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Directional Spreading Effect on a Wave Energy Converter Elliot Hanlin Song NA520 Project

University of Michigan

Abstract

Directional spreading effect on a wave energy converter (WEC) is studied to further optimize the design under particular sea conditions where directionality becomes important, such as hurricanes. The hydrodynamic coefficients are obtained and the equation of motion is solved in frequency domain to obtain response amplitude operators (RAOs) for different motions. The response spectrum and subsequently the expected response are calculated from RAOs and input spectra. The resulting power is shown to be the same as that modelled by one-directional spectrum as the energy of the waves must be the same for both models. The results also demonstrate the importance of tuning the WEC system for specific wave environments to harvest most energy and to avoid potential capsize due to hurricanes etc.

Keywords: Directional Wave Spectrum, Wave Energy Converter, Renewable Energy

1. Introduction

1.1 Motivation

Wave-energy converters (WECs) harvest ocean wave energy to produce electricity in an environmentally friendly manner. In order to maximize power harvest efficiency, the design natural frequency of the WEC system is matched to the peak frequency of the wave spectrum to obtain resonance. Bachynski etc. (2012) analyzed the behavior of a tethered WEC in irregular waves and optimized the configuration of the WEC. The directionality of waves is mostly neglected in current design of WECs in the literature because of the axisymmetric geometry of WECs. In realistic sea conditions, however, the effect of disturbances such as local winds or even hurricanes must be considered as an additional so called directional spreading term in the wave spectra (Latheef & Swan, 2013).

1.2 Objectives

The objective of this project is to investigate how directional spreading effect influences the design of WECs. In order to do so, an input spectrum must be constructed as a function of both frequency and angle. This study derived cosine-fourth angular spreading function and used the modified input spectrum to obtain response of the WEC system. The power takeoff from the WEC also needs to be studied to provide an estimated power for potential investors.

2. Methodology

2.1 Assumptions

The WEC is treated as a smooth vertical cylinder with free motions in surge, heave, and pitch. Figure 1 shows the simplified model of WEC by Bachynski etc. Based on linear assumptions, the equation of motion is formulated and solved in frequency domain. The fluid is considered to be incompressible, inviscid, irrotational, and subject to linearized free surface and body boundary

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conditions. Motion caps are applied to response amplitude operators (RAOs) to prevent excessive motion near resonance. Bachynski etc. (2012) showed that the probability of mooring line breaking during normal operation of WECs is negligible and does not constrain the design, since the mooring line system introduces an additional surge-pitch resonance with such low frequency that it is unlikely to be excited by wind-generated waves. Therefore in this project the WEC is considered to be unmoored to simplify the solution, with an emphasis on the directional spreading effect on the WEC only.

Figure 1 Wave energy converter (Bachynski,

Young, & Yeung, 2012).

2.2 Mathematical Formulation

2.2.1 Equation of motion

Bachynski etc. have derived the equation of motion for the WEC in frequency domain as

Figure 2 Equation of motion derived by

Bachynski etc.

In this study, the only difference is the absence of terms that are related to mooring lines such as Mt, It, TMt, K11, K33, and K55.

The free-floating cylinder experiences resonance at two frequencies. Resonance frequency in the decoupled heave motion is

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333 33

CM

ωµ

=+

(1)

And the coupled surge-pitch resonance frequency can be found by

( )( ) ( )55 11 11

51 211 11 55 55 51 51

(M )CM M M

µωµ µ µ

+=+ + − +

(2)

2.2.2 Response amplitude operators (RAOs)

RAOs are defined as the response magnitude in a particular motion due to unit amplitude waves. RAOs are equivalent to transfer functions in linear systems. In this project, because of the axisymmetric geometry of the WEC, RAOs in all three motions are independent of wave directions.

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To account for viscous effects in realistic sea conditions, motion caps are applied to the RAOs to prevent excessive motions near resonance. In current design, motion caps are taken as constants: 5 m/m in surge and 10 deg./m in pitch. For heave motion, however, the motion cap is assumed to be T/Hs, which increases linearly with draft.

2.2.3 Directional wave spectrum

For the purpose of directional spread, the directional wave spectrum density function is assumed to be (Gilloteaux & Ringwood, 2009)

( ) ( ) ( ), ,S S Dω θ ω ω θ= (3)

, where ( )S ω is the one-directional energy

spectral density function. Since the directional spreading effect is due to locally generated wind or hurricane, the waves are not fully developed. JONSWAP spectrum is selected in this study. ( ),D ω θ is the

parametric angular spreading function. The formulation of ( ),D ω θ requires that

( ) ( ) ( )

0 0

,S D d d S dπ

π

ω ω θ ω θ ω ω∞ ∞

−

=∫ ∫ ∫ (4)

hence ( ),D ω θ must satisfy

( ), 1D d

π

π

ω θ θ−

=∫ (5)

so that the total energy in the directional spectrum is the same as the total energy in the corresponding one-directional spectrum. There are several idealized spreading functions such as cosine-squared model (Hughes, 1985), cosine-2s model (Gilloteaux & Ringwood, 2009) (Hogben & Cobb, 1986), etc. In this study, the cosine-fourth spreading

function which is recommended by CERC (Costal Engineering Research Center) to represent locally generated waves is derived and applied. It is simple because it is independent of frequency and it is more accurate than cosine-squared model (Hughes, 1985). Cosine-fourth spreading function is constructed as

( ) ( )40 00

8 cos2 23

0D

otherwise

π πθ θ θθ θθ π

⎧− + < < +−⎪= ⎨

⎪⎩

(6) , where 0θ is the mean wave direction in radians. The coefficient 8/3π is determined from Eq. (5) by simple trigonometry identities.

2.2.4 Spectral response

From input spectrum and RAO of the WEC, the spectral response can be calculated as

( ) ( ) ( )2, ,RS RAO Sω θ ω ω θ= ⎡ ⎤⎣ ⎦ (7)

And the expected amplitude for a particular motion can be found as

( )exp

0

,RS d dπ

π

ζ ω θ ω θ∞

−

= ∫ ∫ (8)

Similarly, the expected power takeoff can be calculated as

( ) ( )2

exp0

,pP RAO S d dπ

π

ω ω θ ω θ∞

−

⎡ ⎤= ⎣ ⎦∫ ∫ (9)

, where the power takeoff RAO is calculated by Fitzgerald & Bergdahl’s method (2008) as

( )( )2

0

12 PTO

p

BRAO

ζ ωω

ζ= (10)

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2.3 Numerical Implementation

2.3.1 Wave environment input

Station 41004 from National Data Buoy Center near South Carolina is chosen as the design location (NDBC, 2015). The averaged wave environment data is tabulated in Table 1.

Table 1 Wave environment for input spectrum

Water depth (H) 38.4 m

Significant wave height (Hs) 1.3 m

Mean wave direction ( 0θ ) 120 deg. S.E.

Wind speed (U10) 12.7 m/s

Fetch (LF) 100000 m

The directional wave spectrum ( ),S ω θ

matrix can therefore be obtained from Eq. (3) for each frequency and angle.

2.3.1 Dimensional input

Bachynski etc. (2012) proved that for a given sea environment, the mean annual power increases with diameter and decrease with draft within design criteria Hs ≤ T ≤ H/2 and 0.1 ≤D/T ≤0.8. In this study, the ratios are D/H=0.3 and D/T=0.8 for optimum output.

Fitzgerald and Bergdahl (2008) provided a solution to obtain the center of gravity and radius of gyration for pitch motion. The power takeoff, which is treated as a linear damping term in the equation of motion, is tuned to be equal to the resonant heave hydrodynamic damping (Yeung, Peiffer, Tom, & Matlak, 2011) but without consideration of viscous effect so it is essentially heave radiation damping B33PTO. The power takeoff damping in surge and

pitch is neglected. The proposed dimensional input of the WEC is specified in Table 2.

Table 2 Dimensional Input

Diameter (D) 11.5 m

Draft (T) 15.3 m

Center of gravity (ZG) -8.1045 m

Radius of gyration (RP) 11.1423 m

Power takeoff damping (B33PTO) 1100 Ns/m

2.3.2 Hydrodynamic coefficients calculation

Non-dimensional hydrodynamic coefficients and exciting forces in surge, heave, and pitch are calculated using the method by Yeung (1981) and Johansson (1986) for each frequency with interval of 0.01 rad/s over 0 to 2.5 rad/s. Linear interpolation is used to obtain finer hydrodynamics and forcing coefficients. At the end, all non-dimensional coefficients are dimensionalized for the calculation of response.

2.3.3 Solution of equation of motion

The equation of motion is solved by iteration because wave radiation forcing coefficients are frequency dependent. The solution is a 3-dimension complex vector. The magnitude and phase of the responses are calculated by MATLAB built-in functions abs and atan2, respectively. Similarly, the solution for each frequency is obtained in a loop.

3. Results and validation

3.1 Directional wave spectrum

The directional wave spectrum based on JONSWAP one-directional wave spectrum and cosine-fourth parametric angular spreading function is shown in Figure 3. The directional wave spectrum peaks at frequency

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around 0.95 rad/s and at angle of -1.04 rad and 2.1 rad, corresponding to the mean wave direction 120 deg. S.E. and 180 deg. after that, i.e., 300 deg. N.W., respectively. These two directions are essentially the same as they are both along the main wave direction. As a pictorial verification, Figure 3 has the same trend as Figure 4 by Hogben & Cobb (1986), which is based on cosine-2s model. The difference between Figure 3 and Figure 4 is due to the different directional spreading function models, but the peak frequencies are located at the same frequencies and angles.

Figure 3 Directional wave spectrum

Figure 4 Directional spectrum based on cosine-2s model from buoy DB1 (48°43’ N 8°58’ W) in 1979 (Hogben & Cobb, 1986)

3.2 RAOs and spectral responses

The resulting RAOs with motion caps based on the solution of the equation of motion are shown in Figure 5. As discussed in 2.2.2, RAOs are independent of angles. As a verification, the RAOs in heave and pitch are both 1 at zero frequency and 0 at infinite frequency. For surge motion, however, the RAO at zero frequency can be very large because there is no restoring force and little system damping in surge for an unmoored WEC. This peak is not due to resonance, and it is not applicable to the design of WECs.

Figure 5 RAOs with motion caps

The spectral responses results are shown in Figure 6, which indicates that the input spectra are significantly modified by RAOs in heave motion and less modified in surge and pitch motions. This is because the peaks of input spectrum and RAO in heave motion

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are located at very close frequencies (0.95 rad/s and 0.72 rad/s, respectively), whereas the RAOs of surge and pitch climax at around 0.3 rad/s. Since the input spectrum is based on JONSWAP one-directional spectrum, significantly large values of are present only at frequencies close to 0.95 rad/s.

Figure 6 Spectral responses

3.3 Power takeoff RAO and power response spectrum in heave

Since only heave motion results in significant response from the waves, the power takeoff estimation is based on heave motion only. The resulting power takeoff based on Eq. (10) is shown in Figure 7. It can be shown that the power takeoff RAO peaks at 0.72 rad/s, same as heave resonance frequency because from frequency domain analysis, velocity is ( )3iωζ ω .The maximum velocity

also locates at the resonance frequency. The resulting response spectrum is shown in Figure 8, which is similar to heave response spectrum in Figure 6 with concentrated response near the resonance frequency. From the two plots we know that maximum power RAO is about 20 kW/m and the maximum power response spectrum is 35 kW2s, both of which are reasonable values.

Figure 7 Power takeoff RAO in heave motion

Figure 8 Power response spectrum in heave

3.4 Expected power and energy harvest

The expected power of the WEC is calculated from Eq. (9). The double integral is essentially the volume under the power

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response surface shown in Figure 8. It is calculated in MATLAB by numerical interpolation and integration. The resulting expected power for the current configuration and wave environment is 0.9204 kW.

As a verification, this power must agree with the results from one-directional wave spectrum since the energy of the waves must be the same for both models. The results from JONSWAP spectrum shows the agreement.

The expected annual energy harvested from the WEC is therefore 29.02 GJ, if the power is assumed to be constant throughout the year. It must be noticed that in reality, the WEC can only harvest few percentage of this amount. The significance of its expected power and annual energy harvest, shown in Table 4, can be compared to the results from Bachynski etc. (2010). in Table 3.

Table 4 Expected power and energy

Draft (T) 15.3 m

Expected power (P3exp) 0.9204 kW

Expected energy (J3) 29.02 GJ

Table 3 Heave energy extraction for a WEC located near NDBC BUOY 46026, Northern California (Bachynski, Young, & Yeung, 2010)

4. Conclusion and discussion

The directional spectrum was developed based on the derivation of cosine-fourth spreading function and JONSWAP spectrum. RAOs and response spectra are obtained and plotted with respect to both wave frequency and direction. The response spectrum is shown to peak along the mean wave direction.

The results show that the expected power takeoff and annual energy harvest are in good match to the literature. As a verification for the directional wave spectrum model, an agreement in energy has been found between both directional spectrum and one-directional JONSWAP spectrum. From the response spectrum we can draw the conclusion that in order to harvest the most energy, the WECs in a wind farm should be collectively placed along the main wave direction, for those non-axisymmetric WEC models. To prevent hurricane attack, on the other hand, WECs or other offshore structures must be placed perpendicularly to the incoming hurricane direction.

For future students, the time domain simulation of the response of the WEC is preferred, since locally generated wind and hurricanes may cause large responses which violate linear approximations. Furthermore, viscous damping is not considered in this project due to its non-linear nature. However, it may be even exceed radiation damping in reality.

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References Bachynski, E. E., Young, Y. L., & Yeung, R. W. (2012). Analysis and optimization of a tethered

wave energy converter in irregular waves. Renewable Energy, 133-145. Bachynski, E., Young, Y., & Yeung, R. (2010). Performance of a Tethered Point Wave Energy

Absorber in Regular and Irregular Waves. ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels (pp. 1133-1142). Montreal: American Society of Mechanical Engineers.

Fitzgerald, J., & Bergdahl, L. (2008). Including moorings in the assessment of a generic offshore wave energy converter: a frequency domain approach. Marine Structures, 23-46.

Gilloteaux, J., & Ringwood, J. (2009). Influences of wave directionality on a generic point absorber. National University of Ireland Maynooth, Department of Electronic Engineering. Kildare: Maynooth, Co. Kildare, Ireland.

Hogben, N., & Cobb, F. (1986, January). Parametric modelling of directional wave spectra. Offshore Technology Conference.

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Johansson, M. (1986). Transient motions of large floating structures. Tech. Rep. A:14., Chalmers University of Technology, Department of Hydraulics, Gothenburg.

Latheef, M., & Swan, C. (2013). A laboratory study of wave crest statistics and the role of directional spreading. London: Royal Society Publishing.

NDBC. (2015, December 14). National Weather Service. Retrieved from National Oceanic and Atmospheric Administration: http://www.ndbc.noaa.gov/station_page.php?station=41004

Yeung, R. (1981). Added mass and damping of a cylinder in finite-depth waters. Applied Ocean Research, 119-133.

Yeung, R. W., Peiffer, A., Tom, N., & Matlak, T. (2011). Design, Analysis, adn Evaluation of the UC-Berkeley Wave-Energy Extractor. Journal of Offshore Mechanics and Arctic Engineering, 134(2).