Stochastic modelling of OWC device and power production

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(1)PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE ESCUELA DE INGENIERIA. STOCHASTIC MODELLING OF OWC DEVICE AND POWER PRODUCTION. CARLOS I. MALDONADO RIVERA. Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the Degree of Master of Science in Engineering. Advisor: ROLANDO A. REBOLLEDO BERROETA. Santiago de Chile, January 2017 c MMXVII, C ARLOS I GNACIO M ALDONADO R IVERA.

(2) PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE ESCUELA DE INGENIERIA. STOCHASTIC MODELLING OF OWC DEVICE AND POWER PRODUCTION CARLOS I. MALDONADO RIVERA. Members of the committe: ROLANDO A. REBOLLEDO BERROETA SEBASTIÁN RÍOS MARCUELLO CLAUDE PUECH GUSTAVO LAGOS CRUZ-COKE Thesis submitted to the Office of Research and Graduate Studies in partial fulfillment of the requirements for the Degree of Master of Science in Engineering. Santiago de Chile, January 2017 c MMXVII, C ARLOS I GNACIO M ALDONADO R IVERA.

(3) To my friends and family.

(4) ACKNOWLEDGEMENTS. I would like to express my deepest gratitude to my family for their unconditional support. They have been there for me in all times, regardless situations were good or bad. Thank you for never losing the faith that I would be able to find my way, even in the darkest moments. I would like to thank my advisor Rolando Rebolledo, for guiding me in this long process. Especially, I would like to thank him for all the opportunities he has granted me, like traveling to congresses and workshops, and for all the fruitful discussions we had about science, but also for all those when we talked about vocational and life issues. I would also want to thank deeply to my friends, for being with me in this journey. I would especially like to mention Alfonso Lobos, Daniel Arancibia, Lorena Moraga and Sergio Rojas. They were always giving me strength and in several occasions they generously helped me in different topics related to my work. This research was supported by FONDECYT Project ACT#1112 and the CIRIC project.. iv.

(5) TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. iv. TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. v. LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. vii. LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi. Introduction and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.. 2.. 1.1. Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.2. State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.1. Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. Real waves as stochastic processes . . . . . . . . . . . . . . . . . . . . .. 8. 2.2.1. Sea waves potential . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.2.2. Linear sea waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.2.3. Integral representation of a monochromatic wave . . . . . . . . . . .. 10. 2.2.4. Stochastic sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.2.5. Fixed Structures responses . . . . . . . . . . . . . . . . . . . . . . .. 13. Stochastic processes with memory . . . . . . . . . . . . . . . . . . . . .. 13. Memory in the OWC . . . . . . . . . . . . . . . . . . . . . . . . . .. 14. The OWC dynamics and power extraction . . . . . . . . . . . . . . . . .. 16. 2.4.1. OWC dynamics for monochromatic waves . . . . . . . . . . . . . . .. 16. 2.4.2. Pressure chamber dynamic in real waves . . . . . . . . . . . . . . . .. 18. 2.3. 2.3.1 2.4. v.

(6) 2.4.3 3.. Wells turbine power extraction . . . . . . . . . . . . . . . . . . . . .. 20. Simulation of Extracted Power. Validation and Application to a Local Chilean Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. Simulation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.1.1. Numerical Scheme for the Euler Maruyama . . . . . . . . . . . . . .. 24. 3.1.2. Stability and convergence . . . . . . . . . . . . . . . . . . . . . . .. 25. Simulation validation . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. 3.2.1. Simulation of a OWC device in Monte-Redondo place . . . . . . . . .. 27. 3.2.2. Power production estimation for the OWC in Monte-Redondo place. .. 28. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 4.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35. 4.2. OWC Prospective and Future Work . . . . . . . . . . . . . . . . . . . . .. 35. REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 3.1. 3.2. 4.. vi.

(7) LIST OF FIGURES 1.1. Draw of LIMPET OWC chamber . . . . . . . . . . . . . . . . . . . . . . .. 4. 1.2. Classification of OWC. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.1. OWC chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2. fP (Ψ) Curve for Wells Turbine NACA 0015 . . . . . . . . . . . . . . . . .. 21. 3.1. Simulation Scheme Flow Chart . . . . . . . . . . . . . . . . . . . . . . . .. 22. 3.2. Γ(ν) and B(ν) functions for circular OWC . . . . . . . . . . . . . . . . . .. 29. 3.3. OWC simulated trajectories . . . . . . . . . . . . . . . . . . . . . . . . . .. 30. 3.4. Monte Redondo place sea state histogram . . . . . . . . . . . . . . . . . . .. 32. 3.5. Histogram of rectangular shaped OWC . . . . . . . . . . . . . . . . . . . .. 33. 3.6. Histogram of circular shaped OWC . . . . . . . . . . . . . . . . . . . . . .. 34. vii.

(8) LIST OF TABLES 3.1. Compared results for Falcã o chamber . . . . . . . . . . . . . . . . . . . .. 27. 3.2. OWC Simulation resumed results . . . . . . . . . . . . . . . . . . . . . . .. 30. 3.3. Simulation results for Monte Redondo OWC . . . . . . . . . . . . . . . . .. 31. viii.

(9) RESUMEN. Este trabajo se enfoca en el modelamiento matemático del dispositivo de columna de agua oscilante, OWC (Oscillating Water Column). Este dispositivo permite transformar energı́a de las olas a potencia eléctrica. Consiste de una cámara de aire parcialmente sumergida con apertura al mar en su parte inferior y provista de una turbina Wells en su aporte superior. Cuando la marea entra y sale provoca un cambio en la presión del aire al interior de la cámara generando un flujo de aire que mueve la turbina Wells produciendo potencia eléctrica. Aunque la estructura de un dispositivo OWC es bastante simple, la naturaleza estocástica de las olas del mar y las turbulencias internas hacen su modelamiento matemático una tarea desafiante. Esta tesis propone un modelo estocástico que considera ambas, la aleatoriedad del mar, ası́ como los efectos de memoria por las turbulencias al interior de la cámara. Se muestra, en particular, que la forma de la cámara juega un rol importante al definir el ruido que afecta el suministro de energı́a. Como resultado, se obtiene un modelo numérico y computacional que captura el efecto de diferentes formas de sección de cámara. Los resultados anteriores se muestran en una serie de simulaciones, comparando nuestra aproximación con la realizada por Falcão and Rodrigues (2002), quien no considera interacciones internas de la presión con la estructura y no alcanza un modelo estocástico adecuado.. Dos hechos principales motivaron esta investigación. Primero el enorme potencial de energı́a marina de nuestro, lo que determina la necesidad de innovar en esta área. Segundo, es auto evidente que las energı́as marinas por si solas no cubrirı́an nuestra demanda de potencia eléctrica. Por lo que es importante combinar las diferentes fuentes de energı́a. ix.

(10) renovable tales como solar, eólica y marina. Como resultado, un modelo matemático adecuado, deberı́a extenderse a dichas conexiones como redes inteligentes, ası́ como incorporar el proceso de almacenamiento de energı́a. Esta tesis presenta una primera contribución para dicho programa. Con este fin, un ejemplo concreto es mostrado. Más precisamente, en Monte Redondo, un lugar al norte de Chile, existe una granja eólica importante con 24 generadores que podrı́an producir hasta 48 MW. Con vista a su inclusión como red inteligente, se simuló la contribución que tendrı́a un dispositivo OWC en el lugar utilizando los estados de marea local.. x.

(11) ABSTRACT. This research focuses on the mathematical modelling of the so called Oscillating Water Column (OWC) device. This device allows the transformation of sea wave energy in electrical power supply. It consists of a partially submerged air chamber opened to the sea at the bottom and endowed with a Wells turbine at the top. As sea water enters and leaves the chamber it changes its inner air pressure which moves the turbine and produces electrical power. Despite the OWC structure being simple, the stochastic nature of the seawater and air turbulences makes its mathematical and engineering modelling a challenging task. This thesis proposes a stochastic model which considers both, the randomness of sea waves, as well as the memory effects of turbulences inside the chamber. It is shown, in particular, that the shape of the chamber section plays an important role in defining the noise affecting the power supply. As a result, a numerical and computational model capturing different shapes of chamber sections is obtained. The above results are illustrated by a number of computer simulations, comparing our approach with that of Falcão and Rodrigues (2002) who did not consider internal interaction of the pressure with the structure and did not reach a proper stochastic model. Two major facts motivated this research. Firstly, the enormous marine energy potential of our country which determines the need to innovate in this area. Secondly, it is self-evident that the marine energy alone would not cover our electric power demand. Thus, it is important to combine different renewable sources such as wind, solar and marine energies. As a result, a suitable mathematical modelling of those, should be extended to consider their smart grid connection as well as the corresponding storage processes. This thesis represents a first contribution of the author to this program. To this end, a concrete example is shown in the text. More precisely, in Monte Redondo, a location in the northern Chile, there is an important wind farm of 24 generators that could produce up. xi.

(12) to 48 MW. We considered the local data of sea waves to simulate the contribution of an OWC there, in view of its inclusion in a smart grid power plant.. Keywords: Stochastic Modelling, Stochastic Simulation, Gaussian Processes, Non Markovian Processes, Waves Spectral Density, Wave Energy Converters, Power Production xii.

(13) 1. INTRODUCTION AND OBJECTIVES For almost 40 years there has been active research on how to produce power from ocean waves. Wave energy converters (WEC) devices has been developed and studied during those years but hasn’t attracted much industrial interest. This situation has been changing in the last 15 years because of the increasing concern with climate change and clean energy interest. An Oscillating Water Column (OWC) is a power production device consisting of a rigid chamber partially submerged in which an air volume is trapped. Thanks to an opening in the bottom of the chamber, the sea waves produce a compression/expansion of the inner air mass creating a flow through a turbine at the top of the chamber. The power take-off mechanism usually consist of a bidirectional turbine, named Wells turbine, which has symmetric blades allowing the flow of the air in and out without changing the rotation sense. Chile is a great place for WEC devices. Its long coast, high energy currents and steeped ocean floor creates very good conditions for WEC devices. In particular, off-shore protective walls of ports are perfect places for OWC devices, and these devices can be a good choice for electric supply in small coastal towns. In a smart grid deployment, an OWC can diversified the power source, mixing it with solar, wind and energy storage. This diversified generation may create reliable and stable power source. The difficulty in the installation of a OWC device relies in cost estimation and the hazards related to the sea waves, on the other side the power generation is persistent with some short terms variability. This work aims to be a contribution to the study and industrial application of OWC devices which would be used in a smart grid project (CIRIC, Inria Chile). Smart grids can combine wind farms, solar panels and energy storage like batteries or water pumping. The particular study requires to estimate the energy production by means of knowing the. 1.

(14) probability distribution of the different power sources. In this context our contribution to the project is to estimate the energy production of a fixed structure OWC device with a Wells turbine for a given sea state distribution and setup parameters. This problem involves deciding the chamber design, in the sense of dimensions and shape, as wells as the Wells turbine design, size and operation. The production estimation implies to know the short term probability distribution of the power generation due to the sea variability and the optimal turbine operation. The first chapter presents the objectives of this work and a short literature review. The second chapter shows the theoretical background, in which this work is based, where the stochastic modelling of the water and chamber interaction is shown in the context of the OWC system dynamic. In the third chapter a simulation scheme is presented, validation of the scheme and an application to the case of a particular wind park deployed very close to the coastal side called Monte-Redondo with its own sea states, obtaining the expected power production for several chamber shapes and settings. 1.1 Objectives The main objective of this work is to obtain a theoretically better representation and formalization for the simulation of an OWC device, presenting a probabilistic model that is capable of simulating the OWC power extraction, in time and frequency, and test the performance for several given conditions. The proposed stochastic model exploit the Airy waves theory, using a spectral representation of waves and the known system response to represent processes in time. The sea is modeled as a Markovian Gaussian process using an integral representation in frequency, and the chamber pressure as a stochastic process with memory depending on the chamber geometry.. 2.

(15) The secondary objective is to obtain a good tool for a virtual laboratory, creating a scheme for the simulation of an OWC device over different settings, which allows explore better/optimal configurations. The third objective of this work is an initial application of our virtual laboratory to find how an OWC device would behave for the sea states associated to Monte Redondo, Coquimbo, Chile, and compare some chamber designs, together with simulating the power production considering its long term historic sea state probability distribution. To achieve this objective, two different chamber shapes where used, with several size variations, for this a particular turbine design was performed with optimized size and operating conditions. This work is intended to represent a tool for the economic valuation of an OWC plant and to improve the knowledge about these plant and their applicability. In particular, the results obtained aid in clearing the path for the construction of an OWC plant and some factors of the real operation behaviour. 1.2 State of the art General approaches for the modelling of WEC may be found in Newman (1974); Evans (1978); Li and Yu (2012); Evans and Porter (2012); Falnes and McIver (1985); Evans (1987, 1981, 1982); McCormick (2007); Falcão and Henriques (2014). An Oscillating Water Column (OWC) is a power production device consisting of a rigid chamber partially submerged in the sea, in which an air volume is trapped, see Figure1.1. Thanks to an opening in the bottom of the chamber, the water surface variation produce a compression/expansion of the air pocket creating an air flow through a turbine at the top of the chamber. The modelling of a OWC requires to model: • the incident wave field and its interaction with the structure, 3.

(16) F IGURE 1.1. LIMPET OWC chamber, Islay, Scotland. Source : Wavegen. • the elevation of the water in the chamber, • the air pocket dynamics, • the power take-off system. As shown in Figure 1.2 there are several types of OWC devices. For more information about these please refer to Heath (2011) OWC Fixed Sea-bottom-fixed. Floating Shoreline. Isolated. Backward Bent Duct Buoy. Spar buoy. Sloped Buoy. In breakwater F IGURE 1.2. Classification of OWC.. 4.

(17) Fixed OWC devices extract energy from the water column variation, and floating OWC extract energy from the structure buoyancy and the water column. There are several advantages of the OWC devices compared to other WEC’s. First, OWC devices have a low cost of testing/installation, high reliability (the latter because the use of air turbine eliminate gearboxes). Regarding to shoreline devices, they have low impact on marine media, a efficiently sea space use, can be integrated with shore structures, easy to maintain and few moving parts in the water Several variations can be made in shoreline fixed-structure OWC for electricity production, depending on the chamber shape • the most studied, rectangular shaped horizontal section, • Circular shaped, studied by Martins-Rivas and Mei (2009a,b) • With a vertical duct (U-OWC), developed by Boccotti (2007b,a); Boccotti et al. (2007); Malara and Arena (2013). Variations depending on the OWC chambers distribution • OWC Arrays, one turbine per chamber in a single frame Falcão (2002b), • Segmented OWC, one turbine per chamber and multiple chamber per frame Dorrell et al. (2010); Hsieh et al. (2012), • Modular OWC, many chambers with valves to produce unidirectional airflow Georgiou et al. (2012). Depending on the electricity extraction mechanism • wells turbine or bidirectional turbines, Darabi and Poriavali (2007); Gato et al. (1996); Dhanasekaran and Govardhan (2005); Brito-Melo et al. (2002), • unidirectional turbines Jayashankar:2009vh, • using dielectric elastomer Vertechy et al. (2013), • using hydraulic accumulator Bakar et al. (2011). 5.

(18) With the current state of the technology, OWC devices can reach 1MW of power production, this is observed in the installed and tested OWC. Some of these fixed-structure shoreline OWC are: • European Pilot Plant on the island of Pico in the Azores Sarmento et al. (2006); Neumann et al. (2006); Neumann and Le Crom (2011); Ano (2000); Rudd (2003); Neumann et al. (2008); Trust (2005). • Wavegen LIMPET (Land Installed Marine Powered Energy Transformer), located on Islay, Scotland. Power production of 75 kW .Whittaker et al. (1993); Heath et al. (2000); Folley and Whittaker (2002); Trust (2005); Delmonte et al. (2014) • Sakata port, Japan. Power production of 60 kW. Ohneda et al. (1991); Trust (2005) • Vizhinjam OWC, Trivandrum, India. Production of 125 kW. Ravindran et al. (1989); Ravindran and Koola (1991); Trust (2005) • Bergen, Norway. Power production of 350 kW and 500 kW. Bønke and Ambli (1986) • Kvaerner OWC, Norway. Falnes (2009) • Twin-Chamber OWC at the mouth of Douro river, northern Portugal. Power production of 750 kW. Martins et al. (2005) • Mutriku, Basque country, Spain. Production of 16×18.5 kW, aproximated power production of 300kW. Torre-Enciso et al. (2009) • REWEC3 in the strait of Messina, southern Italia. Shoreline U-OWC, Boccotti et al. (2007); Boccotti (2007a). 6.

(19) 2. THEORETICAL BACKGROUND 2.1 Framework The stochastic modeling of a shoreline fixed OWC, see Figure 2.1 will consider that the sea is a stochastic process η and its interaction, q, with the chamber, and a stochastic process for the inner chamber pressure p.. wv (t). w(t). p(t). η(x, t) q(t). F IGURE 2.1. OWC chamber with rigid boundary representing the structure, coastal breakwater and sea floor. p(t) represent the inner chamber pressure, w(t) and wv (t) represent the air mass flux through the Wells turbine and the valve, q(t) is the water flux in the oscillating water column and η is the water free surface. A deterministic generation of water waves will be considered and extended to stochastic waves. The sea modeling uses the Airy Waves Theory (See subsection 2.2) and short term sea states. The Airy waves theory considers irrotational, incompressible and inviscid fluid flow. The short term sea states are considered for the spectral representation of the sea. The assumption of fully developed sea is used with the Airy Waves Theory, however it’s not necessary when considering linear interaction with a structure and the use of local sea states. The theory considered was developed by Krée and Soize (1986). 7.

(20) In our study we will assume that the sea waves have a known spectrum corresponding to wind waves in fully developed sea. Any kind of gravity waves, or other waves generated by equilibrium’s restoration due to gravity, can be considered and its spectrum added. The water waves interact with the structure changing the water level inside the chamber. This oscillating water column will activate the chamber dynamics, changing the inner pressure and creating a mass flux of air through the turbine. The inner pressure will be modeled as a process, where the infinite spring effect will affect the inner chamber pressure. This process will assume no heat transfer, and an isentropic process. The air mass flow through a Wells turbine, as a power take-off mechanism, and a valve, bounding the pressure to the linear range of the turbine. The OWC device transform hydraulic power to electric power via a Wells turbine, where a linear behaviour for the extracted power versus the pressure is considered. For modelling purposes the difference between the inner chamber and the atmospheric pressure needs to remain in a given range, and therefore a perfect valve is considered, bounding the pressure and keeping it under the linear behaviour of the Wells turbine. With this, the extracted power is measured by a parameter function, which will define the power transform rate depending on the pressure. This function is the power extraction curve and assumed to be known. 2.2 Real waves as stochastic processes According to Krée and Soize (1986) the waves observed in the sea correspond to a propagation of the deformation made by the wind on the water free surface. The deformation is a random perturbation exerted by the wind in a finite area, transferring energy and a chaotic vertical displacement of the water. Outside this area the sea is considered as fully developed if energy balance is attained. In such conditions sea waves correspond necessarily to a stochastic process and this sea state is called short-term sea state.. 8.

(21) The ocean waves correspond to a wave propagation phenomena and it will be presented in next subsection. The non-linear equations for a fully developed sea is shown and then linearized for monochromatic waves. The monochromatic waves will be presented with an integral form and extended to stochastic waves. 2.2.1 Sea waves potential Considering η the water free surface in a laterally unbounded sea Ω, with water elements M (x, y, z) ∈ Ω located in x, y, z, SL = {M (x, y, z) : η(x, y, t) = z} the points in the water free surface. Supposing incompressible water, perfect fluid, irrotational motion, almost infinite sea depth D and ignoring thermal phenomena, the equation for the water velocity potential φ can be written as. ∂φ ∂z. 4φ = 0 = 0 z=−D. ∂φ 1 + ||Oφ||2 + gη = 0 ∂t 2 ∂η ∂φ ∂η ∂φ ∂η ∂φ + + − = 0 ∂t ∂x ∂x ∂y ∂y ∂z. ∀M ∈ Ω, ∀t. (2.1). ∀x, y, t. (2.2). ∀M ∈ SL , ∀t. (2.3). ∀M ∈ SL , ∀t.. (2.4). Where (2.1) is the continuity equation for incompressible fluid, (2.2) represent an impermeable sea floor, (2.3) comes from the Bernoulli equation for conservation of energy and (2.4) is the kinematic boundary condition for the elements in the free surface. See Krée and Soize (1986) for more details. 2.2.2 Linear sea waves The equations (2.3), (2.4) are not linear, however they can be linearized when η is close to 0. In this case we say we are in the infinitesimal harmonic plane wave related to the free surface elevation η = −g∂φ0 /∂t at z = 0 with amplitude a > 0, frequency ω/2π 9.

(22) and propagation direction θ. Thus, the new velocity potential is given by the function φ0 (M, t) = Re {ϕ0 (M )eiωt } solving 4φ0 = 0. ∂φ0 = 0 ∂z ∂ 2 φ0 ∂φ0 = 0 +g 2 ∂t ∂z. ∀M ∈ Ω0 , ∀t. (2.5). z = −D, ∀x, y, t. (2.6). z = 0, ∀x, y, t,. (2.7). where (2.7) correspond to the linearization of (2.3), (2.4) when η = 0 and then |∇φ| = O(η). The free surface elevation and velocity potential produced by (2.5) – (2.7) is given by D E cosh(kω (z + D)) ~ ω , X + ωt + α), cos( K ω cosh(kω (D)) D E ~ ω , X + ωt + α), η(x, y, t) = a sin( K φ0 (M, t) = ag. (2.8) (2.9). ~ ω,θ = kω (cos(θ), sin(θ)) with X = (x, y) the horizontal coordinates, α a phase shift, and K is in the Airy Surface (AS) defined by the kω following the dispersion relation ω 2 = gkω tanh(kω D). (2.10). This is a monochromatic wave solution and the idea is to explore its generalization 2.2.3 Integral representation of a monochromatic wave The monochromatic solution for the water velocity potential (2.9) and (2.8), can be ~ ω,θ , ω), a measure written using an integral representation. Defining Q(θ, ω) = (K mθ,ω = ~ ∈ R2 , then (2.8) is with K Z φ0 (~x, z, t) = −. 1 ~ ω 0 ) − δQ(θ,ω) (K, ~ ω0) , δ−Q(θ,ω) (K, 2. ~ (K,ω)∈R×T. ag. cosh(|K|(z + D)) i(hK,~ ~ xi+ωt) ~ ω), e mθ,ω (K, ~ ω cosh(|K|D). (2.11). (2.12). 10.

(23) and η = −g −1 ∂φ0 (X, 0, t)/∂t. The same analysis can be expanded to another measures, in particular in the next section a stochastic measure will be considered. 2.2.4 Stochastic sea For a given short term sea state of a fully developed sea and η(~x, t) being stationary with respect to ~x = (x, y) ∈ R2 , t ∈ R+ , centered Gaussian process of second order, almost surely continuous and with a power spectral measure with density S(ω) and finite spectral moments of order 4. Then φ and η will admit a integral representation as Z ~ cosh(|K|(z + D)) i(hK,~ ~ xi+νt) ~ ν) e Mη (K, (2.13) φ(~x, z, t) = ig ~ ~ ν cosh(|K|D) (K,ν)∈AS Z ~ ~ ν) η(~x, t) = ei(hK,~xi+νt) Mη (K, (2.14) ~ (K,ν)∈AS. n o ~ ν) ∈ AS, ν ≥ 0 . AS is symmetric with respect to the frequency ν = 0 then AS + := (K, The deterministic measure in (2.11), representing a monochromatic incident wave, is replaced by a stochastic spectral measure Mη , such that for every subset A, A0 ⊂ AS + . E[Mη (A), Mη (A0 )] = m(A ∩ A0 ). (2.15). where m is the power spectral measure of the process η(t). The measure M can be restricted to AS + and interpreted as Z p Mη (A) = S(ν, θ)dW,. (2.16). A. with W : R+ × [0, 2π] → R is a Brownian Sheet or W : R+ → R is a Wiener process, and the integral is in sense of Itô integral. This formula is very useful due the direct simulation method allowed by this representation.. 11.

(24) On the other side, the power spectral measure m, depending on M , is defined by Z m(A) = S(ν, θ)dνdθ (2.17) A. S(ν, θ) is the one-sided sea spectrum, where ν is the frequency and θ the incidence angle. It represents the available wave energy in A ∈ AS + . S(ν, θ)/2 is an even function in ν when considering ν ∈ R and called two-sided sea spectrum. Following Krée and Soize (1986) and Ochi (2005) the covariance of η(~x, ·) can be written as h i Rη,η (t, t + τ ) = E η(~x, t)η(~x, t + τ ) Z T 1 η(~x, t)η(~x, t + τ )dt = lim T ↑∞ 2T −T Z S(ν, θ) e−iντ = dνdθ 2 θ,ν∈AS. (2.18) (2.19) (2.20). where relation of (2.18) with (2.19) is due to the Ergodic Theorem, and (2.18) with (2.20) by the Wiener-Khintchine Theorem. If wave direction θ is ignored, i.e. AS + = R+ , then Z cos(νt)S(ν)dν. (2.21) Rη,η (t, t + τ ) = ν∈R+. The process η(~x, t) has mean wave energy given by the variance Rη,η (0, 0) Z Rη,η (0, 0) = S(ν, θ)dνdθ. (2.22). AS +. Wave spectrum can be different depending on the local sea conditions, however S(ν, θ) = S(ν)D(θ) has general characteristics which can be approximated by a directional PiersonMoskowitz spectrum (Falnes (2002)) with H2 S(ν) = 2 · 131.5 4 s 5 exp Te ν. . −1054 Te4 ν 4. ,. for ν ≥ 0. (2.23). 12.

(25) with an even extension to R as S(−ν) = S(ν) and has a maximum at the modal frequency ωm . Hs is the significant wave height, Te the energy period, D(θ) is a distribution function related with the incoming wave direction, these data is assumed to be known by experimental measures. An example of D(θ), given by Falnes (2002), is   2 cos2 (θ − θ0 ) for |θ − θ0 | < π/2 π D(θ) =  0 e.o.c.. (2.24). In further the sea will be assumed as in (2.14) with a Pierson-Moskowitz spectrum (2.23) and AS + = R+ . 2.2.5 Fixed Structures responses Following Krée and Soize (1986) and assuming linear water waves, the interaction of η with a structure, in particular the water flux through an area due to the wave diffraction of an incident monochromatic plane wave in the sea is represented by the frequency response Γ(ν). Using the linearity and the probabilistic description of η, the diffracted flux can be written as Z. Γ(ν)e−iνt dM. qd (t) =. (2.25). AS +. Assuming Γ(ν) to be positive an even function, the qd is a real Gaussian process and has energy given by the covariance function Rqd ,qd (0, 0) given by Z Rqd ,qd (0, 0) = S(ν)|Γ(ν)|2 dν. (2.26). AS +. 2.3 Stochastic processes with memory Complex interaction of environment requires dynamics including memory. In particular, Langevin processes with property of being Gaussian and non-Markovian may be. 13.

(26) useful in presence of memory. These has been studied by Lizama and Rebolledo (2010) in Langevin equations for classical mechanical systems embedded in a reservoir. In particular, in a mechanical system where a great number of harmonic oscillators collides with the main system, the dynamic can be characterized by a process Xt = (Qt , Pt ), with Q denoting position and P momentum of a particle. The moment variations are described by a memory kernel or random force assumed to be Gaussian, but not Markovian. The evolution equation of such Langevin equation used in Lizama and Rebolledo (2010) can be stated as: dQt = Pt dt. (2.27). dPt = −(∂q V (Qt ) + (v ∗ P• )(t))dt + dBv (t),. (2.28). where V is the potential energy, Bv is the Gaussian transform associated to the function v and can be represented by a stochastic integral in frequency Z ∞ Z ∞ ψ2 (ν, t)dW2 (ν), ψ1 (ν, t)dW1 (ν) + Bv (t) = 0. (2.29). 0. with s Z 2 ∞ f (x) = v(t)cos(νt)dt π 0 f (ν) (1 − cos(νt)) ν f (ν) ψ2 (ν, t) = sin(νt), ν ψ1 (ν, t) =. (2.30) (2.31) (2.32). where v(t) must be even with positive cosine transform 2.3.1 Memory in the OWC The influence of the random sea and its interaction with the chamber structure through the pressure, may generate noise in the OWC chamber. In particular, the radiated flux. 14.

(27) qr has an associated stochastic process directly related with the radiation conductance B(ν) = f 2 (ν) acting like infinite harmonics oscillators exerting a noise force bh (ν, t) on the air mass [ν/h] h. b (ν, t) =. X√. hf (jh)(cos(jht)ξj + sin(jht)ζj ),. (2.33). j=1. with ξj , ζj independent and identically distributed (i.i.d.) standard normal random variables . The integral of (2.33) is Bh (ν, t) and represent the effect of the noise in the system X f (jh) √ [ν/h] B (ν, t) = h (sin(jht)ξj + (1 − cos(jht))ζj ) jh j=1 h. (2.34). According to Lizama and Rebolledo (2013, 2010), in the limit, when ν → ∞ and h → 0, 2.34 defines a Gaussian process with memory Bgr Z ∞p B(ν) Bgr (t) = (1 − cos(νt)) dW1 (ν) ν 0 Z ∞p B(ν) sin(νt)dW2 (ν) + ν 0 The process Bgr is centered with memory and has covariance Z ∞ B(r) KBgr (s, t) = (1 − cos(rt))(1 − cos(rs))dr r2 0 Z ∞ B(r) + sin(rt) sin(rs)dr, r2 0. (2.35). (2.36). then the radiated flux qr can be written as dqr (t) = gr ∗. ∂p dt + dBgr (t), ∂t. (2.37). in that way qr (t) = gr ∗ p(t) + Bgr (t).. (2.38). Further details are explained in the next section and correspond to the OWC dynamics.. 15.

(28) 2.4 The OWC dynamics and power extraction 2.4.1 OWC dynamics for monochromatic waves The next analysis was made formerly by Evans (1976), Falnes (2002) and others, under the assumptions of linear water-waves theory. An Oscillating Water Column (OWC) consists of a rigid chamber partially submerged in which an air pocket is trapped. Thanks to an opening in the bottom of the chamber, the inner water surface variation produces a compression/expansion of the inner air mass flowing through a Wells turbine at the top of the chamber. Considering a shoreline bottom fixed OWC device with an internal water free surface Si , external water free surface SF , rigid boundary SB , water free surface elevation η(x, y, t), atmospheric pressure as 0 point for the inner pressure p(t), the water velocity potential Φ(x, y, z, t) equation for the OWC device is given by ∇2 Φ = 0 in the fluid,    −p(t) on Si , ∂Φ ρ = gη +  ∂t  0 on SF ∂ ∂η = Φ(x, y, 0, t) ∂t ∂z ∂Φ = 0 on rigid boundaries , SB , ∂n. (2.39) (2.40). (2.41) (2.42). where equation (2.40) is the Bernoulli energy balance equation relating the pressure inside the OWC chamber with the water potential, giving origin to the flux radiated by the pressure to the water. Equation (2.39) is the continuity equation for incompressible liquids, (2.42) represent the impermeable layer of the sea bottom and coast bounding the water, (2.41) the vertical water velocity on z = 0 matches the free surface elevation variation in time and k satisfies the dispersion relation (2.10).. 16.

(29) It is convenient to separate the water velocity potential by Φ = Φ 0 + Φd + Φr ,. (2.43). where Φd denotes the scattered water in absence of inner pressure, i.e when p = 0, Φr is a radiated potential due to the action of p(t) over the water, and Φ0 the incident potential satisfying (2.5)–(2.6) (i.e. without rigid boundaries or p). Now, consider the water flux inside the chamber separated in two, q(t) = qd (t)+qr (t). The diffracted flux qd (t) is generated by the incident and diffracted potential Φ0 + Φd . The radiated flux qr is generated by the refraction of the pressure represented in the potential Φr . The water flux q(t) is given by Z ∂η dS q(t) := Si ∂t Z ∂Φ dS = Si ∂z Z Z ∂ ∂Φr = (Φ0 + Φd )dS + dS Si ∂z Si ∂z = qd (t) + qr (t).. (2.44) (2.45) (2.46) (2.47) (2.48). The inner water volume variation q(t) induce an air mass flux dm/dt through the turbine inside the chamber, following the equation dm d(ρa V ) = dt dt V0 ρa dp = + ρa q(t), γpa dt. (2.49) (2.50). where following Falcão and Justino (1999), a linear isentropic relation was assumed, with V0 being the equilibrium air chamber volume, ρa the air mass density, and γ the isentropic expansion factor of air.. 17.

(30) Now, assuming linear water waves and monochromatic incident waves, time independent quantities are convenient to be defined as harmonic oscillators η, p, q, qd , qr = Re (η̂, p̂, q̂, qˆd , qˆr )e−iωt. (2.51). With this, a convenient decomposition, by Evans (1982), can be made relating (2.43) to (2.50) dp + Bp, dt. (2.52). qˆr = (B + iωC)p̂,. (2.53). qr = C In that way. with B real and even and C real. qr will depend on the mass flux rate through the valve an wells turbine. However qd depends only on the incident wave interacting with the fixed structure without pressure variation, then it can be written as: qd = Re η̂Γ(ω)e−iωt. (2.54). The function Γ(ω) , B(ω) and C(ω) are frequency responses and can be known by analytical models as in Martins-Rivas and Mei (2009a) or Falcão and Justino (1999); with numerical testing in equations (2.39)–(2.42), obtaining Γ by testing with p̂ = 0, η̂ = 1 for given ω, and B, C by setting p̂ = 1, η̂ = 0; or with empirical measurements. 2.4.2 Pressure chamber dynamic in real waves Considering an oscillating water column wave energy converter fixed with respect to the sea bottom and coast, the motion of the water free-surface inside the device will produce an oscillating pressure inside the air chamber, generating a mass flow of air leaving the chamber through a turbine and/or a valve. To represent this we have:. 18.

(31) • the oscillating pressure equation w(t) = ρa q(t) −. V0 ρa dp , γpa dt. (2.55). where w is the air mass flux, q the water flux, p the inner pressure assumed uniform inside the chamber. The linear water wave theory and a linear isentropic relationship is assumed, where V0 is the initial OWC air chamber volume, ρa is the air density, pa the atmospheric pressure, and γ is the isentropic expansion factor. This equation was presented by Falcão and Justino (1999). • Water flux: defined in (2.44), can be separated in the sum of the diffracted and radiated flux q(t) = qd (t) + qr (t), with qd (t) the diffracted water flux due to the sea waves interaction with the structure at constant atmospheric pressure, and qr (t) the radiated flux due to the air pocket pressure variation interaction with the water free-surface. The flux qd has been defined in (2.25) and depends on Γ(ν) and the sea state. Following Falcão and Justino (1999) the radiated flux qr can be written as: Z t qr (t) = gr (t − τ )p(τ )dτ + Bgr (t) (2.56) −∞. , assuming B(ω) an even function, gr is defined as the cosine transform of B(ω) Z 2 ∞ B(ν) cos(νt)dν (2.57) gr (t) = π 0 Depending only on the chamber geometry. Bgr (t) is defined in (2.35). The functions Γ(ν) and B(ν) are supposed to be known. • Mass flux: the chamber air mass variation dm/dt = w(t) is given by w(t) = wD (t) + wv (t) =. d (ρa V ) , dt. (2.58). 19.

(32) With wD the mass flux through the turbine and wv the mass flux through a valve. For a Wells turbine a linear relation can be supposed between the pressure p and the air flux wD , then for a given Wells turbine geometry wD (t) =. KD p(t), N. (2.59). with D the rotator diameter, K a design factor depending only on the turbine geometry (Falcão and Justino (1999)), N the rotational speed. The valve will bound the pressure keeping the turbine working in a optimal range of pressure, a critical pressure pcrit is defined such that |p| ≤ pcrit , then   − KD p − d(ρa V ) if |p| >= pcrit N dt . wv (p) =  0 e.o.c.. (2.60). Hence, the valve will lower the pressure and keep it under the critical value pc rit. This will depend of the Wells turbine power curve fp defined in subsection 2.4.3. Reordering (2.55) and replacing terms (2.56), (2.58), (2.59), (2.25)), the main equation for the pressure p(t) is V0 p(t)+ γpa. Z. t. KD p(s) − gr ? p(s)ds 0 ρa N Z t V0 1 = p(0) + qd (s) + Bgr (s) − wv (s)ds γpa ρa 0. (2.61). A unique and weak solution will exist for p(t). It can easily be demonstrated, first without considering wv , the solution of the system is Gaussian process with memory and has first and second finite spectral moments, then by adding wv the process will be bounded. 2.4.3 Wells turbine power extraction In the considered OWC, the power is extracted using a Wells turbine. Power extraction curves are known from turbo-machinery studies. For these purpose the non-dimensional. 20.

(33) Π, adimentional power. 0.0025. Wells turbine fp for NACA0015. 0.0020 0.0015 0.0010 0.0005 0.0000 −0.0005 0.00. 0.02 0.04 0.06 0.08 Ψ, adimentional pressure. 0.10. F IGURE 2.2. fP (Ψ) Curve for NACA 0015 Model of Wells Turbine,Falcão and Justino (1999). coefficients for power Π and pressure Ψ are defined as Pt , ρa N 3 D 5 |p| , Ψ = ρa N 2 D 2 Π =. (2.62) (2.63). and the power function relating Π and Ψ is given by Π = fP (Ψ),. (2.64). this function is a parameter for the model. Then by knowing the pressure inside the chamber, the power output can be obtained. In the Figure 2.2 the extraction curve of a Wells turbine is shown. An approximately linear behaviour can be seen before the power maximum Πmax , after this the power extraction decrease fast. The inflection point Πmax will define the critical pressure Ψcrit , and the pressure range |Ψ(t)| < Ψcrit used in (2.60). 21.

(34) F IGURE 3.1. Simulation Scheme Flow Chart. 3. SIMULATION OF EXTRACTED POWER. VALIDATION AND APPLICATION TO A LOCAL CHILEAN CASE We consider a general simulation scheme which can be used in different chamber shapes and sea states distribution using the spectral representation of different involved processes, the objective is to develop a tool to create a virtual laboratory 3.1 Simulation Scheme The simulation scheme is based in known coefficients for the chamber shape, sea spectrum and turbine characteristics. Where the Airy waves theory assumption is fundamental, the air chamber dynamic following (2.55), the valve dynamic (2.60) and Well turbine air flow properties (2.59). 22.

(35) The chamber shape configuration will define the functions of radiation coefficient B(ν) and the diffraction coefficient Γ(ν). The sea model and data will define the spectrum S(ν) or S(θ, ν) , the shape of S must be selected according to the best fitting with the local measured sea spectrum, usually depending on the significant wave height Hs and wave modal period Tm , with this the stochastic measure M and the process η will be completely defined. The Wells turbine characteristic will define the air flux through the turbine wD , (2.59), and the power extraction curve Π = fP (Ψ) , (see Figure2.2 and equation (2.63)). Then for a given input of diffracted flux, radiation coefficient , geometric noise, valve dynamic and turbine mass flux, the radiated flux qr (t) and the pressure p(t) can be obtained. Finally the power extraction is obtained using the power curve fp (Ψ) , with Ψ as in (2.63). The chamber dynamic, Diffracted Flux and the Geometric Noise where simulated using a Euler-Maruyama scheme for the simulation of stochastic processes, following the Figure 3.1 The step by step for the simulation scheme is as follows 1. Obtain the function Γ(ν), B(ν). 2. Define the used sea spectrum S(ν) and obtain the diffracted flux qd by (2.25). 3. Calculate gr as in (2.57), and Bgr with (2.35). 4. Define the turbine design and operation with K, D, N , the power function fp , and pcrit . 5. Calculate p(t) and qr (t) following the numerical scheme presented in section 3.1.1. 6. Get the power production using fp. 23.

(36) 3.1.1 Numerical Scheme for the Euler Maruyama The equation (2.61) can be rewritten as KD V0 p(t)dt + wv dt = − 2 dp + ρa (qr + qd )dt N ca. (3.1). and will be discretized in time with a forward Euler-Maruyama scheme. Consider a time step ∆t and ti = i∆t KD V0 pn ∆t = − 2 (pn+1 − pn ) + ρa (qr (tn ) + qd (tn ))∆t, N ca. (3.2). rearranging pn+1. c2a KD c2a )pn + ∆tρa (qr,n + qd,n ). = (1 − ∆t N V0 V0. (3.3). Conveniently defining C1, C2 > 0 we get pn+1 = (1 − ∆tC1 )pn + ∆tC2 (qr,n + qd,n ).. (3.4). The term qr,n is calculated in the next way qr,n =. n X. pn−i gr (ti )∆t + Bgr (tn ).. (3.5). i=1. wv was included bounding the pressure as: for each n do pn+1 = (1 − C1 )pn + C2 ∆t(qr,n + qd,n ) if |pn+1 | > pc rit then pn+1 = pn. The presented qd and Bgr (tn ) are exact. However for the simulation qd and Bgr are replaced with approximated processes. Bgr is approximated in (2.34). qd is approximated. 24.

(37) with Riemann sums in frequency up to a big enough frequency Ω [Ω/h]. qdh (t). X. =. j=0. Γ(jh) cos(jht) [Mj+1 − Mj ] .. (3.6). 3.1.2 Stability and convergence Defining pn = (p0 , p1 , p2 , . . . , pn ) to be the pressure solution from time t0 to tn and its Euclidean norm ||pn ||, and assuming gr , S ∈ L2 ∩ L1 A sufficient condition for the system stability is ||pn || ≤ K||p0 ||,. ∀n ≤ N.. (3.7). If K depends on the discretization it will be called conditionally stable, which is the case of our equation, then ||pn+1 || ≤ ||pn |||1 − C1 ∆t| + ||pn ||||gr ||C2 ∆t + ||qdn ||∆t. (3.8). ≤ ||pn ||(|1 − C1 ∆t| + ||grn ||C2 ∆t) + ||qdn ||C2 ∆t. (3.9). ≤ ||pn ||Kgr + ||qdn ||C2 ∆t. (3.10). ≤ ||p0 ||Kgnr + ∆t. (3.11). ≤ ||p. 0. ||Kgnr. n−1 X i=0. + ∆tKqd. ||qd,i ||Kgn−i r n−1 X. Kgir ,. (3.12). i=0. With Kqd = α maxω S(ω) and Kgr = |1 − C1 ∆t| + ||grn |||C2 |∆t, α > 0 is a constant and can be ||S(ω)||L2 . The system will be conditionally stable if Kgr ∆tKqd. n X i=0. ≤. 1. (3.13). Kgir −−−→ 0.. (3.14). ∆t→0. 25.

(38) If condition 3.13 is assumed, the second became Kqd. n X i=0. Kgir −−−→ Kqd ∆t→0. 1 . 1 − Kgr. (3.15). Then 3.13 is sufficient to grant conditional stability, i.e. Kgr = |1 − C1 ∆t| + ||gr,n |||C2 |∆t ≤ 1 ∀n ≤ T /∆t.. (3.16). Note that only the stability in the sense of Lax and Rychmeyer or deterministic stability, is needed in SPDE, Lang (2010). With this result, the mean square convergence is granted √ for an SPDE like 3.1, with a rate of ∆t In the practice an easier expression was considered to reach stability, ∆t '. 1 C1 + C2. (3.17). 3.2 Simulation validation The validation of the simulation scheme was made comparing the results obtained by Falcão (2002a) for the OWC located at Pico, Azores in Portugal. The considered chamber dynamic is (2.55), the diffracted water flux qd defined by (2.25), the radiated water flux qr by (2.38) relating the pressure p(t) and geometric noise Bgr given in (2.35), and a valve dynamic given in (2.60) This chamber has a rectangular horizontal section with diffraction coefficient 2νb sin(Ka), K. (3.18). νbµ sin2 (Ka), ρw gK. (3.19). Γ(ν) = and radiation coefficient B(ν) =. where µ = (1 + ν 2 hcsch2 (Kh)/g)−1 and K following the dispersion relation. This considers a two-dimentional geometry with rectangular section of a × b with a = b = 12 m 26.

(39) TABLE 3.1. Results for the chamber proposed by Falcão (2002a). Simulated sample: 4096 Sea state j Hs (m) Te (s) E(kW). 1. 2. 3. 4. 5. 6. 7. 8. 9. 0.8. 1.2. 1.6. 2. 2.4. 2.9. 3.4. 4. 4.5. 9. 9.5. 10. 10.5. 11. 11.5. 12. 12.5. 13. 33.17. 76.41 138.64 220.52 322.57 477.53 664.49 929.81 1188.35. NF (rad/s). 75.3. 95.4. 112.8. 128. 141.8. 157.5. 165.2. 165.2. 165.2. Falcao P t (kW). 14.1. 30.2. 51.1. 76.1. 105. 145.8. 190.3. 234.4. 260.9. 13.28. 28.04. 47.27. 69.38. 95.76 132.07 175.95 230.04. 269.07. 4.48. 9.15. 15.19. 21.6. Simulated E(P t )(kW) σPt (kW) Tsim (s) E(Ef f )(%). 55.62. 58.79. 100.03 105.59 111.15 116.71 122.26 127.82 133.38 138.94. 144.49. 40.05. 36.73. 34.21. 31.47. 30.01 29.65. 41.11 27.65. 50.72 26.56. 24.77. 22.73. , mean water depth h = 8 m, initial volume V0 = 1050 m3 , isentropic expansion factor γ = 1.4, air density ρa = 1.25 kg/m3 , water density ρw = 1025 kg/m3 , g the gravity constant and K following the dispersion relation 2.10 . The Sea spectrum model is a Pierson Moskowitz spectrum (2.23) not including directional component, and for a given sea state distribution of significant wave height Hs and energy period Te . The considered Wells turbine is NACA0015 with diameter D = 2.3 m with power extraction curve fp (Ψ) obtained from Falcão and Justino (1999). The results are given in Table 3.1,with similar in results. Is important to note that the difference between results may be attributed to the different methodology and power curve fp . 3.2.1 Simulation of a OWC device in Monte-Redondo place One of the objective of this work is to show an OWC device would behave for the sea states conditions of Monte Redondo place and compare some chamber design. For that sea state data from Explorador Marino was used and a rectangular shape was compared with several circular configuration of the OWC device.. 27.

(40) The simulated sea states where extracted from 10 years of hourly data from the coast aside Monte Redondo wind farm. These data was processed through a cluster analysis method based on density and neighborhood grouping, selecting 17 points for simulation. The sea states histogram for (Tm , Hs ) is shown in Figure 3.4, where the selected 17 sea states are shown. The used sea states data is publicly available thanks to the Geophysics department of the Universidad de Chile in an online software Explorador Marino. The study of different chamber shapes requires the use of different frequency response functions. Simulation of the rectangular shaped OWC uses the diffraction coefficient (3.18) and radiation coefficient (3.19) obtained from Falcão and Justino (1999). The circular shape requires another parameters functions Γ(ν) and B(ν), obtained with a methodology presented in Martins-Rivas and Mei (2009a), the obtained diffraction and radiation coefficients are shown in Figure 3.2. 3.2.2 Power production estimation for the OWC in Monte-Redondo place To obtain a power estimation is necessary to choose the device design, there are different level to analyze. First the rotor speed, secondly the turbine diameter, and finally the chamber shape and size. The rotor speed can be optimized in an operational stage depending on the instantaneous sea stage. A turbine can be changed in mid-term, then considered a strategic decision. The chamber shape and size is definitive and a long term decision. No economic valuation was made, following only a criteria to obtain the maximum power output for a given sea states distribution, in particular for Monte Redondo for two different shapes. The optimization was implemented with a subgradient scheme searching for local maximum of the turbine power extraction, then for each chamber shape the turbine rotor diameter is tested, for each turbine diameter D is needed to optimize the rotor speed at each sea state. In each step is needed to apply the simulation scheme given in Figure 3.1. 28.

(41) Parameter functions for circular OWC 4. ×102 Diffraction coef. Γ(ν). 0.5. ×10−2 Radiation coef. B(ν). 0.0 3 −0.5 −1.0. 1 0 −1 −2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Frequency ν. B(ν). Γ(ν). 2. −1.5 −2.0 −2.5 −3.0 −3.5 −4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Frequency ν. F IGURE 3.2. Diffraction Γ(ν) and radiation B(ν) coefficients for the simulated circular section OWC. The continuous line correspond to radio dimension a = 7.6 m, the point-dashed line to a = 6.77 m and the dashed line to a = 6.0 m. These results were obtained by solving the equations presented by Martins-Rivas and Mei (2009a). The obtained results, shown in Table 3.3, outlines how behaves an OWC device with circular shape with respect to a rectangular shape. A simulated trajectory for the pressure p(t), η, and Pt can be seen in Figure 3.3. The simulated rectangular OWC has dimensions a = b = 12 m ( a is the coastal length, b is the length of the projection over the water ), h = 8.0 m, V0 = 1050 m3 , the comparison was made taking different dimensions for the circular section, fixing h = 8m and V0 = 1050m3 and considering a submerged wall d = h/5 , the first experiment considered equal chamber perimeter as the rectangular section, so that a circular section with diameter a = 12/π m ; the second experiment was √ made considering equal chamber section area, with a = 12/ π = 6.7 m ; finally equal coastal length was considered, then a diameter a = 6.0 m was considered. 29.

(42) TABLE 3.2. Resumed results for the OWC simulations presented in Table 3.3. shape section size (m) E[Pt ](kW ) σPt (kW) Efficiency % rectangular a/2=6.0 216.77 98.1 40.87 % a=7.6 152.07 66.38 24.42 % circular a=6.77 98.53 56.46 17.89 % a=6.0 68.06 31.32 14.29 % Simulated trajectory for a rectangular OWC. Hs = 5.73 m , Tm = 8.67 s 2 η incoming wave. m. 1 0 −1. −2 ×103 3. Pt power. kW. 2 1 0 −1 ×101 2. p(t) pressure. kPa. 1 0 −1 −2. 0. 10. 20. 30 Time (s). 40. 50. 60. F IGURE 3.3. Simulated trajectories for an OWC. The incoming waves η (m), extracted power Pt (kW), and pressure p(t) (kPa) are shown.. The results show that the rectangular chamber is the most effective in extracting wave energy from Monte-Redondo, as can be seen in Table3.2 The probability distribution can be found by means of histograms. The simulated histogram for rectangular shaped OWC is shown in Figure 3.5, and for the circular section OWC in Figure 3.6. Two selected sea states are shown.. 30.

(43) 31. 65.9. 45.6. D = 4.51 m. 41.1. 12.0. 24.0. 52.5. D = 4.214 m σPt (kW). a = 6.0 m. 17.6. D = 4.65 m 70.7. σPt (kW). a = 6.77 m. NF (rad/s) 71.4 E P t (kW) 98.8. 79.5. Circular. 23.3 67.3. 27.9. 7.53. 2.66. 68.5. 39.3. 7.58. 3.15 8.24. 3.31. 73.1. 74.2. 132.0 45.3. 7.80. 5.73. 75.0. 26.3. 8.41. 2.51. 10. 44.4. 97.5. 80.4. 69.4. 76.9. 72.2. 8.57. 4.14. 11. 77.3. 45.9. 8.79. 3.28. 12. 88.8. 15.2. 9.29. 1.86. 13. 89.6. 41.2. 9.84. 3.03. 14. 90.0. 79.2. 9.89. 4.20. 15. 17 2.05. 95.0. 95.0. 106.1 20.6. 12.31 13.24. 4.69. 16. 78.9. 53.1 64.4. 54.0 66.8. 61.3 65.6. 69.9. 78.9. 74.6. 70.2. 85.9. 63.1. 85.9. 81.7. 62.1. 122.4 53.9. 67.4. 37.8. 70.2. 26.3. 176.8 58.0. 89.9. 98.7. 24.2. 60.0. 64.1. 38.2. 88.8. 63.0. 56.9. 81.7. 77.9. 128.1 67.0. 59.2. 41.2. 33.3. 83.5. 71.6. 61.5. 61.4 78.4. 23.5. 248.8 72.0. 89.9. 108.0 52.9. 14.3. 39.0. 54.2. 26.6. 113.7 360.4 106.2 59.7. 81.7. 64.4. 66.6. 94.0. 45.0. 55.8. 54.9. 64.3. 29.3. 34.4. 22.1. 103.7 60.2. 71.5. 60.3. 162.7 90.5. 73.6. 93.2. 5.4. 14.8. 40.4. 9.6. 23.6. 42.0. 15.4. 13.5. 35.6. 57.4. 24.7. 55.5. 61.8. 37.5. 93.5. 62.6. 63.1. 34.8. 64.7. 37.1. 27.5. 67.0. 76.0. 39.6. 12.2. 40.9. 56.2. 25.2. 113.0 67.3. 66.4. 66.0. 1.8. 4.5. 32.8. 2.7. 7.8. 29.8. 5.2. 181.3 119.1 14.6. 73.5. 143.9 141.8 155.3 210.3 159.8 246.6 145.7 38.5. 78.9. 90.7. 122.6 128.8 145.1 152.9 187.3 202.3 207.2 225.6 230.7 364.3 368.2 381.1 660.9 424.2. 65.0. 138.9 257.0 84.6. 88.8. 24.6. Circular. NF (rad/s) 78.3 49.7 E P t (kW) 131.1 34.1. 82.2. σPt (kW). a = 7.6 m. 70.0 63.8. 7.14. 2.48. 9. 198.7 345.8 119.5 128.7 175.2 475.9 170.5 91.9. 75.6. Circular. NF (rad/s) 71.2 50.8 E P t (kW) 175.8 48.3. 65.1. 54.6. 37.4. 26.4. 21.3. 53.4. 9.4. 47.7 28.0. D = 2.3 m. σPt (kW). NF (rad/s) a = b = 12 m E P t (kW). Rectangular. 7.14. 4.29. 8. 0.051 0.073 0.048 0.067 0.046 0.059 0.067 0.009 0.066 0.093 0.089 0.053 0.056 0.087 0.071 0.040 0.026. Esea (kW/m) 28.0. 7.03. 3.15. 7. P(Hs , Te ). 6.92. 1.59. 6. 6.26. 5. 2.85. 4. Te. 3. Hs. 2. 1. Sea State. TABLE 3.3. Simulation results for rectangular and circular shaped OWC placed in Monte Redondo. The first four rows shows the sea state data, significant wave height Hs , energy period Te = 0.8997Tm , the sea state probability of occurrence P(Hs , Te ) and the available waves power per meter of coast, Esea .Then, results for different simulated chamber settings are shown, where N is the optimal rotational speed of the Wells turbine, E P t is the expected value of the time mean extracted power, finally the standard deviation of the power production. The first setting correspond to rectangular shaped OWC has dimensions a = b = 12 m, h = 8 m, V0 = 1050 m3 . Then the results for circular shaped OWC with common dimensions h = 8 m, V0 = 1050 m3 , d = h/5, studying three cases for the radio of the section a = 7.6 m, a = 6.77 m and a = 6.0 m. For each case 1024 samples where obtained..

(44) 0.045. 6. 0.040 5. 0.035 0.030. 4. 0.025 0.020. 3. 0.015 0.010. 2. Probability density %. Significant Wave Height, Hs (m). Monte Redondo place, 10 years sea states histogram. 0.005 1. 0.000 8. 10 12 14 Mean Wave Period, Tm (s). 16. F IGURE 3.4. Monte Redondo sea states for Tm (s) and Hs (m) histogram of years 2000 to 2010 of hourly data and 17 selected points for simulation. Data obtained from Explorador Marino. 32.

(45) Histogram of extracted power by a rectangular OWC , a = 12.0, 1024 samples Hs = 1.86 m , Tm = 10.32 s Expected PowerP = 65.65 kW. 0.06. Hs = 5.73 m , Tm = 8.67 s Expected Power P = 657.24 kW. Frequency %. 0.05 0.04 0.03 0.02 0.01 0.00. 0. 50 100 150 200 Extracted power (kW). 250. 0. 200 400 600 800 1000 1200 1400 Extracted power (kW). F IGURE 3.5. Histogram of OWC power production simulation for two Monte Redondo sea states. The OWC chamber shape is rectangular with dimension a = 12 m. 1024 samples of the process simulated during 4.4Tm s. In the upper side of the plot sea states parameters are shown as the expected power production.. 33.

(46) Histogram of extracted power by a circular OWC , a = 7.6, 1024 samples. 0.06. Hs = 1.86 m , Tm = 10.32 s Expected PowerP = 37.71 kW. Hs = 5.73 m , Tm = 8.67 s Expected Power P = 469.25 kW. Frequency %. 0.05 0.04 0.03 0.02 0.01 0.00 −20. 0. 20 40 60 80 Extracted power (kW). 100. 0 100 200 300 400 500 600 700 800 900 Extracted power (kW). F IGURE 3.6. Histogram of OWC power production simulation for two Monte Redondo sea states. The OWC chamber shape is circular with radio dimension a = 7.6 m. 1024 samples of the process simulated during 4.4Tm s. In the upper side of the plot sea states parameters are shown as the expected power production.. 34.

(47) 4. CONCLUSIONS AND OUTLOOK 4.1 Conclusions This thesis obtained a stochastic model for OWC sea wave energy converter for power production. The model considers both, the randomness of sea waves as well as turbulences inside the chamber. Mathematically, the above considerations are expressed via stochastic differential equations driven by non Markovian noise. This is a new approach to the dynamics since previous studies did not consider simultaneously both types of randomness. A numerical scheme followed the above model construction. Software routines were developed which allow to simulate the power production of OWC’s for different chamber configurations. These computational codes constitute an OWC virtual laboratory. The simulations have been tested against previous results obtained by Falcão et al. An analogical OWC simulator in a wave tank laboratory should be constructed to better validate the proposed numerical model. Simulations using real data from Monte Redondo place were made for several chamber configurations. Using its sea state distribution, and considering perpendicular incoming waves, the power production was obtained. The simulations have shown that a rectangular shaped OWC devices performs better than a circular OWC. The studied rectangular shape has an expected power production of 216.77kW, or an energy of 1.9GWh per year of energy with one OWC. 4.2 OWC Prospective and Future Work The study of an OWC in a smart grid scheme may be of great interest for industry applications. In general OWC devices generate less than 1MW rated power production. The small demand in transmission due to the OWC allows the use of the installed capacity of other renewable power sources. In this way OWC’s installed in a smart grid can. 35.

(48) help diversify the power matrix, reducing volatility and increasing the system generation reliability. In view of designing a smart grid that includes solar, wind and marine energy, the sea of Monte Redondo was considered. This location has a wind farm of 24 generators, each producing up to 2 MW. The installation of the studied OWC, the expected power production per installed unit should be 216.77kW. Interconnection between a wind mill and an OWC chamber is suitable for this small amount of power, diversifying the power matrix and reducing the idle capacity of installed transmission. The installation of single OWC devices is suitable for small coastal towns. The studied OWC can supply 155.5 MWh per month, enough energy for a small town with more than 100 houses, each with a mean consumption of 180 kWh per month, and their commercial or industrial activities. The studied OWC configurations were not exhaustive. The study of other shapes is needed in a more comprehensive analysis. An interesting different OWC design is the U-OWC proposed by Boccotti. The study of new shapes requires the numerical implementation of the frequency response functions Γ and B. The OWC simulation scheme used as a virtual laboratory can help testing different settings in the search for a good/optimal choice. However solving a complete optimization problem for the OWC may require the use of high performance computing (HPC) techniques. An implementation using HPC in C++ or FORTRAN are sugested as future work.. 36.

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