1. Introduction
As wave energy converter (WEC) technology matures from deployments of individual WECs for testing and demonstration, to wave farms consisting of multiple WECs for utility-scale power generation, it will be increasingly necessary to understand and quantify the effect of these WECs on the surrounding wave field. Deployment of WECs in wave farms has the potential to affect both the near-field and far-field and may have an impact on the nearshore environment and processes. In order for wave energy projects to be permitted, and thus for wave energy to be a viable part of our renewable energy portfolio, we must have numerical tools capable of modeling the effects of wave farms on their surrounding environment. Numerical tools are especially important due to the lack of data available from open ocean deployments of wave farms; until data are available, we must rely on simulations.
In the literature, there are many different numerical approaches for modeling WEC devices and wave farms. Each numerical approach has its benefits and limitations, and thus, the best method is highly dependent on the intended application. Folley (2016) provides a comprehensive review of techniques for modeling individual WECs and wave farms [
1]. In his book, Folley splits the numerical approaches into the following categories: frequency-domain models, time-domain models, and spectral-domain models. Frequency-domain models include linear potential flow methods such as the boundary element method (BEM) codes WAMIT [
2] and NEMOH [
3]. Time-domain models include numerical methods based on the Cummins’ impulse response formulation, which allows for the inclusion of nonlinearities; these methods are often referred to as wave-to-wire models [
4]. Spectral-domain models include numerical methods based on the spectral action balance equation, such as the codes SWAN and TOMAWAC [
5,
6]. The rate of change of the action density is governed by the action density balance equation. This equation includes wave kinematics, variations in depth and currents, sources, and sinks but is not phase resolving.
To estimate wave field effects due to the presence of a wave farm in an open ocean environment, the most common approach is to apply spectral-domain models. In spectral-domain models, the spectral action balance equation includes an energy sink term to model the WEC’s energy extraction as an obstacle in the computational domain. Many researchers have used this approach to estimate a wave farm’s potential impact on the wave field and on the nearshore environment. Millar et al. (2007) modeled a planned wave farm at the WaveHub site off the north Coast of Cornwall in the UK using a SWAN obstacle with varying amounts of transmission [
7]. Smith et al. 2012 furthered the WaveHub study by modifying the SWAN code to include a frequency- and direction-dependent WEC obstacle based on the WEC’s power performance [
8]. Silverthorne and Folley (2011) added a similar frequency- and direction-dependent obstacle in the spectral code TOMAWAC and modeled a 40 WEC wave farm using this approach [
9]. Researchers at Oregon State University performed a series of 1, 3, and 5 WEC array wave tank experiments and then modeled the resulting wave field in a modification of the spectral-domain code SWAN, and in the BEM code WAMIT [
10,
11,
12]. Rusu and Onea (2016) used a computational framework coupling SWAN with a circulation model (navy standard surf model) to study the influence of a wave farm’s distance from shore off the Portuguese coast on the nearshore wave propagation and coastal circulation [
13]. Greenwood et al. (2016) compared the performance of spectral and Boussinesq methods in MIKE for modeling small arrays of oscillating surge WECs [
14]. Venugopal et al. (2017) modeled arrays of attenuators and oscillating surge WECs in MIKE 21 Spectral Wave using reflection, absorption coefficients derived from the boundary element method (BEM) code WAMIT [
15].
Researchers at Sandia National Laboratories have developed SNL-SWAN, a modified version of the open source SWAN code to include a WEC Module to extract energy from the model domain based on the WEC’s power performance [
16]. Sensitivity analyses have been performed using the SNL-SWAN code to model the nearshore effects of wave farms by varying the the WEC type, spacing, and number of WECs [
17]. SNL-SWAN has also been used to drive circulation models to understand the effects changes in the nearshore wave environment may have on circulation patterns [
18]. While results from the SNL-SWAN code have been compared to experimental data, the resolution of experimental data from point measurements does not allow for the complete assessment of SNL-SWAN’s performance throughout the near- and far-field [
19].
This paper extends previous work modeling wave field effects produced by individual WECs and multiple WEC wave farms with spectral-domain models by comparing the performance of SNL-SWAN with the linear-wave, BEM code WAMIT. The objective of this work is to identify the conditions where SNL-SWAN and WAMIT agree and differ in order to better understand the limitations of this numerical approach to model the wave field. The work presented in this paper is an extension of the single WEC pitching flap results presented in McNatt et al. (2017), by expanding the analysis to include two addition WEC types (point absorber and hinged raft), and multiple device WEC arrays [
20]. The effort is based on the fact that phase-resolving BEM codes better model the physics of wave-body interactions and thus simulate a more accurate wave field than spectral wave models. Comparing SNL-SWAN to WAMIT, as opposed to using experiments, has the advantages that BEM codes are faster and less expensive to run, results are precise and repeatable, arbitrarily large areas and resolutions can be assessed, and comparison can be made throughout the computational domain (not just at measurement locations). The comparison between SNL-SWAN and WAMIT is made over a range of incident wave conditions (short-, medium-, and long-wavelength waves), with various amounts of directional spreading, and for three WEC archetypes: a point absorber (PA), a pitching flap (PF) terminator, and a hinged raft (HR) attenuator.
3. WEC Types
For the numerical study between SNL-SWAN and WAMIT, three WEC archetypes were selected: a pitching flap (PF) terminator, a point absorber (PA), and a hinged raft (HR) attenuator, shown in
Figure 1. These WECs were selected to represent the breadth of existing WEC technologies due to their varying size, power performance, and modes of motion. A pitching flap was selected to represent terminator type WECs similar to the WaveRoller. The PF is a bottom-mounted WEC with a flap that rotates in pitch relative to seabed, and it is used to generate power. A half-submerged sphere was chosen to represent point absorber-type WECs similar to the Power Buoy. The PA absorbs power in heave motion relative to the seabed. A hinged raft similar was selected to represent attenuator-type WECs similar to the Pelamis. The HR consists of two hulls with a single hinge used for generating power through the relative pitch motion. An overview of the WEC’s dimensions and modes of motion is summarized in
Table 2. For each WEC,
Table 2 lists the active degrees of freedom (DOF), the power take-off (PTO) DOF, the dimensions of the device, the water depth in which the device is located, and an image of the WEC.
3.1. Power Performance
Each WEC’s power performance was evaluated with wave spectra for
${T}_{p}=4,8,12$ s; see
Section 4 for additional details. The damping value for each WEC was tuned to maximize its power absorption for the JONSWAP and Bretschneider spectra at
${T}_{p}=8$ s and
${H}_{s}=2.56$ m. For the array tests, the
${T}_{p}=4$ s spectrum was removed, and the
${T}_{p}=6$ s spectrum was added as a more realistic sea state. The hinged raft and the point absorber absorb more power in the
${T}_{p}=6$ s spectra than the others.
Figure 2 shows the power performance of each WEC as a function of frequency overlaid with the nondimensionalized spectral sea states. This dimensionless characterization of a WEC’s power performance, a function of wave frequency, is referred to as the relative capture width (RCW) curve. The RCW is defined as a ratio of the power absorbed by the WEC, to the incident power available to the WEC; see definition in Equation (
2).
where
$\omega $ is the radial wave frequency,
w is the device width,
${C}_{pto}$ is the PTO damping,
$X\left(\omega \right)$ is the WEC’s motion response,
$\rho $ is the water density, g is gravitational constant, and
${c}_{g}(\omega ,h)$ is the wave group velocity (a function of
$\omega $ and water depth,
h). The diamond markers in
Figure 2 are the RCW values that were provided to SNL-SWAN at 101 discrete wave frequencies. While a WEC’s RCW can be greater than 1 due to the WEC’s antennae effect, where a device absorbs more power than what is available over its diameter, SNL-SWAN requires the RCW to be capped at 1 (corresponding to 100% energy absorption) because the code cannot extract more energy than is available incident to the WEC obstacle width.
3.2. Characteristic Dimensions
The dimensions of each WEC archetype are listed in
Table 2. In the context of this numerical analysis, the characteristic width,
w, refers to the width of the WEC perpendicular to the incident wave direction. This corresponds to the dimension used to calculate the relative capture width (RCW), and the WEC obstacle length in SNL-SWAN, see
Table 2 and Equation (
2). The characteristic diameter,
d, is defined as the diameter of the hemisphere with the equivalent submerged volume, where
V is the submerged volume of the WEC, as shown in Equation (
3).
The characteristic diameter is used in the Results section to nondimensionalize length values and to define the size of the area over which quantitative measures of the wave field are taken. The basis for its definition is that the impact of the WEC on the wave field should be roughly proportional to the volume of the fluid that it displaces.
4. Wave Conditions
The selected wave cases have periods of
$T=4,6,8,12$ s, which represent short- (25 and 56 m), medium- (100 m), and long-wavelength (212 m) waves, respectively. The medium and long wave periods were chosen because they are representative of sea states at potential WEC deployment sites in the U.S. [
21]. The short wave cases were chosen due to their high potential for wave scattering. The ratio of the device size to the wave length is often called the diffraction coefficient [
22]. The smaller the value of the diffraction coefficient, i.e., the shorter the wave, the larger the impact of the device on the wave field due to scattering.
An overview of the wave cases run in this numerical study is provided in
Table 3 for the single WEC and WEC array cases. The selected wave conditions differ between single WEC cases and WEC array cases. The revised wave conditions are based on findings from the single WEC numerical analysis which led to modification of the sea states, and reduction in the number of cases for the WEC array analysis. For example, the 4 s was used to evaluate the single WEC results; however, when running the WEC array cases, 6 s wave was run because it was more representative of real seas. Similarly, JONSWAP spectra were only run for the single WEC cases, because the results from the JONSWAP and Bretschneider results showed minimal differences.
Irregular Waves
An overview of the irregular sea states run in this numerical study is provided in
Table 3 for the single WEC and WEC array cases. For each of the specified wave periods,
${T}_{p}=4,8,12$ s, two sets of wave spectra were defined, a JONSWAP (JS) and a Bretschinder (BS). The wave spectra used for the irregular sea states run in this numerical study are provided in
Figure 3 and shown in nondimensional form with the RCW in
Figure 2. While an extensive analysis of a wide set of irregular wave conditions was run, only a subset of the irregular wave simulations will be included in this paper. These results provide an overview of the findings from the extensive numerical analysis by highlighting a subset the cases that are representative of the overall numerical analysis.
8. Conclusions
This results presented in this paper compared single WEC and WEC array wave fields produced by SNL-SWAN to those produced by the linear wave BEM code, WAMIT. Three WEC archetypes were considered: a pitching flap (PF), point absorber (PA), and hinged raft (HR). These WECs were selected to represent the breadth of existing WEC archetypes due to their varying size, power performance, and modes of motion. One objective of this work is to provide general guidance on modeling of WECs in spectral wave models, such as SNL-SWAN, independent of design archetypes. Wave conditions included wave periods representing short (4 s and 6 s), medium (8 s), and long (12 s) waves, in various spectral and directional spreading conditions. Comparisons were made qualitatively with detailed plots of the wave fields, and quantitatively by considering the root-mean-square difference between the wave fields normalized by the root-mean-square disturbance predicted in the WAMIT wave field. The study was carried out by first considering the single WEC comparison and then the WEC array cases. Conclusions are presented based on results observed from the single WEC cases and the WEC array cases, then overall conclusions are given.
8.1. Single WEC
The best agreement between SNL-SWAN and WAMIT occurred for higher directional spreading, and where the devices were tuned to absorb the most power. The worst comparisons occurred for lower directional spreading and longer waves relative to the device size. The directional spreading and power-absorption results were expected. Although, the longest waves showed the poorest fit, the overall WEC wave field disturbance was the smallest, and so inaccuracies may have less of an impact. Furthermore, the difference generally decreases further from the device, and nearly all cases show the difference in spectral energy content between the two codes is minimal by 500 m downwave.
8.2. WEC Array
Results showed that for the WEC array, in its best case, SNL-SWAN predicted the wave field over a circular arc at a radius 30 d with a difference of 20%, for an array of 5 point absorbers in two rows spaced at 10 d in an 8 s Bretschneider spectrum with a spreading, s = 20. In general, better agreement between SNL-SWAN and WAMIT occurred for longer waves relative to the device size; worse agreement occurred for shorter waves. Additionally, the WEC arrays cases had better agreement than the single WEC cases. For WEC arrays, the fit between SNL-SWAN and WAMIT differs the most in the near-field, and the overall impact of the array on the wave field decreased significantly with distance. This suggests that evaluating the performance of SNL-SWAN at a radial distance of roughly r = 30d is sufficient. The difference generally decreases further from the array, and nearly all cases show the difference in spectral energy content between the two codes is minimal by 30 d downwave.
Future work includes expanding upon this analysis to model larger WEC arrays and assessing the feasibility and importance of coupling of BEM models in the near-field, with a larger scale far-field SNL-SWAN model over varying depths.
8.3. Guidance
Based on the results of this numerical study, the authors recommend the following guidance when using SNL-SWAN (or other spectral-domain codes) to model wave field effects due to WECs:
For arrays, SNL-SWAN shows differences of between 20% and 60% as compared to WAMIT, where the difference is normalized by the impact of the array on the wave field, which are between 1% and 5% at a distance of 30 characteristic diameters from the WECs;
Directional wave spreading in the incident wave spectrum is important for SNL-SWAN accuracy; unidirectional waves are not modeled well;
SNL-SWAN does not model wave reflection or scattering, which is a mechanism (in addition to power absorption) for the creation of a wave shadow. This causes larger errors in shorter waves. Preliminary results show the errors could be decreased by including reflections in SNL-SWAN;
Truncating power absorption, because of RCW values greater than 1, worsens wave field modeling accuracy.