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# A standardised tidal-stream power curve, optimised for the global resource

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## Abstract

Tidal-stream energy can be predicted deterministically, provided tidal harmonics and turbine-device characteristics are known. Many turbine designs exist, all having different characteristics (e.g. rated speed), which creates uncertainty in resource assessment or renewable energy system-design decision-making. A standardised normalised tidal-stream power-density curve was parameterised with data from 14 operational horizontal-axis turbines (e.g. mean cut-in speed was ∼30% of rated speed). Applying FES2014 global tidal data (1/16° gridded resolution) up to 25 km from the coast, allowed optimal turbine rated speed assessment. Maximum yield was found for turbine rated speed ∼97% of maximum current speed (maxU) using the 4 largest tidal constituents (M2, S2, K1 and O1) and ∼87% maxU for a “high yield” scenario (highest Capacity Factor in top 5% of yield cases); with little spatial variability found for either. Optimisation for firm power (highest Capacity Factor with power gaps less than 2 hours), which is important for problematic or expensive energy-storage cases (e.g. off-grid), turbine rated speed of ∼56% maxU was found – but with spatial variability due to tidal form and maximum current speed. We find optimisation and convergent design is possible, and our standardised power curve should help future research in resource and environmental impact assessment.
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A standardised tidal-stream power curve, optimised for the global resource
Matt Lewis, Rory O’Hara Murray, Sam Fredriksson, John Maskell, Anton de Fockert,
Simon Neill, Peter Robins
PII: S0960-1481(21)00199-3
DOI: https://doi.org/10.1016/j.renene.2021.02.032
Reference: RENE 14923
To appear in: Renewable Energy
Revised Date: 3 February 2021
Accepted Date: 6 February 2021
S, Robins P, A standardised tidal-stream power curve, optimised for the global resource, Renewable
Energy, https://doi.org/10.1016/j.renene.2021.02.032.
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Title: A standardised tidal-stream power curve, optimised for the global resource 1 2 Matt Lewis
1*
, Rory O’Hara Murray
2
, Sam Fredriksson
3
4
, Anton de Fockert
5
, 3 Simon Neill
1
, Peter Robins
1
4 5
1
School of Ocean Sciences, Bangor University, UK; 6
2
Marine Scotland Science, The Scottish Government, UK; 7
3
University of Gothenburg, Sweden; Swedish Meteorological and Hydrological Institute, 8
4
JM Coastal Ltd, 9
5
Deltares, Delft, NL 10 11 Abstract 12 Tidal-stream energy can be predicted deterministically, provided tidal harmonics and turbine-13 device characteristics are known. Many turbine designs exist, all having different 14 characteristics (e.g. rated speed), which creates uncertainty in resource assessment or 15 renewable energy system-design decision-making. A standardised normalised tidal-stream 16 power-density curve was parameterised with data from 14 operational horizontal-axis 17 turbines (e.g. mean cut-in speed was ~30% of rated speed). Applying FES2014 global tidal 18 data (1/16° gridded resolution) up to 25 km from the coast, allowed optimal turbine rated 19 speed assessment. Maximum yield was found for turbine rated speed ~97% of maximum 20 current speed (maxU) using the 4 largest tidal constituents (M2, S2, K1 and O1) and ~87% 21 maxU for a “high yield” scenario (highest Capacity Factor in top 5% of yield cases); with little 22 spatial variability found for either. Optimisation for firm power (highest Capacity Factor with 23 power gaps less than 2 hours), which is important for problematic or expensive energy-24 storage cases (e.g. off-grid), turbine rated speed of ~56% maxU was found – but with spatial 25 variability due to tidal form and maximum current speed. We find optimisation and 26 convergent design is possible, and our standardised power curve should help future 27 research in resource and environmental impact assessment. 28 29 Keywords: tidal-stream energy; power curve; resource; optimization; renewable 30 energy 31 32 1. Introduction 33 Tidal energy can be extracted using hydrokinetic devices or “in-stream” tidal-stream energy 34 converters (e.g. Tsai and Chen, 2014; Masters et al., 2015), based on the principle that 35 power (P) is a function of the cube of the volumetrically averaged current velocity (u) over 36 the rotor swept area (A), turbine power coefficient (Cp) and seawater density (): 37
 

[1]. 38
39 As nations look to increase their renewable energy capacity in response to climate change 40 (Neill et al., 2016) or improve access to affordable electricity (Goward-Brown, et al., 2019; 41 Zhang et al., 2019), tidal-stream energy could offer one substantial renewable resource due 42 to the predictability and reported power quality (Lewis et al., 2019). Three main types of tidal-43 stream turbines are in various stages of development (for a review, see Rourke et al., 2010): 44 (1) horizontal axis turbines; (2) vertical axis turbines; and (3) rotating and reciprocating 45 devices. This paper shall focus on the horizontal axis turbine, used for the majority of test 46 and operational deployments; hence much data is available to inform and constrain our 47 analysis – such as estimation of device efficiency and the device power coefficient (Cp: 48 extracted power relative to the available power), alongside turbine behaviour parameters 49 including turbine cut-in and rated speed (see Mason-Jones et al., 2012; 2013).
50 51 The potential of tidal-stream energy for a sustainable future is immense (~2.5TW M2 52 tidal energy is dissipated globally see Egbert and Ray, 2001), with diverse applications: 53 predictable contributions of renewable electricity to a national grid (Neill et al., 2016) to 54
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energy solutions for remote communities and industries (e.g. Nielsen et al., 2018), such as 55 contributing to UN sustainability goals and reducing energy poverty (e.g. Lozano and 56 Taboada, 2020). However, the costs associated with tidal energy (e.g. Vazquez and Iglesias, 57 2015) such as cost reduction through economies of scale (e.g. Johnstone et al., 2013), and 58 deployment constraints (e.g. Lewis et al., 2015), need to be reduced for the true potential of 59 tidal energy to be realised. As power is proportional to the cube of tidal current, industry has 60 predominately focused on turbines with high rated speed (>2.5m/s) at so-called “first 61 generation sites (Lewis et al., 2015). It is unclear if mass-produced lower resource tidal-62 stream turbines for “high-value markets” could provide another route to cost reduction for the 63 industry, and the motivation for this study. 64 65 As discussed in the US Dept. Energy “Powering the Blue Economy” (LiVecchi et al., 66 2019), there is a diverse range of potential power demands (e.g. both in size and timing of 67 power required) and higher value markets (thus economic viability). We hypothesise that 68 previous focus on MegaWatt-scale contributions from tidal-stream turbines (with high rated 69 speeds above 2.5 m/s) is creating uncertainty and may not be suitable for all potential 70 renewable energy markets (LiVecchi et al., 2019). For example, there has been a reported 71 need for power curves to aid resource mapping studies with one (1 m/s cut-in and 2.7 m/s 72 rated) predominately being applied tidal turbine design (e.g. Hardisty 2012; Vennel et al., 73 2015; Robins et al., 2015) which may introduce bias in resource assessment (Fairley et al., 74 2020). Furthermore, Robins et al. (2015) proposed that turbines suitable for lower flows 75 would reduce temporal variability to the resource and increase resultant net power. Tidal-76 stream energy resource therefore appears uncertain, in part, due to uncertainty of end-user 77 power needs and device design. 78 79 Mapping the tidal resource for a region relies on validated hydrodynamic models, 80 which numerically solve versions of the Navier-Stokes equations to fully capture tidal 81 dynamics. Theoretical resource estimates for a region calculate tidal power from the ocean 82 model output variables to be applied in equation 1. Tidal resource has been shown to be 83 affected by the power extracted (e.g. Garrett and Cummins, 2005; 2007; Yang et al., 2013), 84 hence technical resource assessment often explicitly include power extraction of tidal 85 turbines to further improve potential yield estimates (e.g. Vennell et al., 2010; Goward-Brown 86 et al., 2017). Environmental impact assessments to the deployment of tidal turbines also 87 require power extraction to be explicitly resolved in the ocean model simulations; for 88 example, impacts to circulation and associated processes (e.g. Kadiri et al., 2012), sediment 89 transport pathways (Robins et al., 2014) and morphodynamics (Neill et al., 2009). 90 91 The drag force (Fd) of a tidal turbine is represented within hydrodynamic model 92 simulations applying equation 1 as: 93
 
[2]; 94
hence the impact of tidal energy conversion can be explicitly resolved in environmental 95 impact and resource assessments (see Yang et al., 2013). Tidal-stream turbine behaviour is 96 predominately based on first generation technologies (Lewis et al., 2015); where cut-in 97 speed (Vs), and rated speed (Vr: the current speed where maximum or “rated power” (Pr) is 98 extracted, with power “capped” or “shed” for current speeds above Vr) – must be resolved to 99 adequately represent turbine behaviour (e.g. Goward-Brown et al., 2017). First generation 100 tidal-stream turbines are defined by Lewis et al. (2015) as having a rated speed ~2.5 m/s, 101 and, whilst many devices indeed have high rated speeds, a number of lower flow devices 102 (e.g. Kites – see Buckland et al., 2015) and applications (O’Donncha et al., 2017) have been 103 discussed. Indeed, in many resource assessments, power curve information has been stated 104 as necessary for future work (e.g. Lewis et al., 2015; Vazquez and Iglesias, 2015; Guillou et 105 al., 2018). 106 107
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The lack of data to parameterise turbine behaviour presents a significant challenge 108 due to uncertainty in the parameterisation of tidal-stream turbine behaviour. The impact of 109 various tidal turbine power curves to the technical resource assessment is shown in Figure 110 1; where a 15-day time-series of two harmonics (M2 amplitude of 2 m/s and S2 amplitude of 111 0.5 m/s) is applied to estimate theoretical power density (P/A using Eq. 1), and the 112 theoretical power curve of two devices: Vr=2.5 m/s and Vs=1 m/s (from Lewis et al., 2015), 113 and Vr=2 m/s and Vs=0.5 m/s (from Encarnacion et al., 2019). Although rated turbine speed 114 (Vr) differs by 0.5m/s between the two devices of Figure 1, with mean power and mean daily 115 energy difference of 18% and 23% respectively, the maximum drag (thus impact, estimated 116 from Eq. 2) differed by 41%. Moreover, the Capacity Factor (CF), defined here as the ratio of 117 energy converted relative to the maximum energy that could be converted (i.e. if at rated 118 power throughout the time-series), varied by 14% between the two devices of Figure 1; with 119 a 19% difference in the time of zero power (so called downtime) and a 2 hour difference in 120 the longest duration window of zero power output, which has implications for storage design 121 and whole system costs. 122 123 124 Given that tides are almost entirely deterministic (e.g. Lewis et al., 2019), and the 125 wide variety of potential markets globally (from large-scale power contributions to national 126 electricity distribution networks to remote “off-grid” industries and communities): are the 127 present range of tidal-stream turbine designs suitable for all global markets, and can a 128 scalable convergent solution be found? This paper aims to firstly consolidate the diverse 129 range of horizontal axis tidal turbines to a scalable power curve for unbiased resource and 130 impact assessments. The standardised power-density curve can then be applied to explore 131 convergence based on the global tidal-stream resource. We do not include the swept area in 132 our analysis as this is likely to be based on local bathymetric constraints, life cycle 133 assessment and cost optimisation. Instead our objective is to establish a method, which can 134 be applied in the future to include cost optimisation based on future markets and mass-135 production principles (Junginger et al., 2004; Johnstone et al., 2013): providing a 136 constructive step towards a resource-led globally-optimal engineering solution for the 137 renewable energy industry. 138 139 2. Method 140 This study is composed of three parts: firstly, power curve data is compiled for the majority of 141 published horizontal tidal-stream turbines (i.e. all that could be found). Rated power (Pr) and 142 flow speed (Vr) allow the power coefficient (Cp) and thrust coefficient (Ct) to be estimated, 143 using variables from equations 1 and 2, because: 144
 


[3]; 145
  


[4]. 146
Consolidating the data, a normalised theoretical mean power density curve relative to rated 147 power (i.e. P/Pr and u/Vr) can be established (i.e. swept area removed), and also compared 148 to observed variability in a grid-connected tidal-stream turbine (published in Lewis et al., 149 2019). Here, density of seawater () is assumed to be 1027 kg/m^3 and the turbine is 150 operated at constant Tip Speed Ratio (TSR) irrespective of swept area (A) or flow speed (u): 151 i.e. that Cp does not vary with flow speed and Tip Speed Ratio (Mason-Jones et al., 2012; 152 2013). 153 154 The second part of our method will apply the average power density curve 155 information (which we call the normalised power curve) to resolve optimal power curve 156 characteristics for the diverse range of potential markets and tidal energy sites globally: for 157 example, does the optimal power curve for a remote island/industry differ to an optimal tidal 158 power curve for electricity supply to a grid? 159 160
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Depth averaged tidal current information was based on the FES2014 dataset (Finite 161 Element Solution data assimilated global tide model), which has a global grid resolution of 162 1/16° (Carrere et al.,, 2015). The FES2014 dataset was masked using the NASA distance-163 to-coast dataset (resolution 1/25°) which was created using the Generic Mapping Tools 164 (GMT) coastline. Global tidal data of the four principal semi-diurnal and diurnal tidal 165 constituents (M2, S2, K1, O1), between latitudes 70°S and 70°N and included only ocean 166 grid cells that were within 25 km from land were extracted. We assume tidal energy 167 development beyond 25 km is not economically feasible based on challenges with 168 connecting to shore, and have removed tidal analysis from high latitude (>70°) due to ice 169 interaction challenges and uncertainties. 170 171 Applying the normalised power curve to a wide range of rated speeds (Vr discretised 172 in 0.1 m/s bins between 0.3 m/s and 6 m/s) allows power density curves for all potential 173 tidal-stream turbines to be applied to one year tidal current time-series (5 min frequency). 174 The tidal current time-series at each location was calculated using the “t_tide” toolbox: a 175 harmonic tidal prediction method, where a time-series is described from the sum of sinusoids 176 at frequencies specified from astronomical parameters (Pawlowicz et al., 2008). Global tidal 177 harmonics data were used from the FES2014 product (Carrère et al., 2015; Lyard et al., 178 2020) for all resolved coastal locations (<25 km from land). An optimal power density curve 179 was selected for each site using three scenarios (A, A2 and B) to represent the diversity of 180 end user needs; from weighting the optimal tidal turbine power density curve based on firm 181 and constant power, or maximum possible yield. Hence, the range between high yield and 182 firm power (scenarios A and B) should therefore represent all potential optimal tidal turbine 183 solutions; providing a sensitivity test to power curve choice in resource assessment, but also 184 the potential for current technologies and concepts to be scaled for the more globally 185 prevalent, lower flow and power demand markets. 186 187 Scenario A (maximum yield): the power density curve that gave the highest annual energy 188 yield for each site (irrespective of storage and end user needs). We assume such a scenario 189 useful in free-market economic systems with national electricity distribution networks. 190 191 Scenario A2 (high yield): the highest Capacity Factor (CF) for power density curves that 192 gave the top 5% of annual yields per site. Therefore, although Scenario A2 does not bound 193 the range of potential optimal tidal power curves, it is assumed to represent a likely choice 194 given other resource uncertainties (e.g. higher order tidal harmonic effects, or the impact of 195 waves (Lewis et al., 2014) and weather windows). 196 197 Scenario B (Firm yield): the highest yield power density curve that had a maximum gap in 198 power generation below 2 hours and consistent peak power (within 2%). We assume such a 199 firm power tidal turbine beneficial for users where likely storage potential is low, or the 200 storage costs are high (for example the use of fly wheels instead of batteries). 201 202 The third part of method aims to resolve convergence in an optimal power curve 203 based on the global tidal data; producing simplified rules for industry and researcher to 204 follow (e.g. can we assume tidal turbine rated speed to be equivalent to the peak spring tidal 205 current speed for a given site?) Finally, we investigate the impact of tidal data quality by 206 comparing our 1/16° FES2014 results to that derived from tidal harmonics calculated using a 207 much higher resolution ocean model at 1/100° (~1km instead of ~7km spatial resolution) for 208 the UK domain (14°W to 11°E, and 42°N to 62°N). Data were interpolate onto the higher 209 resolution grid and the data of the UK ROMS model details given in Robins et al. (2015). 210 211 3. Results 212 Horizontal-axis tidal-stream turbine power density curves were normalised and standardised 213 (Section 3.1), which can be applied to idealised tidal current time-series with increasing 214
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complexity in tidal harmonics (Section 3.2), and applied to the global tidal harmonic data in 215 Section 3.3. 216 217 3.1. Power curve analysis results 218 Horizontal axis tidal turbine information was gathered from published data of 14 devices that 219 are in commercial development or deployment (Table 1). We believe data in Table 1 to be 220 the most comprehensive, up-to-date list compiled thus far. We acknowledge that Table 1 is 221 incomplete, with some prototypes and models missing, however convergence of the 222 normalised power-density curve in Figure 2 is clear and the addition of devices likely to 223 only impact parameters that are not considered here (e.g. swept diameter mean rated 224 power). Where key variables are missing (noted in Table 1 with *), data were extrapolated 225 using equations 1 and 4. 226 227 The rated power density and speed (Pr and Vr respectively) of the tidal-stream 228 turbines are shown in Figure 2a, compared to the theoretical (black dash line). Normalised 229 power-density curves of these devices are shown in Figure 2b, using the mean device 230 power coefficient (Cp) of Table 1, assuming Cp constant through all flow speeds, alongside 231 the measured power variability (at 0.5 Hz frequency) for a “grid connected” tidal-stream 232 turbine (taken from Lewis et al., 2019). Measured fine-scale power fluctuations of Figure 2b, 233 likely due to fine-slow flow variability and turbulence (see Lewis et al., 2019), were found to 234 be much larger than variability in mean device characteristics (cut-in and rated speed) for the 235 14 devices. Therefore Figure 2 indicates a normalised mean power curve can be used to 236 represent all horizontal axis tidal turbines currently being developed, and apply the power-237 density curve to global tide data in Section 3.2. Finally, Figure 2a shows there is no trend in 238 diameter of the swept rotor area, especially considering the size range of turbines, shown by 239 the large standard deviation in Table 1, hence further justification to use power density in our 240 analysis - as rotor size is likely to depend on local site charactersitics and cost-benefit 241 analysis (which is beyond the scope of this work). 242 243 A standardised and normalised power curve for horizontal axis tidal-stream turbines 244 was established using the mean value of Table 1: Cut in speed of the turbine (Vs) was found 245 to be 30% of the rated speed (Vr) on average with a standard deviation (STD) of 7%, and we 246 assume power coefficient (Cp) is constant, at a mean value of 0.37; which allows the power 247 density (P/A) to be described relative to the rated power of a device (where Pr is expressed 248 as P/A relative to the rated, thus between 0% and 100%). It should be noted that the power 249 coefficient (Cp) is likely to be affected by a number of variables: flow speed and site 250 turbulence characteristics (including waves), as well as blade design and Tip-Speed-Ratio 251 (see Mason-Jones 2012; 2013) – however the variability does not significantly affect our 252 results (based on unpublished sensitivity test – varying section 3.2 with Cp with one STD: 253 0.04). 254 255 The standardised power curve, based on mean values of Table 1, is shown in Figure 256 3a and is described in equation 5, using three conditions: 257
   !"# $% !& '( ) *  !" # ( )  "#  # [5]. 258 Moreover, the normalised drag and thrust coefficient (Ct) can now also be described (using 259 Equation 2) which allows a tidal-stream turbine, unbiased in technology choice, to be 260 represented for future resource and environmental impact assessment hydrodynamic 261 modelling methods. The device agnostic power curve of Figure 3 therefore only needs a 262 rated power (Pr) and swept area (A) to be assumed, and we shall explore an optimal Vr, 263 based on tidal resource, in Section 3.2 264 265 3.2. Power density curve optimisation 266 Journal Pre-proof The standardised normalised power curve of Figure 3 was applied to a tidal current time-267 series for a range of rated turbine speeds (Vr), with the Capacity Factor (CF) and the yield 268 for each theoretical device compared. Capacity Factor (CF) was calculated as the 269 percentage of energy captured compared to energy captured if a turbine was at rated speed 270 throughout the timeseries: 271   + , - . + , -/0 [6]). 272 273 Here, we consider power-density in our analysis, as bathymetry likely to be uncertain in the 274 spatially coarse global data of FES2014 (1/16° see Section 3.3) and we assume swept area 275 (A in Eq. 1) to be controlled by cost and array-design optimisation. Furthermore, the scaling 276 of depth-averaged current (u) to hub-level flow is not included but cannot be represented in 277 global tide data due to sub-scale temporal and spatial variability. The swept area (A) can be 278 removed from our CF calculation (of Eq. 6), as it is a constant in the numerator and 279 denominator integral; therefore our optimisation is independent of swept area, and instead 280 our analysis focuses on the rated speed of a turbine relative to the temporal variability of the 281 tide for a given site. 282 283 The mean power density and mean daily yield (kWh/m^2 per day) were also 284 calculated as metrics of power curve performance for each theoretical power curve at each 285 site. To demonstrate the method, Figure 4 shows the optimal power density curve (Figure 3, 286 with rated turbine speeds between 0.3 m/s and 6.0 m/s in 0.1 m/s increments) for an 287 idealised tidal current, with a single M2 (principal lunar semi-diurnal tidal harmonic) of 288 amplitude 2 m/s (hence each peak current is 2 m/s with no variability between tides). The 289 optimal power curve for the simplified case of Figure 4 is a turbine with a rated speed at 2 290 m/s (as expected), with an optimal mean power and yield density of ~0.6 kW/m^2 and 15 291 kW/m^2 per day respectively (corresponding CF of 41%). 292 Increasing the complexity of an idealised tide example, we demonstrate the power 293 density optimisation for a site with two harmonics in Figure 5: M2 and S2 ( principal solar 294 semi-diurnal harmonic), which together simulate the fortnightly “spring-neap” cycle that 295 describes 75% of UK tidal variability (Robins et al., 2015). Figure 5 demonstrates the optimal 296 power curve for an extreme case, where the S2 amplitude is 60% of the M2 signal (M2 297 amplitude = 1 m/s), such that peak current of 1.6 m/s occurs when M2 (period 12.42 hours) 298 and S2 (period 12 hours) are in-phase (spring tide), and 0.4 m/s peak current speeds occur 299 when M2 and S2 are out-of-phase (neap tide). Optimal yield for Figure 5 was found when 300 the turbine rated speed was that of the peak spring tide (Vr=1.6m/s) but with a much 301 reduced Capacity Factor (17%), due to the extreme nature of the M2/S2 ratio. The 302 importance of weighting the optimal tidal power curve to either yield (i.e. Scenario A or A2) 303 or consistent power (i.e. Scenario B) is demonstrated in Figure 6. 304 305 Variability in choice of an “optimal” power curve, described here as rated turbine 306 speed (Vr) relative to the M2 current amplitude (thus Vr/UM2), is demonstrated in Figure 6 307 for the range of M2/S2 ratios (M2/S2 of 0 has only an M2 tide, whilst equal M2 and S2 308 current amplitudes has a ratio of 1), with four metrics of turbine performance that were 309 calculated applying the idealised power density curve of Figure 3 to a rated turbine speed 310 between 0.3 m/s and 6 m/s (in steps of 0.1 m/s): hence Figure 6 is independent of resource 311 magnitude. The four metrics of turbine performance in Figure 6 were based on yield 312 performance relative to the maximum (Capacity Factor in Figure 6a and yield as a 313 percentage of the maximum possible yield Figure 6c), and the persistence of power supply: 314 percentage of time no power is produced in Figure 6b (as opposed to percentage of time at 315 rated power of Figure 6a) and the largest “power gap” where no power is produced (Figure 316 6d). The choice of what an “optimal” tidal-stream turbine is clear at the extremes of the 317 M2/S2 ratio in Figure 6, where, although extreme, a turbine with a relatively high rated speed 318 Journal Pre-proof would produce large/largest yield but with a low CF and large gaps in power production (thus 319 having consequences in the design and cost of storage and power distribution). 320 321 A large number of constituents are needed to describe the complex processes which 322 give realistic tides (hour-to-hour and day-to-day variability in current speed); for example, the 323 K1 and O1 constituents together describe the diurnal inequality (one tide bigger than another 324 in a given day for semi-diurnal tidal systems), which, with the M2 and S2 constituents, can 325 describe tidal form (F value) and thus the diurnal (one tide per day), semi-diurnal (two tides 326 per day) or “mixed” nature of a tide at any site (Robins et al., 2015). The complexity of the 327 power curve optimisation, based on resource, is further developed from Figure 6 by using 328 these four principle constituents (see Figure 7). Figure 7 shows theoretical turbine 329 performance for yield (panel a) and persistent power (panel b) for all possible turbine power-330 density curves (Vr 0.3-6 m/s) when varying an idealised tidal current based on the tidal dorm 331 (F value), calculated as the relative magnitude of diurnal and semi-diurnal principle 332 constituents (see Robins et al., 2015): 333 1234  567 869 [7]. 334 335 Unlike Figure 6, the result of Figure 7 was found to be affected by the M2/S2 ratio as 336 multiple combinations of four constituent amplitudes can produce the same F value: 337 Therefore, the result of Figure 7 is based on a tide with a M2 amplitude of 1m/s and S2 338 amplitude being 0.1 m/s (M2/S2 = 0.1). Hence, it should be noted that the result of Figure 7 339 would be different if the F value was the same but the M2/S2 ratio were different (based on 340 sensitivity test, an example of which is shown in Appendix Fig. A1). The tidal-stream power 341 density curve optimisation algorithm, which selects the rated speed (Vr) for Scenarios A, A2 342 and B (see Section 2), must therefore be explicitly resolved for each tidal energy site resolve 343 in the global data (Section 3.3). Nevertheless, the uncertainty of optimal rated speed (Vr) is 344 clear in Figure 7 as the divergence of the optimal power-density curve (described as relative 345 rated turbine speed Vr/UM2) for maximum and high yield (Scenario A and A2) or firm power 346 (red line of Scenario B) as the F value increases and the tidal dynamics change from a 347 regular semi-diurnal (F value<0.25) to a mixed (between 0.25 and 3) or diurnal (F value>3) 348 system (i.e. one tide per day tide). 349 350 3.3. Optimal power curve analysis for the world 351 Spatial variability of tidal dynamics are shown in Appendix A2 as details from data are not 352 clear. The variability of global tidal dynamics is shown in Figure 8 relative to resource, 353 calculated here as maximum tidal current speed (maxU) using the sum of the four major tidal 354 dynamics M2, S2, K1 and O1. Probability exceedance (Prob Exc.) of resource (maxU) 355 resolved in FES2014 data up to 25 km from a land mass is shown in Figure 8a; ~12.8% of 356 sites have maxU>1 m/s, 3.6% of sites have maxU>1.5 m/s, ~1.1% sites have maxU>2 m/s 357 and ~0.3% of global sites resolved have maxU>2.5 m/s. The majority of sites have a 358 dominant M2 current amplitude ~70% of maxU; however some potential tidal energy sites 359 (e.g. maxU>2m/s) have a much lower M2 contribution (see Figure 8b), which can also be 360 seen in Figure 8c. Grouping the tidal data of Figure 8c: 53% of sites resolved had F value 361 below 0.25 (semi-diurnal tides) and 46% were “partial” (F value between 0.25 and 3), with 362 relatively large contributions of K1 and O1 constituents. Some “high tidal resource” (e.g. 363 maxU>3 m/s) of Figure 8c exhibit F values above 3 (one tide per day), but account for ~1% 364 of the sites resolved. Figure 8 therefore indicates tidal dynamics at potential tidal-stream 365 energy sites, and thus the temporal variability of resource, will vary greatly around the world, 366 and any analysis that considers low flow sites (e.g. maxU<2.5 m/s) will have an 367 exponentially greater number of sites with varying tidal dynamics to consider (see Figure 7). 368 369 Applying the standardised power curve method (see Section 2), the optimal rated 370 turbine seed (Vr) for Scenarios A, A2 and B were computed (e.g. shown for an idealised tide 371 in Figure 7), and are shown in Figure 9. Optimal rated tidal-stream turbine speed (Vr) using 372 Journal Pre-proof global data is shown in Figure 9 as absolute (Fig. 9a) and as a percentage of annual 373 maximum tidal current speed (Fig 9b). Both maximum (scenario A) and high (scenario A2) 374 optimisation solutions showed little variability with the exception of low resource sites (where 375 maximum current speed was below 1m/s), with a good linear regression fit (panel a) and 376 small standard deviation (shaded region of panel b) of Figure 9 (values given in Table 2). 377 Optimal Vr for Scenario B (firm power) had a large amount of variability and some trend 378 apparent with tidal resource (Table 2 and Figure 9), likely because the result was greatly 379 affected by tidal form (i.e. the relative contribution of diurnal constituents K1 and O1). 380 381 Annual maximum current speed (maxU) was based on the peak current speed 382 simulated at a given site in 2020 using the sum of four tidal constituent amplitudes (e.g. 383 UM2), calculated the major axis length of each tidal constituent ellipse (CMAX), i.e. 384 :2;<  <=% > <?% > <@$ > <A$ BC:2;D EFGFHFI [8]. 385 Hence, optimal rated speed for maximum yield (Scenario A) will below 100% of maxU as this 386 rarely occurs (when the four considered constituents are in-phase). Two measures of Vr are 387 given (% of max U and absolute). The linear regression statistics, and discretised mean Vr 388 (as % of maxU) for grouped site current speeds, are given in Table 2 alongside respective 389 performance metrics of the mean trend line fit (RSQ for absolute) and Pearson correlation 390 (RHO) associated P-value is not shown as all <0.001 at 5% significance. The standard 391 deviation (STD of Table 2) and convergence of shaded area in Figure 9b show variability in 392 an optimal rated turbine speed (relative to resource), and clear convergence can be seen in 393 the optimal yield scenarios (Scenarios A and A2). 394 395 Optimal rated speed (Vr) for scenario B (firm power) varied with resource (i.e. current 396 speed climatology at a site); with relative mean Vr found to increase with maximum current 397 speed (see Figure 9 and Table 2) but with a similar amount of variability (STD of Table 2). 398 This increase in scenario B relative rated turbine speed (Vr as % of maxU) is likely the 399 significant decrease in sites resolved when increasing maxU (see Figure 8a) as well as the 400 tendency for a semi-diurnal (Fig 8b) and dominant M2 amplitude (Fig 8c) in the tidal 401 dynamics. Furthermore, spatial variability in Scenario B was found when Vr (relative to 402 maxU) were grouped into 6 continents - see Table 3 and are shown in Appendix (Figure A3). 403 Therefore, our analysis shows an optimal tidal-stream turbine rated speed (Vr) based on firm 404 power supply spatially varies due to the nature of the tide and the magnitude of the 405 resource. 406 407 4. Discussion 408 Complex analysis involving a large amount of data resulted in a simple set of rules 409 researchers and engineers can use in renewable energy resource assessment: 410 (1) Tidal-turbine cut-in speed (Vs) was found to be ~30% of rated turbine speed (Vr) on 411 average; 412 (2) For a deployment concerned with near-maximum yield aspirations, rated tidal-stream 413 turbine speed (Vr) at a given site will be ~87% to 97% of site maximum flow respectively 414 (where max flow is assumed as the sum of current speed amplitude of M2, S2, K1 and 415 O1 constituents: see Robins et al., 2015), with little global variation found; 416 (3) Deployments concerned with firm, constant power and small amounts of storage, may 417 aim to deploy tidal-stream turbines with much lower rated speeds (~56% of site maximum 418 flow), with spatial variability due to resource (maximum current speed) and the tidal form 419 (F value) – due to the nature of the tide at a given site (see Robins et al., 2015); 420 (4) Average values of normalised data from fourteen horizontal axis tidal-stream turbines 421 (Table 1), alongside our estimation of optimal cut-in and rated speed, allows a 422 standardised power curve and device behaviour (Figure 3) to be implemented in resource 423 and environmental impact assessment, without bias to one specific design (e.g. Fairley et 424 al., 2020) to allow tidal energy resource mapping for future technologies (e.g. Lewis et al., 425 2015). 426 Journal Pre-proof 427 To ensure the result is not affected by the tidal harmonics data, the analysis of the 428 two scenario extremes (maximum yield and firm power: Scenarios A and B) were compared 429 to the result from tidal data at higher resolution: latitudinal resolution of 1/100°(~1 km) 430 instead of 1/16°(~7 km) in the FES2014 global data. The higher resolution tidal data was 431 taken from the Robins et al. (2015) hydrodynamic model of a UK domain (14°W to 11°E, and 432 42°N to 62°N), using the same four tidal constituents (M2, S2, K1 and O1) computed from a 433 30 day simulation. FES2014 data were interpolated to the Robins et al. (2015) computational 434 grid and domain, and Vr optimisation (of section 3.3) repeated; the comparison of the 435 optimisation algorithm, using tidal harmonic data from these two spatial resolutions, is shown 436 in Table 4. 437 438 To compare sensitivity of turbine optimisation to tidal model data accuracy (Table 4), 439 Root Mean Squared Error (RMSE) and Linear Regression score (RSQ) were estimated 440 assuming the higher resolution data accurate, alongside Scatter Index and the mean 441 downscaling value to convert between model spatial resolutions (e.g. M2 amplitude of 442 coarse data was 66% of the higher resolution model on average). Therefore, the tidal 443 resource data may differ between the two model resolutions (coarse data under-predicting 444 flow speed), but the optimal rated turbine design was found to be constant and independent 445 of tidal flow speed (see Table 4) likely because the relative size of the four tidal constituents, 446 used in this study, slowly spatially vary whilst tidal current magnitude is enhanced by 447 bathymetry – and thus dependant on model spatial resolution. 448 449 Indeed, the tidal data sensitivity test (Table 4) showed that although spatially coarse 450 data under-predicted tidal current speeds (both maximum and the main M2 constituent – see 451 Table 4), the optimal rated turbine speed (Vr as a % of maxU) was independent of tidal data 452 resolution. Anecdotal verification of optimal turbine rated speed, using the coarse data, can 453 be assessed by comparing our optimal rated speed result to an industry driven solution; for 454 example, the Meygen site (Pentland Firth) has a maximum current speed ~3.5m/s (Goward-455 Brown et al., 2017) giving an estimated rated speed (Vr) of 2.9m/s to 3.4m/s (for A2 and A: 456 high to maximum yield scenarios), which is very close to the 2.65m/s to 3.05m/s turbines 457 installed at the site (e.g. Website 2) especially given the extremely coarse global tide data 458 (~7km spatial resolution). 459 460 It is likely that the relative magnitude of the major tidal constituents (i.e. excluding 461 over-tides such as M4), which describe tidal form (F value), has low spatial variability (e.g. 462 Robins et al., 2015; Lewis et al., 2017); therefore, tidal dynamics (i.e. nature of tide) are 463 resolved in coarse models as spatial variation is small, but tidal current amplitudes are 464 under-predicted because coarse models do not resolve bathymetric features that accelerate 465 tidal currents (see Lewis et al., 2015). Therefore, coarse resolution tidal data can be used to 466 resolve tidal dynamics, but not the magnitude of theoretical tidal-stream energy resource 467 hence, future resource mapping efforts must be based on high resolution tidal data (also 468 concluded in Lewis et al., 2017). Higher tidal harmonics (such as the combination of M2 and 469 M4, leading to overtides and flood-ebb asymmetry) can have a significant effect on resource 470 assessment (Neill et al., 2014), and are enhanced by tidal-stream turbine deployments (e.g. 471 Neill et al., 2009), whilst interaction of array-scale tidal energy developments must be 472 included within resource assessment (e.g. Garrett and Cummins, 2008; Vennel et al., 2015); 473 therefore, we hope the standardised power curve presented here will lead to improved 474 understanding of tidal-stream energy potential. 475 476 The approach taken to provide a standardised power curve for use in tidal-stream 477 resource assessment, builds on the work of Hardisty (2012) in the application of an idealised 478 tidal-stream power curve, and device technology reviews of Roberts et al. (2016) and Zhou 479 et al. (2017). In the technologically mature wind energy industry (Lydia et al., 2014), there is 480 reported convergence in wind turbine cut-in speeds (due to insufficient torque to initiate 481 Journal Pre-proof turbine rotation at wind speeds lower than 3m/s) and rated speeds (11-17 m/s), although 482 some variability in design depending on local wind conditions (Carrillo et al., 2013). The 483 knowledge of a common power curve in the wind industry has supported mean resource 484 assessment with much research now focusing on finer-scale variability (Trivellato et al., 485 2012; Lydia et al., 2014). Therefore, our simple set of tidal turbine power curve rules, set out 486 in this paper, would allow improved resource and impact assessments with hydrodynamic 487 models. 488 489 Given the deterministic predictability of tidal-stream resource, and the establishment 490 of a standardised and resource-led power curve (presented here), a convergent tidal-stream 491 energy power curve should be the focus of future research to aid resource mapping (e.g. 492 ”mhkit”; website 4). If we apply technology development of tidal-stream energy (based on 493 Lewis et al., 2015): 1 st to 3 rd generation sites have peak flow speeds >2.5m/s, 2m/s, and 494 1.5m/s respectively. Applying the high yield optimisation (Scenario A2) to the global tidal 495 data: 1 st generation devices should be considered having rated speeds above 2.2 m/s (Vs ~ 496 0.7 m/s), with 2 nd generation rated speed above 1.7 m/s (Vs ~0.5 m/s) and 3 rd generation 497 rated speed above 1.3 m/s (Vs ~0.4 m/s); close to the 0.5 m/s current speed threshold to 498 initiate turbine rotation (Encarnacion et al., 2019). 499 500 Technological learning has led to a reduction in the cost of wind energy devices 501 (Junginger et al., 2004), and a similar cost reduction is expected for tidal energy (Johnstone 502 et al., 2013). Our analysis confirms tidal turbine rated speed optimisation can be achieved. 503 The inclusion of swept tidal-stream turbine area, alongside economies of scale, practical and 504 socioeconomic constraints (e.g. Vazquez and Iglesias 2015), would therefore allow for a 505 convergent resource-optimised tidal turbine design and cost assessment. However, future 506 research must resolve uncertainties in array design choice (e.g. Coles et al.,, 2020); for 507 example, resolving cost of optimised device resilience (maintenance) and yield, will one 508 turbine be installed throughout a country, region or array? 509 510 The predictability of tidal energy, compared to the temporal variability of other non-511 thermal renewable energy resources (see Lewis et al., 2019) and the analysis presented 512 here, indicates the need for develop tools that can perform “whole systems” design of 513 renewable energy systems – where the storage costs and dispatchability of power included 514 in supply-demand analysis (e.g. Stegman et al., 2017; Al Katsaprakakis et al., 2019) as well 515 as resilience and reliability (Johnstone et al., 2013). As power is proportional to the cube of 516 velocity (equation 1), challenges in competitive costed low-flow tidal turbines are clear (i.e. 517 low yields will likely raise LCOE greatly). However, the potential for low-flow tidal energy 518 devices appears great if we consider the persistence of power density achieved with a 519 Scenario B power curve (gap in power <2hours with the highest Capacity Factor), the cost of 520 storage and resilience in an off-grid energy solution: for example, Large lithium batteries 521 (~$500/kWh Nielsen et al., 2018) and the use of back-up diesel generators (e.g. Mala et al., 522 2009). 523 524 Given the prevalence of lower tidal flow sites (e.g. Lewis et al., 2015; 2017), where 525 turbulence intensity (Lewis et al., 2019) and less mean vertical shear (Lewis et al., 2017b) 526 will improve resilience of devices (Encarnacion et al., 2019), the potential cost of low flow 527 tidal-stream turbines appears an important future step. Applying the conservative “firm 528 power” optimisation (Scenario B) to the global data: 1
st
generation devices would have a 529 rated speed of ~1.5 m/s (Vs ~0.5 m/s), 2
nd
generation rated speed ~1.2 m/s (Vs ~0.4 m/s) 530 and 3
rd
generation ~0.9 m/s (Vs ~ 0.3 m/s). Although all rated speeds in our Scenario B were 531 above the 0.5 m/s threshold, novel turbine designs are will be needed to improve tip-speed-532 ratios of turbines at low current speed (0.5 m/s or below: Encarnacion et al., 2019). Indeed, 533 our analysis finds ~12.8% of the world’s coastlines have maximum current speeds above 1 534 m/s (resolved in FES2014 up to 25km offshore and excluding high (> 70°) Latitudes), and 535 3.6% for maxU>1.5 m/s (see Figure 8 and Appendix A2); however absolute currents speeds 536
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are known to be effected by ocean model resolution (Lewis et al., 2015; 2017) and this 537 number is likely to be much higher. Therefore, higher resolution tidal resource data is 538 needed to perform a full tidal-turbine device optimisation assessment, but the analysis 539 presented here shows a suitable method once such data is available. 540 541 Previous research, using high resolution regional models, has shown less energetic 542 flows dominate South East Asia (e.g. Encarnacion et al., 2019), such Malaysia (current 543 velocities reaching up to 1.2 m/s Lim and Koh, 2010) and Philippines (“most areas reaching 544 current velocities of 1.4 m/s” Encarnacion et al., 2019). The development of floating tidal-545 stream devices (Brown et al., 2020) has unlocked the potential for 2
nd
and 3
rd
generation 546 tidal energy sites in the Gulf of California (where peak currents are between 1.0 and 2.4m/s, 547 Mejia-Olivares et al., 2018), and the Kuroshio current where 1m/s to 1.5m/s oceanic currents 548 could be harnessed with floating deep-water, large swept area devices (Liu et al., 2018). 549 Indeed, low-flow rated (1.3 m/s to 1.7 m/s) tidal energy kites, with a large swept area, are 550 also being tested and deployed (Buckland et al., 2015; Roberts et al., 2016). However, there 551 is still a gap in low flow tidal turbines for lower power demand markets and “blue growth 552 economies“ (LiVecchi et al., 2019). For example, incorporation of tidal energy into offshore 553 aquaculture would require tidal-stream devices capable of operating in <1m/s flows (see 554 Gentry et al., 2017), and although some bio-optimisation to accelerate tidal currents is 555 possible (O’Donncha et al., 2017) it may not be required given modest power needs 556 (Aquatera 2014). We therefore, find two tidal-stream turbine markets and designs may be 557 found in the future: (1) larger MegaWatt scale electricity production for grid-connected 558 regions and (2) smaller-scale power systems that provide firm energy for higher value, 559 remote industries and communities. 560 561 5. Conclusion 562 Given the sparsity of published power curves in the literature, and the diverse range of 563 markets tidal energy could benefit, an unbiased power curve characterisation is essential to 564 map tidal-stream energy resource. A standardised tidal-stream power curve was developed 565 so that resource assessment beyond realised technologies can be possible. Our analysis 566 and resource-led optimisation was unaffected by tidal data; finding divergence in rated-567 speed based on weighting of importance: firm power with low amounts of storage, or high 568 yield with larger storage needs. A general rule for turbine power curve of a horizontal-axis 569 turbine was found: cut-in speed was around 30% of the rated speed; and optimal rated 570 speed (tidal current when peak power converted) was either ~50% or greater than 87% of a 571 site’s maximum current speed (based on sum of M2, S2, K1 and O1 harmonic constituents) 572 for firm power or maximum yield respectively - due to the dominance of the major semi-573 diurnal lunar tidal constituent (M2). This paper demonstrates the power” of deterministic 574 predictability with tidal energy, and although temporal variability of the tidal resource appears 575 to be captured by current tidal data products, higher resolution data could transform the tidal-576 stream energy industry by fully mapping the resource. This work also adds to the weight of 577 evidence that a convergent tidal turbine design is needed, and possible, but two tidal-stream 578 turbine types may exist: one for electricity supply to large grid connected communities, and 579 another “lower resource” turbine for remote industry and communities that may have much 580 lower rated speeds. 581 582 6. Acknowledgements 583 584 M. Lewis, P. Robins and S. Neill acknowledge the support of SEEC (Smart Efficient Energy 585 Centre) at Bangor University, part-funded by the European Regional Development Fund 586 (ERDF), administered by the Welsh Government. M. Lewis is funded through the EPSRC 587 METRIC fellowship (EP/R034664/1) and wishes to acknowledge Deltares for hosting his 588 research visit in 2019. It is dedicated to M Lewis’ late friend, and mentor, Dr. Gerbrant van 589 Vledder. This manuscript was developed during conversations with Dr. D Coles, who helped 590 inform the discussion and future work from this research. This paper was developed during 591
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the visit of Dr. S Fredriksson to Bangor University, as part of the PRIMARE SRV grant 592 (Goward-Brown: PI). Dr. Fredriksson wishes to acknowledge supported by the Swedish 593 Energy Agency. 594 595 7. References 596 597 Al Katsaprakakis, D., Thomsen, B., Dakanali, I. and Tzirakis, K., 2019. Faroe Islands: 598 towards 100% RES penetration. Renewable energy, 135, pp.473-484 599 Aquatera 2014.; Renewable power generation on aquaculture siteS, SARF093, report by 600 Aquatera LtD, 2014, www.sarf.org.uk 601 Brown, S.A., Ransley, E.J., Zheng, S., Xie, N., Howey, B. and Greaves, D.M., 2020. 602 Development of a fully nonlinear, coupled numerical model for assessment of floating 603 tidal stream concepts. Ocean Engineering, 218, p.108253. 604 Buckland H, Dolerud E, Baker T. 2015. Application of Standard Tidal Performance 605 Specification and Performance Review to a Non-Standard Tidal Energy Converter. 606 European Wave and Tidal Energy Conference (EWTEC), Spetember 2015. Nantes, 607 France 608 Carrere, L., Lyard, F., Cancet, M. and Guillot, A., 2015. FES 2014, a new tidal model on the 609 global ocean with enhanced accuracy in shallow seas and in the Arctic region. EGU 610 General Assembly, p. 5481. 611 Carrillo, C., Montaño, A.O., Cidrás, J. and Díaz-Dorado, E., 2013. Review of power curve 612 modelling for wind turbines. Renewable and Sustainable Energy Reviews, 21, pp.572-613 581. 614 Coles, D.S., Blunden, L.S. and Bahaj, A.S., 2020. The energy yield potential of a large tidal 615 stream turbine array in the Alderney Race. Philosophical Transactions of the Royal 616 Society A, 378(2178), p.20190502. 617 Egbert, G.D. and Ray, R.D., 2001. Estimates of M2 tidal energy dissipation from 618 TOPEX/Poseidon altimeter data. Journal of Geophysical Research: Oceans, 619 106(C10), pp.22475-22502. 620 Encarnacion, J.I., Johnstone, C. and Ordonez-Sanchez, S., 2019. Design of a horizontal axis 621 tidal turbine for less energetic current velocity profiles. Journal of Marine Science and 622 Engineering, 7(7), p.197. 623 Fairley, I., Lewis, M., Robertson, B., Hemer, M., Masters, I., Horrillo-Caraballo, J., 624 Karunarathna, H. and Reeve, D.E., 2020. A classification system for global wave 625 energy resources based on multivariate clustering. Applied Energy, 262, p.114515. 626 Garrett C, Cummins P. The efficiency of a turbine in a tidal channel. J Fluid Mech 627 2007;588:243e51. 628 Garrett C, Cummins P. The power potential of tidal currents in channels. Proc R Soc A 629 2005;461:2563e72. [7] 630 Brown, A.J.G., Neill, S.P. and Lewis, M.J., 2017. Tidal energy extraction in three-631 dimensional ocean models. Renewable energy, 114, pp.244-257. 632 Goward Brown, A.J., Lewis, M., Barton, B.I., Jeans, G. and Spall, S.A., 2019. Investigation of 633 the Modulation of the Tidal Stream Resource by Ocean Currents through a Complex 634 Tidal Channel. Journal of Marine Science and Engineering, 7(10), p.341. 635 Guillou, N., Neill, S.P. and Robins, P.E., 2018. Characterising the tidal stream power 636 resource around France using a high-resolution harmonic database. Renewable 637 Energy, 123, pp.706-718. 638 Hardisty, J., 2012. The tidal stream power curve: a case study. Energy and Power 639 Engineering, 4(3), pp.132-136. 640 Johnstone, C.M., Pratt, D., Clarke, J.A. and Grant, A.D., 2013. A techno-economic analysis 641 of tidal energy technology. Renewable Energy, 49, pp.101-106. 642 Junginger, M., Faaij, A. and Turkenburg, W.C., 2004. Cost reduction prospects for offshore 643 wind farms. Wind engineering, 28(1), pp.97-118. 644
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753 Fig 1. A demonstration of the effect of two tidal power curves on resource 754 assessment. A spring-neap time-series (2m/s M2 Cmax and 0.5m/s S2 Cmax) of tidal 755 current speed (panel a) is converted to theoretical power density (PD) in panel b and 756 technical power density (panel c) for a power curve rated at 2.5m/s (red line) and 757 2.0m/s (blue line). 758
759
Fig 2. Tidal-stream turbine characteristics from 14 commercially developed devices 760 (panel a), normalised (relative to rated power and speed) and compared to observed 761 variability (grey dots and averaged power curve in red) from a grid-connected device 762 in Lewis et al. (2019). 763
764
Fig 3. A standardised power curve, based on 14 horizontal axis tidal-stream turbines, 765 with the associated Drag (as percentage of maximum drag, Dr) and Thrust Coefficient 766 (CT) normalised curves. 767
768
Fig 4. Single harmonic tidal current (M2 amplitude 2 m/s) over a 2 day period (panel a), 769 and the theoretical power density (PD) of this current (panel b), compared to the mean 770 power density and Capacity Factor (panel c) for multiple tidal-stream turbine power 771 curves, where rated power is capped at rated speed (Vr, and cut-in speed is 30% of 772 Vr), which allows mean daily yield density to be calculated (panel d). 773
774
Fig 5. Spring-Neap tidal current (M2 amplitude 1 m/s, S2 amplitude 0.6 m/s) over a 7 775 day period (panel a), with the theoretical power density (PD) of this current (panel b). 776 Multiple tidal-stream turbine power curves, where rated power is capped at rated 777 speed (Vr, and cut-in speed is 30% of Vr), are applied to resolve an optimal design 778 using mean power density and Capacity Factor (panel c) and mean daily yield density 779 (panel d). 780
781
Fig 6. Performance of multiple tidal-stream power curves, represented here as rated 782 speed (Vr) relative to the resource (amplitude of M2 harmonic: UM2), for a given site 783 where the tidal currents are controlled solely by the spring-neap cycle and the ratio of 784 M2 and S2 amplitude (M2/S2 of 0 has only an M2 tide, whilst equal M2 and S2 current 785 amplitudes has a ratio of 1). Turbine performance is described using Capacity Factor 786 (a), percentage of time no power produced (b), (c) mean yield density (relative to 787 maximum possible) and (d) the longest period of zero power in a 15 day time-series. 788
789
Fig 7. Tide currents harmonic characteristic tidal form (F value), rated turbine speed 790 (relative to M2 current amplitude UM2) and subsequent yield and Capacity Factor 791 (CF) shown in panel a, with mean monthly percentage of zero power and maximum 792 period of no power (max gap) in panel b. Lines of optimal power curve shown in solid 793 white for selection of maximum yield (Scenario A), high yield (scenario A2) as dashed 794 white line and firm power (power gap < 2 h with highest CF: scenario B) as red 795 dashed line. 796
797
Fig 8. Global variability of tidal dynamics, described as maximum flow (maxU) 798 percentage exceedance (a) for sites “coastally” (<25 km offshore) resolved in the 799 FES2014 data, (b) coloured percentage occurrence of M2 amplitude contribution to 800 the maximum flow (as percentage of M2 current amplitude compared to maximum 801
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current speed), and (c) coloured percentage occurrence of the tidal form (F value) that 802 describes the diurnal (F>3) to semi-diurnal (F<0.25) nature of the tide 803
804
Fig 9. Rated tidal-stream turbine speed using standardised power density curve and 805 three optimal solutions: Scenario A (maximum yield density shown in black), Scenario 806 A2 (high yield density shown in blue) and Scenario B (firm power shown in red). 807
808
809 810 Appendix 811 812 813
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814 Fig A1. An example of sensitivity to the tidal-stream turbine optimisation result of 815 Figure 7, when considering tidal dynamics with different M2/S2 ratios but equal F 816 values. Tide currents harmonic characteristic tidal form (F value), rated turbine speed 817 (relative to M2 current amplitude UM2) and subsequent yield and Capacity Factor 818 (CF) shown in panel a and c; with mean monthly percentage of zero power and 819 maximum period of no power (max gap) in panel b and d 820 821
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822 823 Fig A2. Global tidal dynamic variability, described as: (a) maximum current speed; (b) 824 percentage of M2 current amplitude compared to maximum current speed; and (c) 825 Tidal form (F-value) using FES2014 data 826
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827 Fig A3. The optimal rated tidal-stream turbine speed, for three scenarios (a to c for 828 max, high and firm yield respectively), based on FES2014 global data and grouped 829 into the 6 continents: South America (a), North America (b), Asia (c), Europe (d), 830 Africa (e), Australasia (f). 831 832 833
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Table 1. A literature review of 14 horizontal axis tidal-stream turbines, where device
characteristics are published or estimated (marked with *), including: rotor diameter
(); Rated Power (Pr); power coefficient (Cp); cut-in velocity (Vs) when the turbine
starts to produce power; and rated velocity (Vr), the current speed when maximum
power (Pr) is produced. Labels of devices in Fig. 2 are defined in the ID column.
ID device
(m)
Pr
(kW) Vr
(m/s) Vs
(m/s)
Vs
(as %
of Vr) Cp* source
1 MCT 16
600 2.5 1 40 0.37 Lewis et al. (2015)
2 Alstrom 18
1000 2.7 1 37 0.39 Lewis et al. (2019)
3 sabella D-10 10
1000 4 1 25 0.39 Website 1
4 sabella D-15 15
2300 4 1 25 0.4 Website 1
5
seagen
-
2MW twin
rotor 20
1000
(per
rotor) 2.5 1 40 0.4 Website 2
6
Atlantis
AR1000 18
1000 2.65 --- --- 0.41 Website 2; Roberts
et al. (2016)
7
Atlantis
AR2000 22
2000 3.05 <1 --- 0.36 Encarnacion et al.
(2019); Website 2
8 Verdant
gen5 5
35 2.59 --- --- 0.32
Polygae et al
(2010);
Encarnacion et al.
(2019)
9 Nova 8.5
100 2 0.5 25 0.43 Encarnacion et al.
(2019)
10 Voith 16
1000 2.9 --- --- 0.4 Roberts et al.
(2016)
11 openhydro 10
200 2.5 --- --- 0.32 Polygae et al
(2010); Roberts et
al. (2016)
12
schottel
hydro d3 3
70 3.7 0.9 24 0.38 Website 3
13
schottel
hydro d4 4
62 3.1 0.8 26 0.32 Website 3
14
schottel
hydro d5
5
54 2.6 0.7 27 0.31 Website 3
Mean 13 816 2.91 0.88 30% 0.37
Standard
Deviation 6 803 0.6 0.18 7% 0.04
Table 2: Optimal rated tidal-stream turbine speed (Vr) relative to maximum tidal
current speed (MaxU) at any given “coastal” site globally for three optimal power
scenarios, with two methods of representing Vr: absolute with linear regression of
max U and Vr (with linear regression score: RSQ), and Vr relative to maxU at site,
discretised into 0.5m/s groups with mean Vr (as % maxU) and associated standard
deviation (std), with the Pearson correlation score (RHO) is given to indicate strength
of statistical fit at 5% confidence
Optimal Vr scenario:
Max yield
(scenario A) High yield
(scenario A2) Firm power
(scenario B)
Absolute Vr trend RSQ ~100% ~100% 92%
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trend Vr = 0.97*maxU Vr = 0.87*maxU Vr = 0.56*maxU
Mean Vr (as % of
maxU) with
standard deviation
in brackets (std)
maxU Vr as % of maxU (std)
0.5m/s 107 (17) 102 (17) 49 (13)
1.0m/s 99 (8) 93 (9) 48 (17)
1.5m/s 97 (4) 87 (4) 57 (16)
2.0m/s 96 (3) 86 (4) 59 (15)
2.5m/s 96 (2) 85 (4) 58 (16)
3.0m/s 96 (2) 84 (3) 58 (16)
3.5m/s 96 (2) 84 (3) 60 (17)
4.0m/s 96 (2) 85 (3) 64 (16)
4.5m/s 96 (2) 84 (3) 67 (12)
RHO -0.28 -0.43 0.24
Table 3: The linear trend of optimal absolute rated turbine speed (“Trend” Vr in m/s),
with each respective linear regression score (RSQ), for three tidal-stream energy
scenarios (A , A2, and B) and spatially grouped data by continent, using four major
tidal constituents of FES2014 data (latitude <70° and up to 25km offshore)
region Scenario: A (max yield) A2 (high yield) B (firm power)
World RSQ 100% 100% 92%
Trend Vr=0.97*maxU Vr=0.87*maxU Vr=0.56*maxU
Europe: RSQ 100% 100% 93%
Trend Vr=0.96*maxU Vr=0.85*maxU Vr=0.46*maxU
Australasia:
RSQ 100% 100% 91%
Trend Vr=0.97*maxU Vr=0.87*maxU Vr=0.54*maxU
Asia: RSQ 100% 100% 93%
Trend Vr=0.96*maxU Vr=0.85*maxU Vr=0.46*maxU
Africa: RSQ 99% 99% 91%
Trend Vr=maxU Vr=0.92*maxU Vr=0.53*maxU
North America: RSQ 100% 99% 95%
Trend Vr=0.98*maxU Vr=0.89*maxU Vr=0.61*maxU
South America: RSQ 100% 100% 96%
Trend Vr=0.99*maxU Vr=0.89*maxU Vr=0.74*maxU
Table 4: Comparison of optimal tidal-stream turbine rated speed (Vr) based on two
scenarios (max yield and firm power; scenarios A and B respectively) using tidal
harmonic data, giving peak current speed as the sum of the four major constituents
(K1,O1,S2,M2), called maxU, as well as the amplitude of current speed for the M2
constituent (Ua), for high and coarse spatial resolution (Res.) model data comparison
for the UK region. Comparison metrics: Root Mean Squared Error RMSE) and linear
regression score (RSQ) provided alongside scatter and average conversion between
resolutions.
High res. Robins et al.
(2015) (~1km spatial
Coarse res. FES2014
(~7km resolution)
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resolution)
Scenario A linear trend: Vr~0.97maxU+0.01 Vr~0.98maxU
Scenario B linear trend: Vr~0.58*maxU+0.05 Vr~0.56maxU + 0.27
M2 current amplitude
comparison:
RMSE = 0.18 m/s (4%)
RSQ = 71%
Scatter Index = 31%
Coarse(Ua) ~ 0.66*high(Ua)
Maximum current
comparison:
RMSE = 0.23 m/s (4%)
RSQ = 71%
Scatter Index = 28%
Coarse res.
(maxU) ~ 0.68*high res. (maxU)
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Highlights:
Standardised horizontal-axis tidal-stream turbine power-density curve
developed
Convergent power curve characteristics assessed with global tide data
Divergence in rated-speed when selecting for optimal yield or persistent power
Resource-led turbine optimisation is possible but high resolution tidal data
needed
High and low flow designs appear needed to capitalise on resource
predictability
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I declaration no conflicts of interest. This work was funded through EPSRC METRIC
EP/R034664/1. M. Lewis, P. Robins and S. Neill acknowledge the support of SEEC (Smart
Efficient Energy Centre) at Bangor University, part-funded by the European Regional
Development Fund (ERDF), administered by the Welsh Government. M. Lewis is funded
through the EPSRC METRIC fellowship (EP/R034664/1), and wishes to acknowledge
Deltares for hosting his research visit in 2019. Dr. Fredriksson wishes to acknowledge
supported by the Swedish Energy Agency.
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There has been a growing interest in tidal-stream energy, with most past studies focusing on assessing the potential resource of sites with fast tidal currents in relatively shallow water. Regions with less energetic tidal currents, but in deeper waters, have been overlooked. One potential tidal-stream energy region, which fits this categorization, is the Gulf of California. In this paper we quantify the theoretical tidal-stream energy resource in this region. The resource is estimated with an unstructured depth-averaged hydrodynamic model. We find that the highest flow speeds of 2.4 m/s occur in the channel between San Lorenzo and San Esteban Island, and three lower-velocity potential sites are identified in the channels between: (1) Baja California Peninsula and San Lorenzo Island; (2) San Esteban and Tiburon Islands and (3) Baja California Peninsula and Angel de la Guarda Island. Although peak kinetic power density in these regions is found to be relatively low (∼ 3 to 6 kW/m2), the large water depth (100 to 500 m), results in an undisturbed theoretical annual mean power of between 100 to 200 MW. We therefore find the tidal energy resource to be large, but new turbine technologies would be required to exploit these ‘next generation’ resource regions.