where the impact of economies of volume versus diminishing returns from adding more turbines balance out. As the number of turbines increases further the diminishing returns due to blockage have a dominant effect and cause the LCOE to increase. Based on the P50 estimate, the minimum LCOE of £110/MWh is achieved with 38 turbines and an installed capacity of 81 MW. Increasing the installed capacity to 1 GW and 2 GW increases the estimated LCOE to £150/MWh and £180/MWh, respectively.

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. (Online version in colour.)

Table 8.

Optimal LCOE and corresponding array parameters for different scenarios: optimal, pessimistic and typical scenarios and the P10, P50 and P90 generated using a Monte Carlo based analysis with a uniform distribution and upper and lower limits of the optimistic and pessimistic values.

	optimistic	typical	pessimistic	P10	P50	P90
LCOE (£/MWh)	36.6	109.5	209.0	77.7	110.0	146.2
nt	42.0	34.0	39.0	46.0	38.0	33.0
Pavg (MW)	24.0	19.7	22.4	26.1	21.9	19.2
Pavg/nt (kW)	541	554	546	534	547	556
PBE (kW)	541	554	546	534	547	556

The optimal LCOE obtained from the optimistic and pessimistic cases is £37/MWh and £209/MWh, respectively. This disparity between the LCOE estimates is caused as a result of all input parameters being set to their most extreme values. In practice, while there is uncertainty in the associated costs, it is unlikely that all parameters will be at their best or worst at once. This was investigated by estimating the LCOE based on a Monte Carlo analysis, so that a range of CAPEX, OPEX and discount rates could be considered simultaneously. Results in table 8 show that the P10, P50 and P90 minimum LCOE estimates are £78/MWh, £110/MWh and £146/MWh, respectively. While this still provides a broad range, it is significantly reduced in range relative to the pessimistic and optimistic cases. This disparity in estimated LCOE is likely to reduce further once more cost information becomes available.

While the minimum estimated LCOE can vary widely depending on the cost inputs, the number of turbines required to achieve the minimum LCOE remain largely the same, in the region of 33–43 turbines. This is equivalent to a break even power of approximately 550 kW, or a layout with slightly fewer turbines than shown in figure 4a. In this work, the cost reductions due to learning with time are not explicitly modelled, however, the pessimistic scenario would likely correspond to the costs at a very low cumulative installed capacity, and as more arrays are deployed the industry would move towards the increasingly optimistic scenario curves.

The further the costs drop, towards the P10 and optimistic scenarios, the flatter the increase in LCOE projection becomes when the number of turbines exceeds the optimal amount to minimize LCOE. The Offshore Renewable Energy Catapult has predicted that learning rates will cause costs to fall by 7% per 100 MW cumulative deployed capacity [35]. If it is assumed that the optimistic cost inputs can be achieved through learning, as many as 500 turbines with a total install capacity of 1 GW could be installed, while still keeping the LCOE below £50/MWh.

The typical cost case achieves an optimal LCOE of £110/MWh with 34 turbines. This agrees closely with the P50 estimate, which achieves the same minimum LCOE of £110/MWh with 38 turbines. This estimate falls within 15% of the level of subsidy support proposed by the French government for ‘marine hydraulic energy’, of £130/MWh ($=\text{€}\hspace{0.17em}150/\text{MWh}$) [42]. Furthermore, a recent Marine Energy Council report [43] proposed a number of different subsidy schemes to provide a route to market for the tidal energy industry and other innovative clean energy technology types. The typical LCOE predictions are in line with its conclusion that an innovation contract for difference (iCfD) set at an initial level of £150/MWh and falling to £90/MWh could support novel projects in the 5–100 MW installed capacity range.

The typical LCOE prediction is just below the strike price of £115–120/MWh won by offshore wind farm projects at the Contract for Difference (CfD) auctions in 2014–2015 in the UK [44]. Since the time of the 2014–2015 CfD auctions in the UK, the LCOE of offshore wind projects in the UK has reduced to around £40/MWh [45], facilitated by the CfD scheme that has enabled economies of volume, economies of scale and technology innovation to drive down costs [35]. This supports the argument that with similar subsidy support, the cost of tidal energy could move from the pessimistic to optimistic levels set out in table 7.

After 10 MW of installed capacity, the pessimistic LCOE estimate is £295/MWh. The Offshore Renewable Energy Catapult has published figures indicating the LCOE achieved by tidal stream energy was approximately £300/MWh after 10 MW of total installed capacity worldwide (based on 2012 pricing), aligning closely with the pessimistic LCOE approximation in this paper [35]. Likewise, this model prediction is in line with the Marine Energy Council report [43] proposal for an Innovation Power Purchase Agreement (IPPA) set at a level of £290/MWh to support novel small-scale projects.

5. Conclusion

A new, validated Thetis model set-up for the English Channel with non-uniform mesh resolution has been used to minimize the LCOE of tidal stream turbine arrays in the Alderney Race. This work considers arrays using turbines with a rated power of 2 MW and a rotor diameter of 16 m. In general, the optimal array solutions consist of ‘fence-like structures’ of turbines orientated perpendicular to the flow. These array layouts help prevent the flow from diverting around the array as a result of the added array drag and are consistent with previous array optimization studies [41]. Results demonstrate that in the early stages of array development in the Race (0–100 MW of installed capacity), steep reductions in LCOE are achievable through optimization of turbine placement, and economies of volume. Based on the turbine specification considered in this paper, the optimal location for turbines at this early stage of development is in the East Race.

Given the early stage of tidal stream energy industry development, cost information that provides the inputs to the array optimization is limited, with a wide range of estimated CAPEX/OPEX and discount rates found in the literature. This uncertainty in associated costs is reflected in the LCOE estimates presented in this work, where the minimum LCOEs based on assuming pessimistic, typical and optimistic cost scenarios are £36.60/MWh, £109/MWh and £209/MWh respectively. A Monte Carlo analysis is conducted to reflect that it is unlikely that all costs will be at their most extreme (i.e. pessimistic or optimistic) values. Results from this study yield P10, P50 and P90 minimum LCOE estimates of £78/MWh (with 46 turbines), £110/MWh (with 38 turbines) and £146/MWh, respectively (with 33 turbines). This provides a significantly reduced range of LCOE estimates, where the number of turbines required to achieve the minimum LCOE are closely aligned.

There are some notable similarities between the LCOE estimates presented in this work and published data from industry. For example, in this paper, the pessimistic LCOE estimate after 10 MW of installed capacity (i.e. when economies of volume are limited) is £295/MWh. This aligns closely with information published by the Offshore Renewable Energy Catapult, who estimate that the LCOE after 10 MW of installed capacity worldwide was £300/MWh, based on 2012 prices and aggregated data from multiple tidal stream turbine developers.

Additional cost reduction mechanisms such as learning rates are not considered explicitly in this work. However, it could be argued that learning will allow costs to drop from the more pessimistic end towards the optimistic levels found in the literature. If this is possible, it is estimated that a 1 GW array could achieve an LCOE of £49/MWh, based on optimistic cost inputs.

6. Limitations and further work

Aspects of the work presented in this paper can be developed further to reduce uncertainty in the results. This includes improvement to model validation in order to reduce error in modelled velocities, owing in part to the coarseness of the mesh required to provide acceptable run times when using the model within an optimization loop. This can translate into large errors in power. At the settings used in this model, the mean velocity is generally underestimated, so it is likely that the conclusions are conservative relative to what can be achieved in reality. However wave effects, which have been shown to lead to a net reduction in tidal power, have not yet been included in this work. Since wave effects can have a notable impact, especially in sites more exposed to waves, they should be included in future studies [11,12].

Another area of further work is consideration of a suitable range of turbine scales for the Alderney Race. The current work considers 16m rotors only, however, it has been shown that LCOE could reduce significantly if turbine diameter is increased, even by a few metres [32]. It will be important in future array optimization work to identify the rotor diameter limits within different regions of the Race to establish how best to harness the resource with different turbine sizes. This also requires an understanding of any constraints that would prevent turbines from being installed in specific regions of the Alderney Race. For example, local areas with uneven bathymetry may prevent turbines from being installed [46].

The impacts of local blockage and accelerated bypass flow on the yield of tidal stream turbines in large-scale arrays should also be considered [47]. Given that resolving turbines individually requires a much finer mesh and is computationally very expensive, a number of approaches would need to be investigated. A two-stage optimization as described in [2] could be implemented, such that continuous optimization provides a good initial layout, so that the discrete turbine micro-siting optimization requires overall fewer costly iterations to converge.

The LCOE calculations should be updated once additional cost information becomes available. Additional cost information will also be important to validate the assumptions made in this paper, such as the linear increase in OPEX with number of turbines, for example.

Data accessibility

The data accompanying this paper are available on Figshare via: 10.6084/m9.figshare.12449225.

Authors' contributions

Z.L.G. carried out the numerical modelling, with assistance from DSC on the set-up and validation. M.D.P. and D.S.C. provided supervision throughout. All authors read and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

Z.L.G. acknowledges the support of an EPSRC PhD studentship, through the Mathematics of Planet Earth CDT and the support of the National Productivity Investment Fund through NERC and the Grantham Institute. M.D.P. acknowledges the support of EPSRCunder grant nos. EP/M011054/1 and EP/R029423/1. D.S.C. acknowledges the support of the International Centre for Infrastructure Futures (ICIF) under grant no. EP/K012347/1.