- Email: [email protected]
Stochastic wave energy calculation formulation
Renewable Energy 29 (2004) 1747–1756 www.elsevier.com/locate/renene
Technical note
Stochastic wave energy calculation formulation ¨ zger , Abdu¨sselam Altunkaynak, Zekai S˛en Mehmet O Istanbul Technical University, Civil Engineering Faculty, Hydraulics Division, Maslak 34469, Istanbul, Turkey Received 18 April 2003; accepted 19 January 2004
Abstract The wave energy potential is directly proportional to the wave period and second power of wave height averaged over a suitable time period. The wave height and period have temporal and spatial stochastic variations. It is the main purpose of this paper to derive the most general wave energy formulation by considering simultaneously the temporal variations both in the wave height and period. The correction factor is derived explicitly in terms of cross-correlation and the coefficients of variation. The application of the methodology is performed for wave measurement stations located in the Pacific Ocean off the west coast of the US. # 2004 Elsevier Ltd. All rights reserved. Keywords: Wave energy; Stochastic variables; Significant wave height
1. Introduction The possibility of extracting energy from ocean waves has intrigued people for centuries. Although there are a few concepts over 100 years old, it is only the past two decades that viable schemes have been proposed. In a wave energy plant, unless a large energy storage interface follows the waves in the energy conversion chain of the plant (as the on-shore water reservoir of the tapered channel or TAPCHAN device), the power take-off equipment is required to convert an energy flux that is oscillatory, highly irregular and largely random. In the basic studies as well as in the design stages of a wave energy plant, the knowledge of the statistical characteristics of the local wave climate is essential, no matter whether physical or theoretical/numerical modelling methods are to be employed. This information
Corresponding author. Tel.: +90-212-2853726; fax: +90-212-3280400. ¨ zger). E-mail address: [email protected] (M. O
0960-1481/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2004.01.009
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1748
may result from wave measurements, more or less sophisticated forecast models or a combination of both, and usually takes the form of a set of representative sea states, each characterized by its frequency of occurrence and by a spectral distribution. The wave data emphasize that it is important when designing a plant to ensure that it will convert the energy efficiently over a sufficient wave period range while accommodating the large distribution of powers. Notably, the most commonly occurring wave conditions do not necessarily correspond to the largest energy contributions. On the contrary, the most energy intensive regions shift increasingly to the higher power bands as the wave period increases [1]. A stochastic formulation has been presented to evaluate the average performance of an OWC wave power plant in a wave climate characterized by a set of sea states whose one- or twodimensional power spectra are known [2]. The presented method is expected to be a powerful tool in the analysis and design (especially at the level of basic studies) of Wells-turbine-equipped oscillating-water-column OWC plants. A stochastic model is applied to devise an optimal algorithm for the rotational speed control of an OWC wave power plant equipped with a Wells turbine and to evaluate the average power output of the plant [3]. On the other hand, Setoguchi et al. [4] simulated the characteristics in order to clarify the usefulness of the impulse turbine under irregular flow conditions generated by the oscillating water column in the irregular sea conditions. Similarly, S˛en [5] has derived the most general wind energy potential formulation by considering simultaneously the temporal variabilites both in the wind speed and air density. The main purpose of this paper to take into consideration stochastic variabilities not only in the wave height but also in the wave period. A new formulation for the wave energy potential calculations is presented which accounts for the cross stochastic properties of wave height and period.
2. Stochastic structure of wave power In a plan view of an infinitely deep ocean the mean (over time and local space) energy of the water (kinetic þ potential) per unit area is Eu ¼ qgH
2
ð1Þ
where q is the specific density, g is gravitational acceleration and H is the mean square surface elevation. The result is true in all cases and it will be flowing in all directions. It is not possible to tell how from observations at a single point. Thus for a monochromatic plane wave of amplitude A the energy density (crest) velois: q g A2 2 J=m2 which is transported at half the phase city, c ¼ k=T. Hence, the power delivered by such a wave is q g kA2 2T, which is equal to qg2 TA2 8p using dispersion equation w2 ¼ kg.
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1749
In a polychromatic sea, spectral analysis of a time trace of the surface z(t) will tell us the contribution A2 dT 2 from waves of periods between T and T þ dT, again with no information about directionality. Waves of the same period from different directions will add up to a ‘‘wave’’ of the same period in the observed elevation which implies that one can calculate the mean period of the waves at a point P, weighted by energy content, the so-called energy period ð A2 T dT 2 Te ¼ ð 2 : A dT 2 It follows that the ‘‘point’’ power (not following in any particular direction), is from (2) and (3) ð g2 g2 2 q TA2 dT ¼ q Te H 8p 4p which has the unit of ½W=m. If the waves were all travelling in the same direction this would be the power crossing a transverse line. Te and H are parameters which can be calculated rigorously from a single point elevation time trace, available from a buoy or nowadays satellite radar. The question then arises as to how Te and H are related to the old measures of Tz and H s which are, the zero up-crossing and significant wave height, respectively. By convention, it is usually accepted that H s ¼ 4H and Tz is approximately Te in practice, although both these relations will depend on the spectrum. (One could easily do simulation using synthetic wave traces with specified spectra). Then power can be calculated approxiamately in [kW/m] as 2 g2 Hs 2 q ¼ 0:490605Tz H s ð2Þ Tz 4000p 4 Herein, the approximation of HsTz2 =2 is accepted as just a convenient simplification. Larger waves contain more energy per metre of crest length than small waves. It is usual to quantify the power of waves rather than their energy content. The power P (in kilowatts per metre) in an idealised ocean wave is approximately equal to the square of the wave height H (m) multiplied by the wave period T (sec) as follows [6]. P¼
qg2 TH 2 32p
ð3Þ
where q is density of the sea water and g is the gravity acceleration. The typical sea state is actually composed of many individual components, each of which is like the idealised, ‘monochromatic’ wave. Each has own properties, i.e. its own period, wave height and direction. It is the combination of these waves that we observe when we view the the surface of the sea. The total power in each metre of wave front is of course the sum of the powers of all components. It is obviously imposs-
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1750
ible to measure all the heights and periods independently, so an averaging process is used to estimate the total power. By deploying a wave-rider buoy, it is possible to record the variation in surface level during some chosen period of time. Significant wave height Hs is equal to the average of the highest one-third of the waves and the zero up-crossing-period Te, defined as the average time between upward movements of the surface through the mean level. For a typical irregular sea, the average total power is then given by, Ps ¼ as Te Hs2
ð4Þ
where as is a constant which conveys information about q, g and p. It is equal to 0.49 kWs 1 m 3. In the derivation, one of the basic assumptions is that zero upcrossing-period and significant wave height is constant at its average level. Hence, it is implied that they are independent from each other. However, in practice none of these assumptions is valid exactly. Besides, as it stands, the application of Eq. (4) is not possible over finite time duration. Primitive thoughts of application with finite time series in the form of zero up-crossing-period and significant wave height s as lead to average wave power, P e H 2 s ¼ 0:49 T P s e and H 2 are the arithmetic averages of the zero up-crossing-period and where T s significant wave height square, respectively. In stochastic terminology this expression can be rewritten with the concept of expectation operation as EðPs Þ ¼ 0:49 EðTe Þ EðHs2 Þ
ð5Þ
where E(.) corresponds to the expectation of the argument which is equivalent to the arithmetic averages for the long time series. However, in such an approach there is an implied assumption as the independence if Te from Hs. Such an assumption is not considered in the following.
3. Expected wave energy formulation According to the theory of dependent random variables in a stochastic process, if the zero up-crossing period and significant wave height are considered as dependent on each other, then the expectation of both sides in Eq. (4) leads by definition to EðPs Þ ¼ 0:49 EðTe Hs2 Þ
ð6Þ
However, in general the multiplication of the two independent random variables can be written in terms of the expectations of their multiplication and the multiplication of their individual expectations by stochastic covariance definition as CovðTe ; Hs2 Þ ¼ EðTe Hs2 Þ EðTe ÞEðHs2 Þ:
ð7Þ
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Furthermore, the cross-correlation coefficient, r, between the zero up-crossing period and significant wave height is defined as r¼
CovðTe Hs2 Þ STe SHs2
ð8Þ
where STe and SHs2 are standard deviations of the zero up-crossing period and significant wave height square time series, respectively. The elimination of CovðTe ; Hs2 Þ between Eqs. (7) and (8) yields after some algebraic manipulations to EðTe Hs2 Þ ¼ EðTe Þ E Hs2 þ rSTe SHs2 : ð9Þ Finally, the substitution of this expression into Eq. (6) yields EðPs Þ ¼ 0:49½EðTe Þ EðHs2 Þ þ rSTe SHs2 :
ð10Þ
This expression reduces to some simple approaches that are available in the practical applications. The second term in the brackets makes the major difference from the previously presented formulations in the literature. This term vanishes for constant wave period assumption because r ¼ 0, and Eq. (10) reduces to Eq. (5). For instantaneous wave period and height measurements, there is no cross-correlation, and hence Eq. (4) becomes valid. In the general stochastic formulation of Eq. (10), r plays a significant role depending on its actual values between 1 and +1. Logically, the greater the zero up-crossing period the greater the significant wave height. Hence, cross-correlation is expected to have positive sign between the zero up-crossing period and significant wave height. This argument indicates that Eq. (5) yields a lower value than Eq. (10) which can be rewritten as
STe SHs2 2 EðPs Þ ¼ 0:49EðTe Þ EðHs Þ 1 þ r ð11Þ EðTe Þ EðHs2 Þ Hence, the term within the brackets can be defined as the correction factor, a, and it is dimensionless. a¼1þr
STe SHs2 EðTe Þ EðHs2 Þ
ð12Þ
It is also possible to rewrite this expression by considering the coefficient of variation, C, which has definition as the ratio of the standard deviation to the arithmetic average. Since, two of such ratios appear in Eq. (12) as the coefficient for zero up-crossing period and significant wave height square. Finally, one can write succinctly that a ¼ 1 þ rCTe CHs2 :
ð13Þ
This expression indicates that in the case of small and especially less than one coefficient of variations, the second term on the right hand side becomes negligible. For such situations, the traditional formulation is quite acceptable. Furthermore,
1752
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
there are so many uncertainties that the effort involved for a correction does not seem worthwhile. On the other hand, for relatively big coefficients of variation the second term may amount to a significant level which means that traditional formulation underestimates the wave energy potential. It is obvious that only for r ¼ 0 the relative error becomes equal to zero. The formulation in the paper provides a basis for assessing the size of the correction factor for a given pair of wave period and height time series records.
4. Data The historical wave data used for application is obtained from NDBC (National Data Buoy Center) web site (www.ndbc.noaa.gov/hmd.shtml). NDBC-reported wave measurements are not directly measured by sensors on board the buoys. Instead, the accelerometers or inclinometers on board, the buoys measure the heave acceleration or the vertical displacement of the buoy hull during the wave acquisition time. A fast Fourier transform (FFT) is applied to the data by the processor on board the buoy to transform the data from the temporal domain into the frequency domain. Note that the raw acceleration or displacement measurements are not transmitted shore-side. Response amplitude operator (RAO) processing is then performed on the transformed data to account for both hull and electronic noise. It is from this transformation that nondirectional spectral wave measurements (i.e. wave energies with their associated frequencies) are derived. Along with the spectral energies, measurements such as significant wave height (WVHGT), average wave period (AVGPD), and dominant period (DOMPD) are also derived from the transformation.
5. Application In order to show the random variability in the actual situation, daily period and significant wave height measurements are recorded at California, off west US (Fig. 1). In this study, 11 stations are taken into consideration. Hourly data are used in this study in order to obtain more sensitive results. In the study area, each station exhibits different statistical properties. Twenty-four-hour moving averages for both zero up-crossing period and significant wave height for complete year 2002 at Half Moon Bay station are presented in Figs. 2 and 3. Table 1 presents the statistical properties of zero up-crossing period and square of significant wave height. Additionally, cross-correlation between zero up-crossing period and square of the significant wave height in addition to a coefficients are calculated for these stations as presented in Table 2. Distribution of a coefficients throughout the concerned sites are classified to equal intervals (Fig. 4). Most of the a coefficients is in the interval of 0.07–0.15. However, values at 46053 and 46054
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1753
Fig. 1. Location map of the study area.
Fig. 2. Half Moon Bay (46012) station 24-h moving average significant wave height variation for year 2002.
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
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Fig. 3. Half Moon Bay (46012) station 24-h moving average zero up-crossing period variation for year 2002.
stations deviate from others considerably. In the calculation of power for Pt. Conception station, for the averaged values of Hs ¼ 2:16 m and Te ¼ 7:12 sec, one can obtain wave power according to the classical approach as 19.25 kWm 1. How0 0 ever, by using new formulation, Ps ¼ ð1 þ aÞ; Ps ; Ps ¼ 21:51 kWm 1 is obtained. It is concluded that every station has own correction factor (a) according to their time series. So, in the calculation of wave power potential of an area, one can underestimate actual potential by using traditional formula. However, the new formulation gives more accurate results (Table 2).
Table 1 Station statistics Station ID 46012 46042 46028 46062 46011 46023 46063 46054 46053 46025 46047
Station name
Half Moon Bay Monterey Cape San M. Pt. San Luis Santa Maria Pt Arguello Pt.Conception Santa Barbara Santa Barb. E Santa Monica B. Tanner Banks
Northing
37.45 36.75 35.74 35.10 34.88 34.71 34.25 34.27 34.24 33.75 32.43
Westing
122.70 122.42 122.42 121.01 120.87 120.97 120.66 120.45 119.85 119.08 119.53
Average
Standard deviation
Ts ðsÞ
Hs2
7.39 7.40 7.47 7.93 7.44 7.23 7.12 6.72 6.19 6.52 7.47
4.22 5.88 6.61 4.72 5.02 4.35 5.52 4.80 1.98 1.45 5.46
ðmÞ2
Ts ðsÞ
Hs2 ðmÞ2
1.60 1.64 1.82 1.87 1.77 1.60 1.83 3.10 1.80 1.50 1.84
4.44 6.92 7.30 5.18 5.26 3.84 5.82 6.34 2.44 1.40 5.35
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Table 2 a coefficients and cross-correlation between zero up-crossing-period and square of the significant wave height Station ID
46012 46042 46028 46062 46011 46023 46063 46054 46053 46025 46047 Average
Station name
Half Moon Bay Monterey Cape San M. Pt. San Luis Santa Maria Pt Arguello Pt.Conception Santa Barbara Santa Barb. E Santa Monica B. Tanner Banks
rTs ;Hs2
0.40 0.49 0.42 0.39 0.42 0.34 0.43 0.53 0.46 0.34 0.36 0.42
a
1.09 1.13 1.11 1.10 1.10 1.07 1.12 1.32 1.17 1.07 1.09 1.12
Wave energy (kWm 1) Eq (2)
Eq (9)
Relative error (%)
15.29 21.33 24.22 18.33 18.32 15.42 19.25 15.80 6.01 4.64 19.97 16.23
16.70 24.07 26.98 20.18 20.23 16.45 21.51 20.87 7.01 4.99 21.69 18.24
8.43 11.38 10.23 9.15 9.45 6.23 10.50 24.27 14.20 6.91 7.94 10.79
Fig. 4. Equally classified a coefficients.
1756
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
6. Conclusion The stochastic structure of wave power potential formulation has been presented in its general form and its reduction to various classical approximations are shown in detail. It has been observed that in the traditional wave power potential calculations most often zero up-crossing period and significant wave height are assumed averaged values over a period of record time. After the stochastic derivations, it has been found that average wave energy formulation as a function of coefficients of variation of zero up-crossing period and significant wave height records, in addition to the cross-correlation coefficient. When the zero up-crossing period and significant wave height are independent, it simplifies to classical wave energy formulation where only averages of zero up-crossing period and significant wave height are employed. The classical approach leads to underestimations in wave energy assessments. The application of the stochastic wave energy formulation is performed for the climatical wave data recorded at the stations in the Pacific Ocean off California. Further studies can be focused on calculation of wave energy potential of a region by using new formulation. It has been seen that correction factor, a, differs for each station. It is useful to find correction factor separately so as to reach accurate results. Acknowledgements The authors would like to thank the reviewers for their valuable comments. References [1] Curran R, Whittaker TJT, Stewart TP. Aerodynamic conversion of ocean power from wave to wire. Energy Convers Mngt 1998;39:1919–29. [2] de O Falca˜o AF, Rodrigues RJA. Stochastic modelling of OWC wave power plant performance. Appl Ocean Res 2002;24:59–71. [3] de O Falca˜o AF. Control of an oscillating-water-column wave power plant for maximum energy production. Appl Ocean Res 2002;24:73–82. [4] Setoguchi T, Santhakumar S, Maeda H, Takao M, Kaneko K. A review of impulse turbines for wave energy conversion. Renew Energy 2001;23:261–92. [5] S˛en Z. Stochastic wind energy calculation formulation. J Wind Eng Ind Aerodyn 2000;84:227–34. [6] Tucker MJ, Pitt EG. Waves in ocean engineering. New York: Elsevier; 2001.
Technical note
Stochastic wave energy calculation formulation ¨ zger , Abdu¨sselam Altunkaynak, Zekai S˛en Mehmet O Istanbul Technical University, Civil Engineering Faculty, Hydraulics Division, Maslak 34469, Istanbul, Turkey Received 18 April 2003; accepted 19 January 2004
Abstract The wave energy potential is directly proportional to the wave period and second power of wave height averaged over a suitable time period. The wave height and period have temporal and spatial stochastic variations. It is the main purpose of this paper to derive the most general wave energy formulation by considering simultaneously the temporal variations both in the wave height and period. The correction factor is derived explicitly in terms of cross-correlation and the coefficients of variation. The application of the methodology is performed for wave measurement stations located in the Pacific Ocean off the west coast of the US. # 2004 Elsevier Ltd. All rights reserved. Keywords: Wave energy; Stochastic variables; Significant wave height
1. Introduction The possibility of extracting energy from ocean waves has intrigued people for centuries. Although there are a few concepts over 100 years old, it is only the past two decades that viable schemes have been proposed. In a wave energy plant, unless a large energy storage interface follows the waves in the energy conversion chain of the plant (as the on-shore water reservoir of the tapered channel or TAPCHAN device), the power take-off equipment is required to convert an energy flux that is oscillatory, highly irregular and largely random. In the basic studies as well as in the design stages of a wave energy plant, the knowledge of the statistical characteristics of the local wave climate is essential, no matter whether physical or theoretical/numerical modelling methods are to be employed. This information
Corresponding author. Tel.: +90-212-2853726; fax: +90-212-3280400. ¨ zger). E-mail address: [email protected] (M. O
0960-1481/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2004.01.009
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1748
may result from wave measurements, more or less sophisticated forecast models or a combination of both, and usually takes the form of a set of representative sea states, each characterized by its frequency of occurrence and by a spectral distribution. The wave data emphasize that it is important when designing a plant to ensure that it will convert the energy efficiently over a sufficient wave period range while accommodating the large distribution of powers. Notably, the most commonly occurring wave conditions do not necessarily correspond to the largest energy contributions. On the contrary, the most energy intensive regions shift increasingly to the higher power bands as the wave period increases [1]. A stochastic formulation has been presented to evaluate the average performance of an OWC wave power plant in a wave climate characterized by a set of sea states whose one- or twodimensional power spectra are known [2]. The presented method is expected to be a powerful tool in the analysis and design (especially at the level of basic studies) of Wells-turbine-equipped oscillating-water-column OWC plants. A stochastic model is applied to devise an optimal algorithm for the rotational speed control of an OWC wave power plant equipped with a Wells turbine and to evaluate the average power output of the plant [3]. On the other hand, Setoguchi et al. [4] simulated the characteristics in order to clarify the usefulness of the impulse turbine under irregular flow conditions generated by the oscillating water column in the irregular sea conditions. Similarly, S˛en [5] has derived the most general wind energy potential formulation by considering simultaneously the temporal variabilites both in the wind speed and air density. The main purpose of this paper to take into consideration stochastic variabilities not only in the wave height but also in the wave period. A new formulation for the wave energy potential calculations is presented which accounts for the cross stochastic properties of wave height and period.
2. Stochastic structure of wave power In a plan view of an infinitely deep ocean the mean (over time and local space) energy of the water (kinetic þ potential) per unit area is Eu ¼ qgH
2
ð1Þ
where q is the specific density, g is gravitational acceleration and H is the mean square surface elevation. The result is true in all cases and it will be flowing in all directions. It is not possible to tell how from observations at a single point. Thus for a monochromatic plane wave of amplitude A the energy density (crest) velois: q g A2 2 J=m2 which is transported at half the phase city, c ¼ k=T. Hence, the power delivered by such a wave is q g kA2 2T, which is equal to qg2 TA2 8p using dispersion equation w2 ¼ kg.
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1749
In a polychromatic sea, spectral analysis of a time trace of the surface z(t) will tell us the contribution A2 dT 2 from waves of periods between T and T þ dT, again with no information about directionality. Waves of the same period from different directions will add up to a ‘‘wave’’ of the same period in the observed elevation which implies that one can calculate the mean period of the waves at a point P, weighted by energy content, the so-called energy period ð A2 T dT 2 Te ¼ ð 2 : A dT 2 It follows that the ‘‘point’’ power (not following in any particular direction), is from (2) and (3) ð g2 g2 2 q TA2 dT ¼ q Te H 8p 4p which has the unit of ½W=m. If the waves were all travelling in the same direction this would be the power crossing a transverse line. Te and H are parameters which can be calculated rigorously from a single point elevation time trace, available from a buoy or nowadays satellite radar. The question then arises as to how Te and H are related to the old measures of Tz and H s which are, the zero up-crossing and significant wave height, respectively. By convention, it is usually accepted that H s ¼ 4H and Tz is approximately Te in practice, although both these relations will depend on the spectrum. (One could easily do simulation using synthetic wave traces with specified spectra). Then power can be calculated approxiamately in [kW/m] as 2 g2 Hs 2 q ¼ 0:490605Tz H s ð2Þ Tz 4000p 4 Herein, the approximation of HsTz2 =2 is accepted as just a convenient simplification. Larger waves contain more energy per metre of crest length than small waves. It is usual to quantify the power of waves rather than their energy content. The power P (in kilowatts per metre) in an idealised ocean wave is approximately equal to the square of the wave height H (m) multiplied by the wave period T (sec) as follows [6]. P¼
qg2 TH 2 32p
ð3Þ
where q is density of the sea water and g is the gravity acceleration. The typical sea state is actually composed of many individual components, each of which is like the idealised, ‘monochromatic’ wave. Each has own properties, i.e. its own period, wave height and direction. It is the combination of these waves that we observe when we view the the surface of the sea. The total power in each metre of wave front is of course the sum of the powers of all components. It is obviously imposs-
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1750
ible to measure all the heights and periods independently, so an averaging process is used to estimate the total power. By deploying a wave-rider buoy, it is possible to record the variation in surface level during some chosen period of time. Significant wave height Hs is equal to the average of the highest one-third of the waves and the zero up-crossing-period Te, defined as the average time between upward movements of the surface through the mean level. For a typical irregular sea, the average total power is then given by, Ps ¼ as Te Hs2
ð4Þ
where as is a constant which conveys information about q, g and p. It is equal to 0.49 kWs 1 m 3. In the derivation, one of the basic assumptions is that zero upcrossing-period and significant wave height is constant at its average level. Hence, it is implied that they are independent from each other. However, in practice none of these assumptions is valid exactly. Besides, as it stands, the application of Eq. (4) is not possible over finite time duration. Primitive thoughts of application with finite time series in the form of zero up-crossing-period and significant wave height s as lead to average wave power, P e H 2 s ¼ 0:49 T P s e and H 2 are the arithmetic averages of the zero up-crossing-period and where T s significant wave height square, respectively. In stochastic terminology this expression can be rewritten with the concept of expectation operation as EðPs Þ ¼ 0:49 EðTe Þ EðHs2 Þ
ð5Þ
where E(.) corresponds to the expectation of the argument which is equivalent to the arithmetic averages for the long time series. However, in such an approach there is an implied assumption as the independence if Te from Hs. Such an assumption is not considered in the following.
3. Expected wave energy formulation According to the theory of dependent random variables in a stochastic process, if the zero up-crossing period and significant wave height are considered as dependent on each other, then the expectation of both sides in Eq. (4) leads by definition to EðPs Þ ¼ 0:49 EðTe Hs2 Þ
ð6Þ
However, in general the multiplication of the two independent random variables can be written in terms of the expectations of their multiplication and the multiplication of their individual expectations by stochastic covariance definition as CovðTe ; Hs2 Þ ¼ EðTe Hs2 Þ EðTe ÞEðHs2 Þ:
ð7Þ
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
1751
Furthermore, the cross-correlation coefficient, r, between the zero up-crossing period and significant wave height is defined as r¼
CovðTe Hs2 Þ STe SHs2
ð8Þ
where STe and SHs2 are standard deviations of the zero up-crossing period and significant wave height square time series, respectively. The elimination of CovðTe ; Hs2 Þ between Eqs. (7) and (8) yields after some algebraic manipulations to EðTe Hs2 Þ ¼ EðTe Þ E Hs2 þ rSTe SHs2 : ð9Þ Finally, the substitution of this expression into Eq. (6) yields EðPs Þ ¼ 0:49½EðTe Þ EðHs2 Þ þ rSTe SHs2 :
ð10Þ
This expression reduces to some simple approaches that are available in the practical applications. The second term in the brackets makes the major difference from the previously presented formulations in the literature. This term vanishes for constant wave period assumption because r ¼ 0, and Eq. (10) reduces to Eq. (5). For instantaneous wave period and height measurements, there is no cross-correlation, and hence Eq. (4) becomes valid. In the general stochastic formulation of Eq. (10), r plays a significant role depending on its actual values between 1 and +1. Logically, the greater the zero up-crossing period the greater the significant wave height. Hence, cross-correlation is expected to have positive sign between the zero up-crossing period and significant wave height. This argument indicates that Eq. (5) yields a lower value than Eq. (10) which can be rewritten as
STe SHs2 2 EðPs Þ ¼ 0:49EðTe Þ EðHs Þ 1 þ r ð11Þ EðTe Þ EðHs2 Þ Hence, the term within the brackets can be defined as the correction factor, a, and it is dimensionless. a¼1þr
STe SHs2 EðTe Þ EðHs2 Þ
ð12Þ
It is also possible to rewrite this expression by considering the coefficient of variation, C, which has definition as the ratio of the standard deviation to the arithmetic average. Since, two of such ratios appear in Eq. (12) as the coefficient for zero up-crossing period and significant wave height square. Finally, one can write succinctly that a ¼ 1 þ rCTe CHs2 :
ð13Þ
This expression indicates that in the case of small and especially less than one coefficient of variations, the second term on the right hand side becomes negligible. For such situations, the traditional formulation is quite acceptable. Furthermore,
1752
¨ zger et al. / Renewable Energy 29 (2004) 1747–1756 M. O
there are so many uncertainties that the effort involved for a correction does not seem worthwhile. On the other hand, for relatively big coefficients of variation the second term may amount to a significant level which means that traditional formulation underestimates the wave energy potential. It is obvious that only for r ¼ 0 the relative error becomes equal to zero. The formulation in the paper provides a basis for assessing the size of the correction factor for a given pair of wave period and height time series records.
4. Data The historical wave data used for application is obtained from NDBC (National Data Buoy Center) web site (www.ndbc.noaa.gov/hmd.shtml). NDBC-reported wave measurements are not directly measured by sensors on board the buoys. Instead, the accelerometers or inclinometers on board, the buoys measure the heave acceleration or the vertical displacement of the buoy hull during the wave acquisition time. A fast Fourier transform (FFT) is applied to the data by the processor on board the buoy to transform the data from the temporal domain into the frequency domain. Note that the raw acceleration or displacement measurements are not transmitted shore-side. Response amplitude operator (RAO) processing is then performed on the transformed data to account for both hull and electronic noise. It is from this transformation that nondirectional spectral wave measurements (i.e. wave energies with their associated frequencies) are derived. Along with the spectral energies, measurements such as significant wave height (WVHGT), average wave period (AVGPD), and dominant period (DOMPD) are also derived from the transformation.
5. Application In order to show the random variability in the actual situation, daily period and significant wave height measurements are recorded at California, off west US (Fig. 1). In this study, 11 stations are taken into consideration. Hourly data are used in this study in order to obtain more sensitive results. In the study area, each station exhibits different statistical properties. Twenty-four-hour moving averages for both zero up-crossing period and significant wave height for complete year 2002 at Half Moon Bay station are presented in Figs. 2 and 3. Table 1 presents the statistical properties of zero up-crossing period and square of significant wave height. Additionally, cross-correlation between zero up-crossing period and square of the significant wave height in addition to a coefficients are calculated for these stations as presented in Table 2. Distribution of a coefficients throughout the concerned sites are classified to equal intervals (Fig. 4). Most of the a coefficients is in the interval of 0.07–0.15. However, values at 46053 and 46054
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Fig. 1. Location map of the study area.
Fig. 2. Half Moon Bay (46012) station 24-h moving average significant wave height variation for year 2002.
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Fig. 3. Half Moon Bay (46012) station 24-h moving average zero up-crossing period variation for year 2002.
stations deviate from others considerably. In the calculation of power for Pt. Conception station, for the averaged values of Hs ¼ 2:16 m and Te ¼ 7:12 sec, one can obtain wave power according to the classical approach as 19.25 kWm 1. How0 0 ever, by using new formulation, Ps ¼ ð1 þ aÞ; Ps ; Ps ¼ 21:51 kWm 1 is obtained. It is concluded that every station has own correction factor (a) according to their time series. So, in the calculation of wave power potential of an area, one can underestimate actual potential by using traditional formula. However, the new formulation gives more accurate results (Table 2).
Table 1 Station statistics Station ID 46012 46042 46028 46062 46011 46023 46063 46054 46053 46025 46047
Station name
Half Moon Bay Monterey Cape San M. Pt. San Luis Santa Maria Pt Arguello Pt.Conception Santa Barbara Santa Barb. E Santa Monica B. Tanner Banks
Northing
37.45 36.75 35.74 35.10 34.88 34.71 34.25 34.27 34.24 33.75 32.43
Westing
122.70 122.42 122.42 121.01 120.87 120.97 120.66 120.45 119.85 119.08 119.53
Average
Standard deviation
Ts ðsÞ
Hs2
7.39 7.40 7.47 7.93 7.44 7.23 7.12 6.72 6.19 6.52 7.47
4.22 5.88 6.61 4.72 5.02 4.35 5.52 4.80 1.98 1.45 5.46
ðmÞ2
Ts ðsÞ
Hs2 ðmÞ2
1.60 1.64 1.82 1.87 1.77 1.60 1.83 3.10 1.80 1.50 1.84
4.44 6.92 7.30 5.18 5.26 3.84 5.82 6.34 2.44 1.40 5.35
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Table 2 a coefficients and cross-correlation between zero up-crossing-period and square of the significant wave height Station ID
46012 46042 46028 46062 46011 46023 46063 46054 46053 46025 46047 Average
Station name
Half Moon Bay Monterey Cape San M. Pt. San Luis Santa Maria Pt Arguello Pt.Conception Santa Barbara Santa Barb. E Santa Monica B. Tanner Banks
rTs ;Hs2
0.40 0.49 0.42 0.39 0.42 0.34 0.43 0.53 0.46 0.34 0.36 0.42
a
1.09 1.13 1.11 1.10 1.10 1.07 1.12 1.32 1.17 1.07 1.09 1.12
Wave energy (kWm 1) Eq (2)
Eq (9)
Relative error (%)
15.29 21.33 24.22 18.33 18.32 15.42 19.25 15.80 6.01 4.64 19.97 16.23
16.70 24.07 26.98 20.18 20.23 16.45 21.51 20.87 7.01 4.99 21.69 18.24
8.43 11.38 10.23 9.15 9.45 6.23 10.50 24.27 14.20 6.91 7.94 10.79
Fig. 4. Equally classified a coefficients.
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6. Conclusion The stochastic structure of wave power potential formulation has been presented in its general form and its reduction to various classical approximations are shown in detail. It has been observed that in the traditional wave power potential calculations most often zero up-crossing period and significant wave height are assumed averaged values over a period of record time. After the stochastic derivations, it has been found that average wave energy formulation as a function of coefficients of variation of zero up-crossing period and significant wave height records, in addition to the cross-correlation coefficient. When the zero up-crossing period and significant wave height are independent, it simplifies to classical wave energy formulation where only averages of zero up-crossing period and significant wave height are employed. The classical approach leads to underestimations in wave energy assessments. The application of the stochastic wave energy formulation is performed for the climatical wave data recorded at the stations in the Pacific Ocean off California. Further studies can be focused on calculation of wave energy potential of a region by using new formulation. It has been seen that correction factor, a, differs for each station. It is useful to find correction factor separately so as to reach accurate results. Acknowledgements The authors would like to thank the reviewers for their valuable comments. References [1] Curran R, Whittaker TJT, Stewart TP. Aerodynamic conversion of ocean power from wave to wire. Energy Convers Mngt 1998;39:1919–29. [2] de O Falca˜o AF, Rodrigues RJA. Stochastic modelling of OWC wave power plant performance. Appl Ocean Res 2002;24:59–71. [3] de O Falca˜o AF. Control of an oscillating-water-column wave power plant for maximum energy production. Appl Ocean Res 2002;24:73–82. [4] Setoguchi T, Santhakumar S, Maeda H, Takao M, Kaneko K. A review of impulse turbines for wave energy conversion. Renew Energy 2001;23:261–92. [5] S˛en Z. Stochastic wind energy calculation formulation. J Wind Eng Ind Aerodyn 2000;84:227–34. [6] Tucker MJ, Pitt EG. Waves in ocean engineering. New York: Elsevier; 2001.