mathemacal%and%physical%models%for%the%es2maon ......greek%naval%academy%%...

Greek Naval Academy Mathema2cal Modeling Group

Mathema2cal and Physical Models for the Es2ma2on of Wave Energy Poten2al in the Eastern Mediterranean Sea

George Galanis, George Zodia/s, George Kallos, Peter Chu

University of Athens Atmospheric Modeling

and Weather Forecas2ng Group

University of Cyprus Oceanography Center

US Naval Postgraduate School Naval Ocean Analysis

and Predic2on Laboratory

Mathema2cal and Physical Models for the Es2ma2on of Wave Energy Poten2al

THE FRAMEWORK

•  Renewable energy resource assessment is a primary target today for EU

Member States especially under the warnings of the scien2fic community

concerning global warming and the new framework set by the recent

economic crisis

•  Analyses and studies focusing on more than one “clean” energy sources

are of par2cular interest since different ques2ons/problems can be

addressed:

•  Monitoring the renewable energy poten2al spa2ally and temporally

•  Quan2fying and reducing the variability of the available energy

suppor2ng the op2mal integra2on to the general grid


THE FRAMEWORK

•  Demands in this framework are becoming more and more important for

both designing and opera2onal phases.

•  Regularly, data sets and analysis suppor2ng this type of ac2vi2es are not

available from the exis2ng governmental agencies, such as Met Offices.

•  Such informa2on can be provided only by specialized groups with

interdisciplinary type of knowledge and tools based on state-‐of-‐the-‐art

monitoring equipment and numerical systems.


OBJECTIVES

•  In the present work recent ac2ons/projects on the monitoring and analysis of

wave energy poten2al with par2cular emphasis to the area of Eastern

Mediterranean Sea are discussed.

•  A hybrid/mul2-‐parametric approach is adopted based on

Ø  high resolu2on numerical atmospheric-‐wave modeling and

Ø  non-‐conven2onal sta2s2cal techniques

New technologies allow the exploita2on of the energy from waves, currents and sea surface winds

Wave Energy – A new challenge

Oscilla2ng water column devices Point absorbers Overtopping devices


•  It’s easier adopted to the general grid due to its con2nuous behavior.

•  Wave power may be produced even in the absence of local winds by exploi2ng swells

•  Ecological damages or consequences appear negligible

•  Wave energy has the poten2al to meet a significant propor2on of Mediterranean Sea energy demands:

The Interna2onal Energy Agency states that ocean energy has a poten2al to reach 3.6 GW of installed capacity by 2020 and close to 188 GW by 2050

The global wave power resource has been es2mated as around 2.1 TW


Wave Power Es,ma,on

The available (theore2cal) Wave Energy Power density poten2al is es2mated by

where E(f,θ) is the 2-‐dimensional wave spectrum, Hs the significant wave height and Te the mean energy wave period.

22 1 2

0 0( , ) [ / ]

64w s epgP pg f E f dfd T W m

πθ θ

π

∞ −= = Η∫ ∫

Integrated High Resolu,on System for Monitoring and Quan,fying the Wave Energy Poten,al in the EEZ of Cyprus

This was the main target of three recent European projects:

Marine Renewable Integrated Applica,on PlaDorm

Monitoring the Wind and Sea Wave Energy Poten,al

§  State of the art atmospheric-‐wave predic/on system targe2ng at the

high resolu2on monitoring/es2ma2on of the sea state characteris2cs,

essen2al for the energy poten2al.

§  Novel advanced sta/s/cal systems for the local adapta2on of the

results of the models and the es2ma2on of energy distribu2on.

The tools


Wave Power Es,ma,on

The models used

Horizontal Resolu2on 0.05° x 0.05° •  Time-‐step 15 seconds •  45 ver2cal levels up to 50 hPa •  Ini2al and boundary condi2ons: High-‐resolu2on reanalysis (15x15Km) •  Output at: {10, 40, 80, 120, 180} m • Full set of meteorological variables

•  Domain (20–75°N, 50°W–30°E) •  Resolu2on: 0.05° x 0.05° •  Number of frequencies: 25 •  Minimum frequency: 0.055 Hz •  Number of direc2ons: 24 •  Grid points: 1601 x 1101 •  Spectral output at selected loca2ons •  Integrated parameters at every grid •  point: Hs, Te, Swell, Maximum wave

Atmospheric model SKIRON Wave Model WAM


§  The OC-‐UCY in coopera2on with the UOA/AM&WFG has adopted the latest version of the new WAM model from the ECMWF (parallel version).

§  A new advec2on scheme (Corner Transport Upstream) has been adopted providing a more uniform propaga2on in all direc2ons

§  For the E-‐wave project the wave model’s domain cover the whole east Med region in order to capture all the swell informa2on that could reach the study area (Levan2ne Basin)

The numerical models used: The new parallel version of the wave model WAM

The wave model’s domain

Wave model Characteris,cs

WAM ECMWF CY33R1

Model’s domain East Mediterranean (30N – 41N, 15E – 37E)

Study area Levan2ne

Horizontal Resolu,on

1/60 x 1/60 degrees (0.01667 km approximately)

Frequencies 25 (range 0.0417-‐0.54764Hz logarithmically spaced)

Direc,ons 24 (equally spaced) Timestep 45 sec

Wave Model Direct Outputs

§  Significant Wave Height & Direc2on

§  Maximum Expected Wave Height

§  Swell Height and Direc2on

§  Wind driven wave height and Direc2on

§  Mean & Peak Wave Period

§  Wind Speed & Direc2on at sea level

§  Wind and Swell frequencies for the two

dominant spectrum components

§  Full wave spectrum at selected grid points


Evaluation of the wave predictions

The evalua2on results against independent observa2ons

from a wave buoy at the eastern coast of the

Mediterranean (area of Hadera port, Israel) prove

excellent accuracy of the modeling system

Stats Bias (m) -0,09 RMSE (m) 0,32

Evaluation of the wave predictions

The evalua2on results against independent observa2ons from a wave buoy and against

available satellite records prove excellent accuracy of the modeling system

WAM – Buoy WAM – Satellites

Mean BIAS RMSE Mean BIAS RMSE

Test Case 1 0.05 0.16 0.13 0.34

Test Case 2 0.28 0.33 -‐0.02 0.32

Test Case 3 0.10 0.29 0.04 0.46

Test Case 4 0.19 0.42 0.44 0.63

Resource analysis iden,fying “hot spots” areas

§  Wave condi,ons (Hs, T, θ)

§  High spa,al and temporal resolu,on sta,s,cal analysis

§  Temporal variability of wave parameters at different ,me scales

§  Annual §  Seasonal §  Monthly

§  Sta,s,cal indexes for the asymmetry and the impact of non frequent values

§  Mul,variate distribu,on fi^ng

Wave Resource Assessment-‐Methodology


Wind – Wave climatology

•  The results obtained from a 10y high resolu2on

simula2ons provide important informa2on for the

wind-‐wave climatology of the areas under study.

•  Es2ma2ons can be made on the spa2al and

temporal distribu2on of wind-‐wave power

poten2al

Wind-‐Wave energy monitoring and forecas2ng

Wind – Wave climatology

•  The wind speed keeps interes2ng

values over Aegean Sea even in

summer due to the etesian winds.

•  The areas of increased interest are

located over the two sides of Crete

and in a central tunel crossing the

Aegean from north to southeast.

•  The significant wave height is

considerable even in the west parts

of Greece due to the prevalence of

swell, coming from central and

s o u t h e r n p a r t s o f t h e

Mediterranean Sea.

•  The swell dominated southwestern

areas of the Greek seas seem to

keep the primary role in wave

energy poten2al

Wind-‐Wave energy monitoring and forecas2ng in the Eastern Mediterranean Sea

Wind -‐ Wave climatology: Is it enough for suppor,ng efficiently the resource assessment?

Sta,s,cal measures for the asymmetry and the kurtosis of the data could be essen2al

•  Skewness (3rd standardized moment) provides informa2on for the tails of the distribu2ons

•  Areas with poten2al impact from extreme values can be spored based on the kurtosis (4th standardized

moment)

More sta,s,cal measures should be taken into account

Monthly variability of the Mean Hs in Med Sea


Project Results

The results of the E-Wave project indicate higher values for the two sea state

parameters (Hs and Te) that mainly affect the wave energy estimation, in the

west and south offshore Cyprus EEZ

Project results

The impact of extreme/non-frequent values of the sea state is limited, as

revealed by the low kurtosis (4th statistical moment) values

The use of the E-‐wave project results for the wave energy poten2al combined w i t h o t h e r a r e a character i s2cs , as for example the bathymetry, provide informa2on of added value to the decision makers and the industry involved in the energy sector.

Wave Energy distribution

Wave Power Mapping using the characteris,cs of specific wave plaDorms

The use of Hs/Te power transform

matrices are u2lized for es2ma2ng the

wave power poten2al over the Marina

Plasorm domain

(Hs, Te) -‐ surfaces are developed and

compiled into the 10y-‐data base

Some classical myths are shaken down:

Pelamis can be used in Med Sea too

The Med coast of France seems to be

comparable with the most energe2c

Atlan2c coastline, a trend not visible in the

theore2cal resource.


On site sta,s,cal analysis

The 2me distribu2on of crucial parameters over

the whole 10y study period may reveal trends and

(seasonal or other) periodici2es



•  Weibull distribu2on could be a good choice for fitng wind

speed and significant wave height values but Lognormal 3P

provides an interes2ng alterna2ve with even more berer

convergence.

•  Thresholds for extreme values are equivalently es2mated by

the two pdfs as the corresponding 95-‐percen2les.



Different local wave climatology

is depicted in the bivariate plots

describing the joint probability

distribu2on of Hs and Te;

a sta2s2cal informa2on of

primary importance for wave

energy es2ma2on


–  The wave energy poten2al can be also studied by a probability distribu2on fitng point of view.

–  For the present study, a series of independent sta2s2cal tests proved that the Lognormal distribu,on op2mally fits the modeled data

–  Equally good fit can be also succeeded by the Generalized Extreme Value pdf

–  The corresponding parameters have a non trivial spa,al distribu,on and provide informa2on of poten2al value for grid designers and researchers.

Wave Power Poten,al distribu,on

Lognormal m-‐parameter Lognormal v-‐parameter

Probability Density Function

Lognormal (1.5; 0.5)

x1086420

f(x)

0.520.480.44

0.4

0.360.320.28

0.240.2

0.160.12

0.080.04

0

21 ln( )2

( ; , )2

x mvef x

xvµ σ

π

−−

=


The new Wave Model WAM – Evaluation study of the Wave Currents Interaction

with applications to wave power prediction

A decrease of the mean wave

period is noticed as a consequence

of the use of sea surface currents

as a forcing in the wave model.

These changes result to reduced

wave power potential


Operational predictions of wave energy potential

An online operational wave energy system has been coupled with the CYCOFOS WAM and is available at : www.oceanography.ucy.ac.cy/cycofos/offshore.html


Bias analysis and correc,on

Ø  Numerical wind/wave models may result to systematic (or not) biases due to

Ø intrinsic difficulties of the models in simulating subscale phenomena

Ø local area’s peculiarities.

Ø  Main ways out:

Ø Assimilation

Ø Optimization of the outputs of the models by using statistical techniques

(MOS methods, Kalman filters)

Ø The minimization of a “cost-function” that governs the evolution of the error

is needed.



Kolmogorov-‐Zurbenko Filter: An itera2ve moving average aiming at the removal of high-‐frequency varia2ons in the ini2al data.

0

5

10

15

20

0 20 40 60 80 100

wind (m

/s)

,me (h)

obs obs smoothed

Kalman Filter: Simulates the temporal rela2on between observa2onal and forecasted values and improves the larer at a following 2me.


§  Mathema2cal post-‐processes based on Kalman filters have been u2lized suppor2ng the adapta2on of the numerical models to the local area characteris2cs and the elimina2on of possible systema2c biases.

Optimization of the wave predictions

WAM outputs, Kalman improved forecasts and the corresponding buoy wave observa2ons


Kalman combined with assimila,on

Analysis Forecas,ng Period

Ø  Available Observa2ons and WAM forecasts are used by the filter to produce a new improved forecast

Ø  This is assimilated during the forecas2ng period improving significantly the assimila2on impact in 2me and space

SWH


The use of ar2ficial observa2ons

Apart from the real observations assimilated (blue solid line) within the assimilation window, the improved, via the filters, forecasts of the model (dashed line) are assimilated inside the forecasting

period as artificial observations.

The assimilation impact is extended to the whole forecasting period (area of test case: California, USA)



Ø  Recent advances in Statistics and Differential Geometry (Information Geometry) suggests that

the use of distribution fitting should be taken into account.

Ø  The family of two parameter Weibull distributions

form a statistical 2-dimensional manifold with Fisher information matrix

Ø  This matrix defines the inner product, and therefore the geometrical framework, in which the

Weibull manifolds are categorized.

Ø  This distance should be minimized instead of using least square methods (classical Euclidean

Geometry).

(G. Galanis, P.C. Chu, G. Kallos, Yu-‐Heng Kuo and C.T.J. Dodson, Wave Height Characteris/cs in North Atlan/c Ocean: A new approach based on Sta/s/cal and Geometrical techniques, Stoch Environ Res Risk Assess (2012) 26:83–103, DOI 10.1007/s00477-‐011-‐0540-‐2.

G. Galanis, Dan Hayes, George Zodia2s, P.C. Chu Yu-‐Heng Kuo and G. Kallos, Wave height characteris/cs in the Mediterranean Sea by means of numerical modeling, satellite data, sta/s/cal and geometrical techniques, Marine Geophysical Research (2012), DOI 10.1007/s11001-‐011-‐9142-‐0)

𝑝(𝑥; 𝜉) = 𝛼𝛽+𝑥𝛽,𝛼−1

𝑒−+𝑥𝛽,

𝛼

𝐺(𝛼, 𝛽) = (𝛼2𝛽2 𝛽(1 − 𝛾)

𝛽(1 − 𝛾)6(𝛾 − 1)2 + 𝜋2

6𝛼20



Monthly satellite and WAM values as elements of the Weibull distribu,ons sta,s,cal manifold



The minimum distance between two elements 𝑓↓1  𝑎𝑛𝑑 𝑓↓2  of a sta2s2cal manifold S is defined by the corresponding geodesic ω which is the minimum length curve that connects them Geodesics are obtained as solu2ons of 2nd order differen2al systems:

𝜔↓𝑖 ↑′′ (𝑡)+∑𝑗,𝑘=1↑𝑛▒𝛤↓𝑗𝑘↑𝑖 (𝑡) 𝜔↑′ ↓𝑗 (𝑡)𝜔↑′ ↓𝑘 (𝑡)=0, 𝑖=1, 2,…, 𝑛.

under the condi2ons 𝜔(0)= 𝑓↓1 , 𝜔(1)= 𝑓↓2  . The coefficients 𝛤↓𝑗𝑘↑𝑖  are defined by the Fisher informa2on matrix and, therefore, are directly affected by the shape and scale parameter that the data under study berer fit.

An example for the area of Med Sea. The ode system becomes: 𝜔↓1↑′′ −0,82𝜔′↓1↑2 +0,65𝜔↓1↑′ 𝜔↓2↑′ −0,02(𝜔↓2↑′ )↑2 =0 𝜔↓2↑′′ −0,77𝜔′↓1↑2 +0,77𝜔↓1↑′ 𝜔↓2↑′ −0,32(𝜔↓2↑′ )↑2 =0 under the condi2ons 𝜔↓1↑ (0)=3.43, 𝜔↓2↑ (0)=2.30, 𝜔↓1↑ (1)=2.82, 𝜔↓2↑ (1)=2.35 In the plot the numerical solu2on spray of geodesics emana2ng from (3.43, 2.30) is given including the one to (2.82, 2.35) that gives the minimum length curve connec2ng the satellite observa2ons with WAM outputs


Extreme wind/wave values es/ma/on

The credible es2ma2on/forecast of poten2al non-‐frequent/extreme values for

environmental predic2ons is of cri2cal importance for a number of applica2ons

Extreme Condi2ons can be described by different approaches:

Ø  95th percen2les of Significant wave height awer fitng the data on a

Probability density Func2on (e.g Weibull, Weibull 3 parameters, Lognormal

2/3 parameters).

Ø  The maximum expected wave height.

Ø  50 year return period of Significant Wave Height, Wind Speed.

Extreme wind/wave condi,ons

•  Weibull distribu2on could be a good choice for fitng wind speed and significant wave height values but Lognormal 3P provides an interes2ng alterna2ve with even berer convergence.

•  Thresholds for extreme values are equivalently es2mated by the two pdfs as the corresponding 95-‐percen2les.

Significant Wave Height

Wind Speed Calcula,on of the 95th percen,le by fi^ng the 10 year hourly data to PDFs


Es,ma,on of the 50-‐year return period of significant wave height and wind speed For this analysis at least 10-‐year long-‐term data are required

•  Periodic Maximum Method (PMM): based on the 95% confidence intervals

𝑋↓↑𝑇 ±1.96𝜎( 𝑋↓↑𝑇 ) of the Generalized Extreme value distribu2on (GEVD) 𝐹(𝑋)=𝑒𝑥𝑝{−(1−𝑎(𝑋−𝛽))} •  Peak Over Threshold (POT): based on the 95% confidence intervals

𝑋↓↑𝑇 ±1.96𝜎( 𝑋↓↑𝑇 ) of the Generalized Pareto Distribu2on (GPD) 𝐹(𝑋)=1−(1− 𝑘(𝑋−𝑥𝑜)/𝐴 ) ↑1/𝑘 

The 50-‐years return Significant Wave Height or Wind speed

+ Use of local adapta2on methods (eg. bias correc2on) as to op2mize the models output.

Results for Hs


Results for wind speed

Es,ma,on of maximum wave height

Forecas2ng maximum wave height at selected sites based on high resolu2on hindcast wave modeling and local adapta2on techniques

Hmax =𝑎∙𝐻𝑠+𝐹(𝑇𝑚)

The maximum wave height is modeled by the significant wave height and the local mean wave period based on a sta2s2cal approach :

Concluding thoughts

•  The es,ma,on of wave energy poten,al is not always straight forward. The

characteriza,on of an area as suitable for renewable energy exploita,on can not be

based on a Yes/No answer.

•  The lack of a dense observa,onal network over sea areas poses further difficul,es

revealing the increased role that numerical simula,on systems keep.

•  Numerical wind/wave models, supported by op,miza,on post processes, provide an

excellent way out

•  The local wind/wave characteris,cs and the corresponding energy poten,al should

be analyzed on different ,me scales and by employing sta,s,cal indexes measuring

not only averages but also the varia,on, the asymmetry and the poten,al impact by

extreme values as well as the corresponding op,mally filed distribu,ons.

•  The specific characteris,cs of the technology that will be employed for the transla,on

of the wave energy to power are crucial and should be taken into account.

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