Abstract
Potential performance gains from optimal (non-causal) impedance-matching control of wave energy devices in irregular ocean waves are dependent on deterministic wave elevation prediction techniques that work well in practical applications. Although a number of devices are designed for operation in intermediate water depths, little work has been reported on deterministic wave prediction in such depths. Investigated in this paper is a deterministic wave-prediction technique based on an approximate propagation model that leads to an analytical formulation, which may be convenient to implement in practice. To improve accuracy, an approach to combine predictions based on multiple up-wave measurement points is evaluated. The overall method is tested using experimental time-series measurements recorded in the U.S. Navy MASK basin in Carderock, MD, USA. For comparison, an alternative prediction approach based on Fourier coefficients is also tested with the same data. Comparison of prediction approaches with direct measurements suggest room for improvement. Possible sources of error including tank reflections are estimated, and potential mitigation approaches are discussed.
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Notes
Capture width ratio is defined as average power captured divided by the average wave power incident on the device diameter/width.
To obtain the predicted wave elevation at the required time instant in the future, a measurement location that is closer to the device needs a shorter time-series starting at a time determined, such that the prediction time from that location leads to a predicted elevation at the correct time instant in the future as required by the device dynamics. This point is explained further in the space–time diagram of Fig. 1.
Abbreviations
- \({\varvec{\eta }}\) :
-
Vector of wave elevation measurements at M time instants.
- \(\varDelta h_{lp}\) :
-
Small change/error in \(h_{lp}\).
- \(\varDelta T_p\) :
-
Propagation time error due to propagation model approximation.
- \(\varDelta x_p\) :
-
Small error in propagation distance due to propagation model approximation.
- \(\ell \) :
-
Separation along wave direction between the two wave gauges
- \(\eta (x; i\omega )\) :
-
Wave elevation at point x in the frequency domain (includes amplitude and phase information).
- \(\eta _1\) :
-
Wave profile time-series recorded by wave gauge 1
- \(\eta _2\) :
-
Wave profile time-series recorded by wave gauge 2
- \(\eta _\mathrm{appr}\) :
-
Wave elevation approximated using Fourier coefficients.
- \(\varGamma _i\) :
-
Complex amplitude representing the incident-wave field
- \(\varGamma _r\) :
-
Complex amplitude representing the reflected wave field
- \(\omega \) :
-
Angular wave frequency.
- \(\omega _n\) :
-
nth frequency in the Fourier series approximation for the wave elevation at the point of wave measurement.
- \(\omega _{mn}\) :
-
Minimum angular frequency in a realistic spectrum
- \(\omega _{mx}\) :
-
Maximum angular frequency in a realistic spectrum
- \(\varPsi \) :
-
The performance index as defined here for the dispersion relation approximation.
- \(C_n\) :
-
Amplitude of the nth frequency component in the Fourier series approximation at the point of wave measurement.
- \(\theta _p\) :
-
Phase error due to propagation model approximation.
- \(A_1, A_2, \ldots , N\) :
-
Approximate dispersion relation parameters to be estimated with changing sea states.
- \(A_n\) and \(B_n\) :
-
Fourier coefficients estimated using time-series at the point of wave measurement.
- \(A_{nl}, B_{nl}\) :
-
Fourier coefficients estimated at each measurement location l.
- \(C_n\) :
-
Amplitude of the nth frequency component in the Fourier series approximation at the point of wave measurement.
- d :
-
Distance over which prediction is needed \((x_B - x_A)\).
- \(d_n\) :
-
Propagation distance from point \(x_{An}\) to \(x_B\).
- g :
-
Gravity acceleration.
- h :
-
Water depth over which waves propagate.
- \(h_0\) :
-
Nominal depth at which dispersion relation is approximated.
- \(h_l\) :
-
Propagation impulse-response function.
- \(h_{ln}\) :
-
Propagation impulse-response function for prediction from point \(x_{An}\) to \(x_B\).
- \(J_M\) :
-
Performance index for estimation of Fourier coefficients \(A_n\) and \(B_n\).
- \(k(\omega )\) :
-
Wave number
- \(k_0\), \(\omega _0\) :
-
Regular-wave number \(k_0\) and angular frequency \(\omega _0\)
- \(k_n\) :
-
nth wave number in the Fourier series approximation for the wave elevation at the point of wave measurement.
- T :
-
Length of time-series of \(\eta \) needed at \(x_A\) to predict at \(x_B\).
- t :
-
Time variable
- \(t_F\) :
-
Time instant into the future at which prediction is needed; determined by device dynamics (20-30s; same as \(t_p\)).
- \(t_m\) :
-
mth time instant within a wave elevation time-series approximated using Fourier coefficients.
- \(t_p\) :
-
Prediction time into the future.
- \(T_{pmx}, T_{pmn}\) :
-
Propagation times for the smallest and the largest group velocities, respectively.
- \(t_{pn}\) :
-
Prediction duration from point \(x_{An}\) to \(x_B\).
- \(V_\mathrm{gappr}\) :
-
Approximate group velocity from propagation model approximation.
- \(v_{gmn}\) :
-
Minimum group velocity in a realistic spectrum.
- \(v_{gmx}\) :
-
Maximum group velocity in a realistic spectrum.
- \(x_B\) :
-
Point where wave elevation prediction is needed
- \(x_p\) :
-
Propagation distance between point of wave measurement and the point of wave prediction; equivalent to \(d_p\).
- \(x_{An}\) :
-
nth point within the space–time diagram for prediction from \(x_A\) to \(x_B\) in practical spectra.
- \(\mathbf{A}\) :
-
Matrix formed by multiplying the two coefficient matrices \(\mathbf{C}\) and \(\mathbf{S}\).
- \(\mathbf{A}_l\) :
-
Matrix formed by multiplying the two coefficient matrices \(\mathbf{C}_l\) and \(\mathbf{S}_l\) at each location l.
- \(\mathbf{C}\) :
-
One of the coefficient matrices formed with cosine and sine functions in estimation of \(A_n\) and \(B_n\).
- \(\mathbf{P}\) :
-
Vector of coefficients \(A_n\) and \(B_n\).
- \(\mathbf{P}_l\) :
-
Vector of coefficients \(A_n\) and \(B_n\) at each location l.
- \(\mathbf{S}\) :
-
One of the coefficient matrices formed using cosine and sine functions in estimation of \(A_n\) and \(B_n\).
- \({{\varvec{\eta }}}_l\) :
-
Vector of wave elevation measurements at location l.
- \({\mathrm{C}}, {\mathrm{S}}\) :
-
Fresnel cosine and sine integrals.
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Acknowledgements
UAK is grateful to the Sandia National Laboratories for supporting this work. All three authors wish to thank the U.S. Department of Energy Water Power Program for their support. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
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Korde, U.A., Coe, R.G. & Bacelli, G. Deterministic incident-wave elevation prediction in intermediate water depth. J. Ocean Eng. Mar. Energy 6, 359–376 (2020). https://doi.org/10.1007/s40722-020-00177-5
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DOI: https://doi.org/10.1007/s40722-020-00177-5