Edinburgh Research Explorer Sensitivity of tidal range assessments to harmonic constituents and analysis timeframe

Tides exhibit variability over time. This study proposes a methodology for selecting a representative timeframe for tidal range energy analyses, when constrained to a typical, short-term, lunar month-long period. We explore how the selection of particular timeframes skews findings of energy assessments, especially for cross-comparisons across studies. This exercise relies on metrics assessing the magnitude and variability of a tidal signal relative to longer-term nodal cycle quantities. Results based on UK tide gauges highlight that tide characteristics exhibit significant variations temporally within a lunar month. Relative to quantities of tidal elevation standard deviation or average potential energy, values can vary by 15% and 30% respectively. For each lunar month, interquartile range values for tidal height and energy can deviate by 45% from the mean. Spatially, we observe a satisfactory correlation only once sufficient constituents are considered. In that case, a representative timeframe can be identified for comparative tidal range scheme assessments within the same tidal system. In contrast, timeframes with high tidal variability distort individual project performance, particularly under fixed operation. The methodology, if integrated to marine energy resource and environmental impact assessments, would deliver marine power generation insights over a project lifetime that enable robust design comparisons across sites.


Nomenclature
1 2 E M k,j array of theoretical tidal range energy entries E i within M k,j 3 G M k,j array of extractable tidal range energy entries E 0D,i within M k,j 4 R M k,j array of tidal range entries R i within M k,j 5 P E temporally averaged wave potential energy per unit surface area (Wh/m 2 ) 6 P E k temporally averaged wave potential energy per unit surface area for wave ele- There are numerous studies associated with tidal range energy (Table 1) constituents returns on average -10 to -13 kWh/m 2 lower resource. 126 Cornett et al. [ 1. We employ harmonic analysis to extract the most influential constituents across Table 1: Examples of modelling studies related to tidal range resource. Columns include percentage differences of averaged potential energy P E, tidal range energy E variability (IQR), and rating scores of study periods relative to Section 3.5. Tidal signals were reconstructed using the 12 leading constituents and the signal duration was adjusted to the reported timeframe ∆t.
where h is the mean surface level above the datum, f i is a node factor to account Greenwich.

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In this study harmonic analysis is conducted using the Python package uptide

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[45] to reconstruct tidal signals at 46 tide gauge stations across the UK as in Fig.   187 2. Harmonic analysis determines the amplitude and phase of tidal frequencies using  constituents comparison is presented in Fig. 1, where the tidal range R i recorded by 199 the i th transition from high water to low water and vice versa is annotated.
200 Table 2 presents an example of the amplitude (α) and phase (ϕ) of the most influ- where α i , the amplitudes of harmonic constituents for i ∈ {M 2 , S 2 , K 1 , O 1 } is indicated. For F < 0.25 tides are classified as semidiurnal; while, for 0.5 < F < 1.5 as mixedmainly semidiurnal.
(NRMSE), the Mean Absolute Error (MAE) and the coefficient of determination (R 2 ), 219 defined as and where n is the length of data set, µ is the mean of observed water elevations,η i the  in defining the water elevation time series interval η(∆t, k, j) and the notation for 232 representative lunar months as elaborated in the following sections. As tides are very long waves, we adopt some widely used coastal wave statistics.

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For example, tidal range itself corresponds to wave height, and the tide elevation 236 standard deviation from MWL would refer to the significant wave height H m 0 . The tidal range magnitude R i is defined as the difference between high and low water 240 in the i th transition from elevation peaks to troughs or vice versa (Fig. 1a). As in Fig.   241 1b, tide signals of multiple constituents are not sinusoidal, and they vary over short- where µ η = ∞ −∞ ηf (η)dη the mean and f (η) is the probability density function of the 260 tidal signal segment η(∆t, k, j), with arguments ∆t, k, j as defined in Fig. 3.  : where ρ is the fluid density, g is the gravitational acceleration. As with R i , E i relies on discrete points rather than the entire tidal signal. We thus also  given by in which H i = 2α i is the wave height of each constituent. Similarly to the amplitude 295 of constituents we define a participation percentage P E i /ΣP E to account for the 296 influence of constituents on the total average potential energy as in Table 2.  predicted capacity is defined as where η e is the expected generation efficiency, R is the mean annual tidal range and 327 C F is a capacity factor. We set η e = 0.40 following the estimate of 37% by Burrows As with E i and P E, we define equivalent metrics associated with the technically 352 extractable energy. The 0-D energy output prediction over a period ∆t = t i+1 − t i is 353 given by: where P (t) the power output. Each tidal cycle consists of two transitions; one from 355 HW to LW and vice versa. Thus, in correlating E 0D,i comparative to E i we consider 356 the associated energy over half tidal cycles; that is, we set ∆t = T 2 . E 0D,i is in turn 357 aggregated in G consistently with the tidal range and potential energy quantities.

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Next, we define the average 0-D energy output over an arbitrary period ∆t as rendering it comparable to P E (Eq. (9)). In examining the generated energy 360 relative to the available resource over each transition i, we define the efficiency factor and by extension, we denote as η g the lunar-monthly efficiency.
i.e. measures the maximum discrepancy corresponding to empirical CDFs (F X , F X * ) 373 of the samples X and X * (of size m and n respectively). This approach is sensitive 374 to detect differences in both the location and the shape of the empirical cumulative i.e., equal to the area between the two CDFs.
where α and β are weight factors ( in this case α = β = 0.5). M 1 effectively 392 considers the 1-D array X over a particular period (e.g. a lunar month M ) relative 393 to the equivalent X * of a different duration (e.g. a nodal cycle N ). 394 We then consider a second metric, M 2 based on Y ∈ {H m0 , P E, E 0D } as where Y, Y * represent the same quantities over a different timeframe.

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Focusing on tidal range R as an example, X = ⃗ R(M, k, j) and X * = ⃗ R(N, k, 1)     Table 2. Thus, 420 given further data gaps in tide gauge records that add to the uncertainty, we consider 421 k = 12 as the baseline for our analysis. NRMSE and R 2 for the locations of highest 422 range are shown in Fig. 6 with respect to k. The largest NRMSE and the smallest R 2 423 were predicted at Avonmouth where Σα is greatest (8.98 m). This is also expected 424 due to the pronounced non-linear shallow water hydrodynamics present at estuarine    (Fig. 8c). Interestingly, we notice that the tidal range energy variability using IQR 460 exhibits a much higher variation of over 45% (Fig. 8a). Despite the deviation range 461 across gauges, we observe a convergence to baseline predictions for both P E and 462 IQR( ⃗ E), once k ≥ 8.

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The MAE across gauges for k = 8 with regards to P E and IQR( ⃗ E) is 0.7% and 464 4.7% respectively. The latter would be considered acceptable given additional non-465 tidal uncertainties. For k = 16, we obtain equivalent MAEs as for k = 8, affirming 466 the convergence to representative months beyond this point for the UK tidal system.

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While the above results refer to the potential energy content, equivalent results are 468 acquired for tidal range quantities. In Fig. 8    lines are fitted to explore trends between datasets (Fig. 9b). For a fixed operation,  Having established the representative months, we investigate the spatial varia-504 tion for implications to engineering assessments (e.g., tidal range plants). Fig. 11 505 illustrates the spatial behaviour of representative months in Avonmouth. We observe 506 that when these are applied simultaneously across tide gauge sites, corresponding 507 errors for P E, E 0D and associated IQR are confined. Indicatively, the MAE in P E is 508 0.7%, 1.5% and 0.9% for M E k,r , M E k,D and M E k,W respectively. While, the corresponding 509 errors in IQR( ⃗ E) are 5.5%, 5.0% and 6.5% respectively. Additionally, Fig. 11 pro- data. This is indicated in Fig. 6, where we observe that the comparative metrics 526 exhibit no further significant convergence with the addition of constituents beyond 527 around 12. As discussed previously, most uncertainty arises in areas of the greatest re-528 source. This becomes more apparent by observing the RMSE and MAE in Fig. 6a,d.

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We notice that Avonmouth, Portbury and Newport, being closest to the tidal limit     12)), for representative energy months for Avonmouth. Blue bars indicate the range of P E and IQR( ⃗ E); while, black ones display the range of E 0D (M, 12) and IQR( ⃗ G (M, 12)).
The accuracy of predicted water levels is critical in any feasibility assessment of 533 tidal range plant as well as related environmental impact. Apart from historical data, 534 tidal elevation time-series may also be generated from 2-D hydrodynamic models.  Eq. 10). It is defined over recurring signals over long-term periods i.e, a nodal cycle.

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It is expected that the contribution of other constituents becomes more noticeable 547 over constrained periods when phase differences becomes more significant as indicated show encouraging performance compared to ratings in studies from literature (Table   570 3).

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The qualitative performance of representative months based on Avonmouth when 599 rated across the rest of the tide gauge network is statistically explored in Table 3.

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First, for each metric we consider the average value of ratings denoted as RS metric .

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Taking the mean of RS metric for each representative month, we notice that they are of  The technical extractable energy from tidal range plants is closely linked to both 643 the theoretically available resource and the associated variability as in Fig. 9a and 644 the high r s values of Fig. 9b. We observe that operation optimisation primarily 645 benefits energy conversion over high resource tidal cycles (e.g., spring tides). This is  distance and a custom metric that accounts for the magnitude and variability of tidal 671 ranges and energy over prescribed periods. As part of the analysis, a rating score was 672 introduced to evaluate lunar month timeframes within a nodal cycle. We note the [11] E.