3.1. Model Calibration
To accurately assess the tidal energy resources at any specific site or the maximum tidal energy potential (i.e., a theoretical undisturbed resource) using a modeling approach, it is important to calibrate and validate the numerical model using field observations. In this study, model calibration was performed by comparing modeled tidal elevations and velocities with field measurements at various locations during different time periods. Three representative error statistical parameters, the root-mean-square-error (RMSE), the scattered index (SI), and the coefficient of determination (R2), were calculated to assess the model’s ability to reproduce the characteristics in the study domain. Model parameters, such as the bottom roughness, vertical layers, and open-boundary sponge layer (radius, and friction coefficient), were also adjusted iteratively during the calibration runs to achieve an overall satisfactory agreement between the model predictions and field observations. The final calibrated bottom roughness height was 0.005 m. The calibrated radius and friction coefficient of the open-boundary sponge layer were 1500 m and 0.001, respectively. Two model runs with 15 and 30 uniform sigma layers were conducted to evaluate the effect of the total number of vertical layers on model accuracy. The model results with 30 vertical layers showed little improvement in model performance error statistics over 15 vertical layers. Therefore, 15 vertical layers were selected in all the model runs to achieve better runtime efficiency while maintaining the same level of model accuracy.
The simulated water levels at Cutler, Eastport, and Port Greville were compared with the data for a 14-day period in October 2001 (
Figure 4). Overall, the model reproduced the tide-level variations precisely in both the tidal range and phase in the Passamaquoddy–Cobscook Bay archipelago and Bay of Fundy. The water-level time history at all three stations indicated that the tide in the model domain is dominated by semi-diurnal tide (M2) with a strong signal of spring-neap tidal cycle. The tidal wave is amplified as it propagates from the Gulf of Maine (near the open boundary) (
Figure 4a) into the Passamaquoddy–Cobscook Bay archipelago (
Figure 4b) and toward the Bay of Fundy (
Figure 4c). The root-mean-square errors (RMSEs) at the three tidal stations ranged between 0.15 m at Cutler station and 0.41 m at Port Greville station, where the tidal range is greater than 10 m during spring tide (
Table 2). The correlation coefficients (R
2) were persistently higher than 99% at all three stations (
Table 2). The scatter indexes (SIs) were relatively small, indicating that the model was able to reproduce the water level accurately and consistently (
Table 2).
The modeled depth-averaged principal velocities were compared with the measured data at stations EP0003 and EP0004 for July 2000 and at station J02 for October 2001 (
Figure 5). Overall, the model results matched the measurements well, especially during flood tides. However, the model overpredicted the ebb currents at EP0003, where the measurements showed tidal current asymmetry with flood currents larger than ebb currents (
Figure 5a). The error statistics for current predictions at EP0003, EP0004, and J02 are presented in
Table 3. The largest RMSE is 0.38 m/s at station EP0003, which is consistent with the time–series plot shown in
Figure 5a. The RMSEs at EP0004 and J02 are 0.14 m/s and 0.27 m/s, respectively, which are satisfactory for the current predictions. The correlation coefficients (R
2) for the current predictions are all above 0.94, indicating that the model predictions are highly correlated to the measurements at all three stations.
3.2. Model Validation
Once the model was calibrated, model validation was conducted for a different simulation period using the calibrated model parameters. The recent observed data, including water level and velocity collected at station WP1 in the Western Passage, were used for model validation. The model was run for three months, corresponding to the measurement period of April to June 2017.
The long (three-month) record of field measurements at WP1 allowed for accurate harmonic analysis. Therefore, the model skills for predicting water level and currents at WP1 were evaluated by comparing the tidal level harmonic constituents derived from the measurements and model results. Water levels were decomposed into 10 tidal components (M2, N2, S2, K1, O1, P1, Q1, M4, M6, and MK3) using harmonic analysis. Comparisons of the constituents of each observed and predicted tidal level are provided in
Table 4. The maximum difference between modeled and measured tidal constituents is −0.11 m for the principal lunar semi-diurnal tide M2, which is about 3.9% underpredicted relative to the observed M2 constitute (2.62 m). Differences between the model predictions and measurements for other tidal constituents are all small; even the percentage error is high because of the small magnitudes of the constituents. Therefore, the model performed very well in simulating the tidal elevation as part of the model validation.
To better visualize a comparison between the modeled and observed velocity through the water column, velocity contours were generated with respect to water depth and time (
Figure 6). The distribution pattern of the modeled velocity contours for the u (east) and v (north) components (
Figure 6c,d) is very similar to the observations (
Figure 6a,b). Both model results and observations show consistently strong currents through the entire water column, with distinct flood and ebb tidal signals. In general, both the model results and the observed data showed a stronger flood current (negative u and positive v) than an ebb current (positive u and negative v). The velocity magnitudes were mostly uniformly distributed through the water column, except for the modeled v-velocity, which showed a slightly weaker current near the surface during the flood tide (positive value) (
Figure 6d).
Figure 7 shows a comparison of the vertical profiles of the velocity percentiles based on the three-month model results and observed data. Velocity percentiles were calculated as 10%, 25%, 50%, 75%, and 90%. Overall, the model results are similar to the observations in terms of the range of the velocity percentiles. Strong tidal asymmetry in the velocity profiles is seen in both the modeled results and the observations, where currents are stronger during flood tides (negative value) and weaker during ebb tides (positive value) (
Figure 7c,f). The mean error of the predicted mean velocity-magnitude profile is −0.134 m/s, where the negative sign indicates model underprediction, and the percentage error is 11%. Although the model showed good overall skill in predicting the velocity vertical profiles in the Western Passage, it overpredicted the v-velocity component during flood tides (negative value) and underpredicted it during ebb tides (positive value) (
Figure 7b,e). In particular, the model predicted a strong ebb current near the bottom (
Figure 7e), which was not shown in the observed data (
Figure 7b). The cause of such a discrepancy between the model results and observed data is likely due to the accuracy of model bathymetry, which was mainly interpolated from bathymetric datasets digitized from old NOAA navigation charts [
28] that may not accurately represent the real bathymetry in the Western Passage when the ADCP data were collected in 2017.
The error statistics calculated at WP1 show the model’s excellent ability to simulate a tidal current (
Table 5). Similar to the water-level results, the model underpredicted the M2 tidal current by 0.21 m/s, which is 12.4% of the M2 current speed. All other predicted tidal constituents are no more than 0.07 m/s smaller than the observed tidal current constituents. The model validation results in this study are comparable to those presented in [
28]. Because the tide is the dominant force in the system, the model is considered fully capable of reproducing tidal hydrodynamics and characterizing the tidal energy resources in the Western Passage.
3.3. Characteristics of Tidal Hydrodynamics
Once the model was calibrated and validated, the model results could be used to characterize the resource and identify hotspots for tidal energy development in the region.
Figure 8 shows the horizontal distribution of the depth-averaged velocities during peak flood and ebb tides. Clearly, strong currents exist along the main channel of the Western Passage, Cobscook Bay, and Head Harbor Passage. In general, these areas with strong currents during both the flood and ebb tides represent potential hotspots for future tidal energy development. However, tidal asymmetry also exists in some areas between the flood and ebb tides. For example, in the tidal channel between Deer Island and Indian Island, the tidal currents are much stronger during the flood tide (
Figure 8b) than those during the ebb tide (
Figure 8a). On the other hand, the ebb currents (
Figure 8b) are significantly stronger than the flood currents in the Head Harbor Passage (
Figure 8a).
Because tidal turbines are installed in the water column to assess the feasibility of a project site for tidal energy extraction, the vertical structure of the current magnitude and the water depth distribution should be considered. Based on the results of the depth-averaged velocity distribution (
Figure 8), the current magnitudes along the four cross sections with strong tidal currents were generated. Cross sections XS1 and XS2 are located in the Western Passage, cross section XS3 is in the Cobscook Bay, and cross section XS4 is in the Head Harbor Passage. Areas where strong currents occur only during flood or ebb tides, such as the south part of the Head Harbor Passage and the channel between Deer Island and Indian Island, were not selected for cross-sectional velocity analysis.
Figure 9 shows the normal velocity magnitudes along XS1 in the south end of the Western Passage. The water depth in the deep channel is about 120 m. A positive value indicates the flood current flowing away from the reader (
Figure 9a). Strong flood and ebb currents occupy most of the deep channel and the U.S. side of the passage (left side) through the entire water column; the maximum current magnitude is greater than 2.5 m/s during the flood tide. The current distribution on the north end of the cross section is complicated; consistent ebb currents (negative value) occur during both the flood and ebb tides, and reverse flow (positive value) occurs at the bottom layer of the water column during the ebb tide. The consistently strong tidal currents in the deep channel and the U.S. side of the passage indicate that the area is a good candidate for tidal turbine farm deployment.
Strong currents were observed in XS2 (
Figure 10), especially on the U.S. side (the left side of the plot). The current speed in the main channel is approximately 1.75 m/s during both the flood and ebb tidal phases. The maximum current speed (>2 m/s) occurs in the U.S. side of the passage where the water depth is relatively shallow—about 30 m. Again, the current structure on the Canadian side is small and complicated, showing strong vertical shear during the flood tide (
Figure 10a).
Figure 11 shows the current distribution along XS3 in Cobscook Bay. Although strong currents (over 2.5 m/s) are present during both the flood and ebb tides in the main channel, the water depth is very shallow. Most of the cross section is less than 10 m deep, and the deepest water depth is 25 m, which makes the area challenging for the deployment of tidal turbine farms.
XS4 in the Head Harbor Passage (Canada) shows the strongest current magnitude and the largest area of high velocity among the four cross sections. The current speeds in most parts of this cross section are greater than 1.5 m/s during the flood tide (
Figure 12a) and greater than 2 m/s during the ebb tide (
Figure 12b). The seabed in the cross section is relatively flat, but the water depth is shallow (approximately 30 m deep for most of the cross section). Again, this shallow water depth may pose a challenge for the installation of tidal turbine farms, unless small tidal turbines (e.g., rotor diameters less than 15 m) are considered.
3.4. Along-Channel Kinetic Energy Flux
The predicted tidal currents within the potential deployment area can be used to estimate the tidal kinetic-energy flux through a channel cross section. First, the temporal-averaged power density at any location can be calculated using the predicted tidal currents as follows:
where
is the temporal-averaged power density (kW/m
2) at any location or grid cell in the model,
N is the total number of velocity output timesteps over the simulation period,
is the seawater density (1025 kg/m
3), and
U is the current magnitude normal to the cross section (m/s).
Figure 13 shows the distribution of the mean power density at each of the four cross sections (XS1–XS4). XS1 maintains around 1.0 kW/m
2, which is evenly distributed in the middle–bottom layers of the channel that are deep enough for device deployment (
Figure 13a). XS2 and XS3 show a range of 1–2 kW/m
2 power density available near the surface (
Figure 13b,c). The highest power density value, up to 3.0 kW/m
2, was identified at XS4, which presented the strongest current at a magnitude of 2.5 m/s (
Figure 13d).
The mean tidal kinetic-energy flux
Pxs across a cross section can be estimated by multiplying the mean power density
with the grid cell area and integrating it over the entire cross section using the following formula:
where
Ncell is the total number of model grid cells projected along the cross section, and
Acell is the projected area for each grid cell. Based on Equation (3), the tidal kinetic energy flux
Pxs for each of the four cross sections was estimated to be 45.8 × 10
3 (kW) for XS1, 22.3 × 10
3 (kW) for XS2, 9.03 × 10
3 (kW) for XS2, and 48.6 × 10
3 (kW) for XS4, respectively.
3.5. Energy Extraction in the Western Passage
As described in
Section 3.4, the tidal energy fluxes in the Western Passage (XS1) and the Head Harbor Passage (XS4) are much greater than those in XS2 and XS3. However, compared to XS4 in the Head Harbor Passage, the Western Passage (XS1) is more favorable for tidal energy deployment because of its greater water depth. In this section, the tidal energy extraction capacity in the Western Passage was evaluated through a theoretical analysis and the use of a hypothetical tidal turbine farm. The maximum theoretical extractable tidal power can be estimated based on the formula developed by Garrett and Cummins [
5] for a tidal channel connected to two basins:
where
γ is a dimensionless constant in the range of 0.21 to 0.24,
is the largest tidal constituent amplitude (M2) in the channel, and
are the additional
Mt tidal constituent amplitudes. Specifying
γ = 0.22,
= 1025 kg/m
3,
g = 9.81 m/s2,
Qmax = 74,388 m
3/s, and
from
Table 4 into Equation (4) yields
Pmax = 447,825 (kW). The
Pmax value can be used as a reference (upper limit) for the development of tidal energy projects in the Western Passage. It should be noted that Equation (4) likely overestimates the
Pmax value due to the simplification and assumptions made in the derivation of the equation [
5].
A hypothetical tidal turbine farm was considered to simulate the energy extraction at a realistic scale in the Western Passage. The tidal farm is located at the south end and U.S. side of the Western Passage, where strong tidal currents occur. The tidal farm consists of a total of 19 tidal turbines, with along-channel spacing of 160 m and cross-channel spacing of 80 m (
Figure 14). The tidal turbine hub height and diameter are 15 m and 20 m, respectively. The configuration of the tidal turbine farm is listed in
Table 6.
The tidal energy extraction was simulated using the FVCOM-MHK model [
7] based on Equation (1). The extractable energy at any specific site depends on not only the current speed but also the thrust coefficient, which is a function of the turbine design and flow speed for characterizing power extraction efficiency. Many far-field modeling studies showed that a turbine can potentially reach peak efficiency for extracting the maximum power, i.e., 59% of the available power in a system (called the Betz Limit) when the thrust coefficient is specified to be in the range of 0.8–1 [
30,
33,
54,
55,
56]. Based on the values reported in the previous studies, a thrust coefficient of 0.9 was chosen in this study. For simplicity, the cut-in and cut-out velocities were not considered. The simulated temporally averaged extracted energy was 4810 (kW) for the tidal farm or 253 (kW) per turbine. Because the theoretical resources in the Western Passage are estimated to be 447,825 (kW), the extracted energy by the hypothetical tidal turbine farm is 1.07% of the theoretical resources, which is below the 2% threshold specified in the IEC standards [
27].
Another method recommended in the IEC TS for tidal energy resource assessment is to estimate the extractable power of a tidal farm using measured or modeled undisturbed flow when the expected extractable power is smaller than 10 MW or 2% of the theoretical resource in the system. For the purpose of comparison, the extractable power was also estimated using Equation (1) and the modeled undisturbed velocity. All parameters in the calculation were kept the same as those specified in the FVCOM-MHK simulations, including the thrust coefficient (0.9), turbine blade diameter (20 m) in Equation (1), and undisturbed velocity at the turbine hub height (15 m from the seabed) at the 19 turbine locations. The estimated mean power extracted by the tidal farm based on the modeled undisturbed flow field was 5198 (kW), which is 1.2% of the theoretical resources. The extractable power estimated based on the FVCOM-MHK model run with energy extraction is slightly smaller than that directly calculated using undisturbed flow, which is expected because of the flow reduction caused by the turbines and the interactions among the turbines in the tidal farm in the FVCOM-MHK simulations.
Clearly, by doubling the number of tidal turbines in the existing tidal farm, i.e., increasing the turbine distribution density twofold, the estimated extractable power using modeled undisturbed flow will likely exceed the 10 MW and the 2% thresholds. However, the extractable power for the same tidal farm calculated using the FVCOM-MHK model with energy extraction may not exceed both thresholds because the efficiency of energy extraction decreases as the number of turbines increases in the tidal farm [
7,
22]. Therefore, it is better to use the numerical model with energy extraction to accurately estimate the extractable energy, especially when the energy extraction by the tidal farm is expected to exceed 10 MW or 2% of the theoretical resources.