As the wave energy industry develops, there is a clear and distinct need to provide higherfidelity resource assessments and reduce the associated uncertainty. The IEC Technical Specification suggests a methodology that will undoubtedly provide the necessary data for an assessment, particularly when assessing the gross mean annual energy production. However, evidence indicates that the resulting data set may be insufficient to mitigate uncertainty in the hour-to-hour variability in the final WEC power production (Robertson et al. 2015, 2016). Hence, higher fidelity assessment methods are being developed to reduce the uncertainty in the resource characteristics and parameterizations. However, these improvement methods need to continue to create results that are easily relayed to those with a less intimate understanding of wave dynamics.
Prior to detailing these improved methods, it is illustrative to revisit the baseline parameterization of a full directional wave spectrum into a few representative parameters. This process inherently introduces significant uncertainty in the resulting wave resources assessment. The parameterization of the directional wave energy spectrum into significant wave height and energy period parameters
Nov Dec Jan Feb Mar Apr May Jun Jul
Month 2
4 6 8
Significant Wave Height (m)
Fig. 8 Hourly variation of significant wave height in Western Canada
18 B. Robertson
removes all details about the variance density distribution across the frequency and direction axes. As Fig.9illustrates, directional variance density information is lost when using a nondirectional wave spectrum, and the frequency variance density information is lost when using basic parameters. The colored boxes in Fig.9 indicate individual wave systems in the various representations.
As a result of this information loss, it is commonly assumed that any wave condition, drawn from any histogram bin, can be represented with a single-peaked spectrum with a Joint North Sea Wave Project (JONSWAP) or Pierson–Moskowitz (PM) spectrum with an equivalent Hm0 and Te value (Hiles et al. 2014;
Lenee-Bluhm et al.2011; Rusu and Soares2012; Babarit et al.2012; Iglesias and Carballo2011; van Nieuwkoop et al.2013). The generalized JONSWAP spectral form is presented in Eq. (12):
E fð Þ = αg2ð2πÞ− 4f− 5exp −5 4
f fpeak
− 4
" #
γexp −12
f̸fpeak − 1 φ
2
h i
ð12Þ
where α is calculated so thatR
Sðf Þ df = Hmo2 ̸ 16 (Brodtkorb et al.2000),φ is a peak-width parameter (φ = 0.07 for f ≤ fpeak and φ = 0.09 for f > fpeak), and γ is a peak-enhancement factor. For a standard JONSWAP spectrum γ = 3.3, while γ = 1 for a PM spectrum.
A simple method for increasing resource assessment fidelity is to quantify the
“best-fit” peak-enhancement factor, γ, for each histogram bin. The resulting γ value will improve the representation of the wave spectrum with limited nontractable increases in the characterization parameters.
However, as shown in Fig.10, a reconstructed spectrum, even with a“best-fit”
JONSWAP spectral shape based on wave buoy Hm0and Temeasurements, can still vary dramatically from the original measured wave spectrum. It is apparent that the discrepancy is primarily due to the assumption of a single-peaked wave spectrum.
This assumption results in two significant uncertainty sources: firstly, the peak value of the variance density spectrum will be erroneous, and secondly, variance Fig. 9 Parameterization of directional variance density spectra
Wave Energy Assessments: Quantifying the Resource… 19
density will be misappropriated across the frequency axis. The incorrect repre-sentation of the peak variance density results from all of the variance density being constrained to a single spectral peak. If two or three peaks exist, as shown in Fig.10, all of the variance density will be assigned to the single priority peak.
Compounding these uncertainties, the energy period is a variance-weighted mean of frequency-domain variance density, and not a physical component of the spectrum, so the location of the peak variance density will generally fall between subsequent variance density peaks. Expanding this analysis to a fully directional variance density spectrum, a similar misappropriation of variance density in the directional space will result from an assumed single-peaked wave spectrum with θJmax
direction.
While there are bimodal (double-peaked) analytical spectral shapes (Ochi and Hubble 1976; Torsethaugen and Haver 2004), they generally require five to six descriptive parameters for each wave condition and are thus extremely cumbersome to handle and nontractable to the general audience.
The predominance of multi-peaked wave spectra is heavily dependent on location, so relative uncertainties associated with single-peaked assumptions are not homogeneous. For example, locations that are fetch constrained will generally result in single-peaked spectra and minimal uncertainty under the single-peaked assumption. However, locations that have vast geographic fetches will experience multi-peaked wave systems more commonly. For example, the west coast of Canada experiences multi-peaked wave systems 67% of the year (Robertson et al.
2016), and the shape and distribution of the variance density can vary dramatically (Fig.11).
More recently, numerical wave models are using a spectral-partitioning method to improve the representation of the sea state. Spectral partitioning separates the directional wave spectrum and represents each variance density peak, or wave system, individually. The current version of WWIII and the upcoming release of SWAN have built-in spectral-partitioning algorithms to assist with this effort. Wave spectral partitioning has been used in oceanography for over two decades as a Fig. 10 Comparison of
measured and histogram equivalent variance densities spectra
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post-processing technique to add fidelity to sea state characterizations (Boukha-novsky and Soares2009; Gerling 1991; Hanson and Phillips 2001; Portilla et al.
2009). The technique has only recently been applied within the wave energy sector (Robertson et al.2016; Kerbiriou et al.2007).
The spectral-partitioning method uses the complete directional wave spectrum and incrementally increases a threshold variance density value to find regions of lower variance density (troughs) between two higher variance density areas (peaks).
Once a trough is found, the peaks are separated and defined as separate partitions.
These separate partitions are recursively searched tofind additional troughs. Once all of the partitions are located, a series offilters is applied to determine the discrete wave systems. Gerling recommended that the ratio of discrete peak frequencies needs to be greater than 1.25, the difference between peak directions greater than 20°, and the wave mode weights greater than a factor of 10 in order to be classified as a distinct and separate spectral partition (Gerling 1991). Through the spectral-partitioning process, the variance-weighted Teor the peak-related Tp both provide a sufficient representation of the frequency distribution of wave energy around the single peak.
Using the same data used to create the histogram in Figs.4and12provides the histograms for the primary, secondary, and tertiary wave systems. Several important details about the wave climate can be identified in these partitioned histograms:
(1) the full range of incoming wave conditions is represented (Fig.12 correctly identifies waves at 22.5 s, while Fig. 4 has a maximum period of 18.5 s); (2) fi-delity in large significant wave height sea conditions is increased by separating the maximum variance density for individual wave systems (represented by system-specific significant wave height); and (3) both the most frequent and most energetic wave individual wave system occurs at lower wave heights. Given that many WEC technologies maximize power generation within narrow wave height and energy period bandwidths, the noted changes in maximum wave height, wave period, and most frequent sea states may alter the physical WEC design and associated dynamics for WECs to be deployed at the location of study.
Fig. 11 Variation in nondirectional spectral shapes within single histogram bin (measured by the Amphitrite Bank buoy off the west coast of Canada (WCWI2016))
Wave Energy Assessments: Quantifying the Resource… 21