Mathematical modelling and analysis in marine renewable energy · 2012-05-08 · BCAM Seminar...

23rd April 2012

Mathematical modelling and

analysis in marine

renewable energy

Pierpaolo Ricci

Bilbao – 23rd April 2012 – Pierpaolo Ricci

INDEX

About Tecnalia

Introduction: The state of Marine Renewables

Marine Energy technologies: a systemic approach

Resource

Hydrodynamics

Dynamic simulation tools

Integration of Energy Conversion and Moorings

Other aspects

Conclusions

Bilbao – 23rd April 2012 – Pierpaolo RicciBCAM Seminar

ABOUT TECNALIA

Tecnalia is a private technological corporation that arose from the Basque business environment and has become an International benchmark in its field.

We all share a common operating model based on sector-focused Business Units

Progress and well-being for all, generating value through scientific and technological research.


ABOUT TECNALIA

Efficient and Sustainable Energy Carriers

Ocean Energy

Wind Energy

Solar Energy

Biomass

The electricity Network of the Future

Hydrogen and Fuel Cells

Energy Storage

Proximity to the Market

Renewable Energy Sources

Tecnalia-Energía: R&D lines


ABOUT TECNALIA

New concepts for high power electric generators.

Reliable power converters based on advanced topologies and new devices.

Structural materials for blades.

Multipurpose offshore renewable platforms: Hydrodynamics, structural and mooring designs.

Offshore Wind Wave Energy

PTO systems: Hydraulic equipment and linear generators.

Design of moorings systems and umbilical cables.

Power converters and control systems

Dynamic models for performance assessment and grid integration

Environmental Impact Studies

Site assessment and resource characterization

Grid connection of marine renewable parks


INTRODUCTION: THE STATE OF MARINE RENEWABLES

• Tidal range (rise & fall)

• Tidal and ocean currents

• Ocean Thermal Energy Conversion (OTEC)

• Salinity gradients

• Wave Energy



• Marine Geothermal

• Marine Biomass

• Offshore Wind



WavesWaves

OTEC OTEC Tides & salinity gradientsTides & salinity gradients



Offshore Wind

resource





In terms of resource, Wave energy and Offshore Wind are the mostinteresting options for development in Spain

However, both technologies present many difficult challenges to be solved:

Wave energy technology is not yet proven (many device types)

Offshore wind development will involve the design of floating platforms and mooring systems (deep water sites)

Reliability and suitability of the conversion machinery still an issue

Economic feasibility and survivability



Oscillating Water Column (OWC)

Overtopping Devices

Oscillating Bodies(Point Absorbers & SurgeDevices)







Sway (NO)WindFloat

Principle Power(US)

Drijfwind(NL)

HyWindStatoil

(NO)

Blue H (NL)


MARINE RENEWABLE ENERGY: A SYSTEMIC APPROACH

How do we assess the performance of marine energy devices?

• Approximated formulations (theory, practical considerations …)

• Numerical methods (frequency domain, time domain)

• Experimental testing

• Empirical assumptions (data from developers, previous experience)

In any case we need MODELS



• Experimental testing is usually very expensive and difficult

• No standardised method nor database is available (technology is not mature)

Numerical modelling is essential for any development

Ideally, a numerical model would cover the whole

energy chain but energy technologies are

composed by several different elements



Hydrodynamic design and assessment of floating and bottom‐mounted converters

Models of PTO systems and control strategies assessment

CvC2vC1

Cr

Vdc

stabc

vrvsvt

3*Lg

Power electronics and grid connection models


To build a model for marine energy devices means to deal with different physical disciplines:

Hydrodynamics (Fluid-body interaction)AerodynamicsHydraulics/Mechanics/Turbomachinery (PTO)Electric engineeringCable dynamics (moorings)Structural dynamicsElectronics…

It is much likely that our global dynamic model will require a specific toolbox for each of the elements above



RESOURCE

The primary input for any marine energy analysis is the modelling of the environmental input (resource in the broad sense)

Wind resource is generally described by the velocity averaged over 10 minutes and involves specification of different aspects:

• Spectral representation for modelling of gusts• Turbulence intensity

Current velocity needs also characterisation (velocity at the surface and profile over the water column)Waves are modelled as stationary stochastic processes

• Complex processes defined generally by 1-hour duration (sea states)• Spectral analysis in frequency-domain to provide parameterisation• Shallow water conditions require specific modelling and analysis (numerical

modelling of wave propagation)


The Wave Resource: WAVE SPECTRA

Hs=2.5m Ts=6seconds

Wave Period / Frequency

(Wave Amplitude)2

6sec

Regular (extreme narrow banded) wave: Irregular wave (ISSC spectrum):

RESOURCE


RESOURCE

Methods for the wave performance assessment of tested devicesClassical performance indicators:

Normalised power matrix as from IEC TC114Performance assessment based on spectral model

Through this model annual energetic power response can be defined

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

3

4

5

6

Frequency(Hz)

Ener

gy s

pect

ral d

ensi

ty(m

2/H

z)

Target spectrum (Bret.)

Simulated spectrumMinimum parameters:

•Hm0

•Te

•Energy flux J (kW/m)

Many others might be

recommended

These parameters are fundamental for the characterisation of theperformance


The Wave Resource: Site Specific Data

•Wave Conditions at any given time interval are characterised statistically by:

•Hs, significant wave height (defined as peak to trough – i.e. twice wave amplitude)

•Ts, significant wave period

100.006.267.9013.7118.3922.3631.39Totals:

3.901.920.720.460.530.230.044-5m

15.561.823.516.103.050.920.153-4m

21.091.752.373.567.714.651.052-3m

33.820.610.922.674.7313.0511.841-2m

25.640.150.380.922.373.5118.310-1m

Totals:>11s9-11s7-9s5-7s3-5s< 3sHs \ Ts

•Longer term data (eg over a year) can be represented in a scatter plot, representing the percentage of time that the sea state was at a certain Hs and Ts:

Sample Scatter Plot:

RESOURCE


RESOURCE

Spectral analysis and sea state characterisation of the deployment site


RESOURCE

Analysis of the wave, current and other environmental parameters is also required to define design criteria for offshore structure

The usual procedure is to apply Extreme Value analysis

procedures to a long-term time series:

• Whole sample• Annual maxima• Peak-Over-Threshold

The resulting sample is typically fitted with a theoretical distribution

(Generalised Pareto, Gumbel, Weibull)

Statistical analysis and advanced extrapolation techniques are often

needed


RESOURCE

Challenges:

Numerical models for wave propagation and forecast at a local site to assist planning of operation and maintenanceComputationally efficient algorithms for wave prediction at a specified position for control purposesThe reconstruction of the directional spectra from real buoy data is onerous and still subject to many uncertainties (many methods available: maximum likelihood, bayesian …)Methods for extrapolation of longer-term data series for extreme value analysis and weather windows determination (largely a statistical work where estimation of the uncertainties is very important)


HYDRODYNAMICS

The first step of any analysis of the dynamics of marine energy devices is the understanding of the loads due to their interaction with the water

Those loads are of different origin and are associated to different aspects:Hydrostatic restoring forcesCurrent loads due to uniform flow (mostly modelled as drag forces, drag coefficient needs calibration either by testing or numerical codes)Mean wave drift loads (steady loads due to waves, can be found by linear analysis)First-order wave loads (computed by linear potential theory)Low-frequency drift loads (second-order potential is needed)High-frequency drift loads (usually negligible on moored systems)Other dissipative effects (generally modelled as drag forces due to oscillating flow but might require specific treatment)


HYDRODYNAMICS: THE WATER WAVE PROBLEM

The most fundamental description of the flow around the body involves the application of the Navier Stokes equations for incompressible flow

0=⋅∇ u ugzpuutu 2∇+⎟⎟

⎠

⎞⎜⎜⎝

⎛+−∇=∇⋅+

∂∂ ν

ρ

It is well known that, if viscous effects can be neglected then it is generally correct to consider the flow irrotational

2 0∇ Φ =Laplace EquationIf this is case, we can introduce a velocity

potential and the problem reduces to the Laplace equation

The boundary condition for the problem would be generally non-linear. The linearisation of the free surface boundary conditions is one of the most restraining limitations -> Few solvers available for non-linear


HYDRODYNAMICS: LINEAR POTENTIAL THEORY

Linearizing appropriately the boundary conditions and assuming a solution that is sinusoidal ( ):

Vnφ∂=

∂ 0=∂∂

zφ2ω 0

z gφ φ∂− =

∂

,1/2,( ) 0d r

d rkr ikrφ

φ∂⎛ ⎞

− →⎜ ⎟∂⎝ ⎠

Boundary conditions

On the free surface (z=0) On the body surface Sb On the seabed (z=-h)

Radiation condition:

∞→r

{ }Re i te ωφΦ =

We can decompose the potential in different components, respectively incident, diffraction and radiation potential. I D Rφ φ φ φ= + +

Radiation and diffraction potential have to satisfy the radiation condition:


HYDRODYNAMICS: ANALYTICAL SOLUTIONS

It is possible to find solution of this type in some specific regions of the domain. They have the form of an eigenfunction expansion

In some cases boundary-value problems like the one before can be solved by simple separation of variables

)()()(),,( zZrRzr θθφ Θ=

( )( )( )∑∑

∞

=

∞

=−+×

++=

0 0)exp()exp(

)()(sincos),,(

n m nn

nmnm

zikBzikArkFKrkEImDmC

zrθθ

θφ

The boundary conditions need to be homogeneous. If this does not happen then particular solutions should be considered

Since the set of eigenfunctions is complete on the region over which they are defined, the solution can be found by “matching” them for the boundaries

As far as the number of eigenfunctions goes to infinity, it is theoretically exact but it might be slow to converge


HYDRODYNAMICS: THE APPLICATION OF BEM CODES

Since they apply sources on panels derived by the discretisation of the body surface, the results are very sensitive to the total number of panels

Common commercial codes are available based on the solution of an integral equation involving the Green’s function, which is the fundamental solution of

the potential due to a point-source

)()()(),,,,,( 0000002 zzyyxxzyxzyxG −−−=∇ δδδ


HYDRODYNAMICS: MORE SOPHISTICATED APPROACHES

There are few CFD packages for free-surface problem but they are very time-consuming and mainly useful for very specific applications

Some few solvers offer the possibility of solving the second-order problem. Alternatively, Froude-Krylov and hydrostatic components can be computed

in real time based on the instantaneous position of the body

From the point of view of the performance assessment, it is much easier and often relatively accurate to apply linear solutions and make

corrections afterwards


HYDRODYNAMICS: HEAVING SURFACE-PIERCING CYLINDER



The convergence depends on the number of solution modes considered. Clearly a large number of modes is necessary for large frequencies



The computational time is acceptable but a bit large (around 40 s per frequency for the most accurate solution)

R=5 m, d=5 m, h=50 m frequency= 0.1 rad/s.

n1=200, n2=200, n3=400.

R=5 m, d=5 m, h=50 m frequency= 0.1 rad/s. n1=30,

n2=30, n3=50.

The problem is that the solution is singular at the corner and this makes the algorithm relatively slow


HYDRODYNAMICS: THE LINEAR EQUATION OF MOTION

In a first approach linear wave theory may be applied to evaluate the wave-body interactions

Every body hit by a wave experiences a force dependent on its own geometry(diffraction)

Indeed an oscillating body in calm water generates waves itself. This phenomenon is called radiation

The analysis of the body dynamics may then be achieved superimposing this two effects. The diffraction case corresponds to the case of a wave hitting the body held fixed while the radiation one is analogous to the situation of a body freely

floating in absence of waves. If we consider a regular wave we can describe the motion of a body as a sinusoid (frequency analysis)


HYDRODYNAMICS: THE LINEAR EQUATION OF MOTION

Hydrodynamic coefficients A(ω) and Brad(ω) and Excitation forces are obtained from computational methods, eg Boundary Element Method (BEM) or analytical theoretical solutionsWith linear fluid assumptions:•Inviscid•Incompressible•IrrotationalQuestion: Are these assumption sufficient? Non-linear effects may be important, especially in extreme conditions. What about turbulence? What about viscosity?

( )ωηωωωωω ˆ})B)((B))A(M{()(F̂ PTOrad2

E ×++++= Cj

Wave Excitation Forces for 1m

wave amplitude

Mass Matrix Hydrodynamic Added Mass

Hydrostatic Stiffness Matrix (from Cylinder dimensions)

Linear Radiation Damping

Linear PTO

damping

SOLVE FOR RESPONSE

MOTION AMPLITUDES


Increasing draft/radius ratio low resonance frequency, higher maximum power

Decreasing draft/radius ratio moderate motion amplitudes, broader power response curve

Dimensionless variables and parameters to investigate the performance in regular waves: Cylindrical shapes are described by different draft/radius ratio

Each geometry shows a resonance behavior at a certain frequency



HYDRODYNAMICS

Challenges:

Commercial BEM codes relatively established: questions on their applicability to non-linear waves and extreme dynamic situationsAlternatives approaches involve CFD program to solve RANSE: still very slow and computationally intensive for free-surface flowsPossibilities to deal with the potential problem with alternative methods (Finite Elements, DGP)The effect of non-linear hydrodynamics are often difficult to implement: calibration against tank testing is absolutely necessaryDetailed hydrodynamic analysis of survival conditions with higher-order methods still too time-consuming and not efficientOptimisation processes (GA) might be based on simple assessment methods (frequency-domain models). Thorough analysis of their applicability is required


DYNAMIC SIMULATION TOOLS

Marine Energy Dynamic Simulation Methods

•Applicable to linear systems

•Based on stationary processes

•Suitable for optimization

•Simple and reliable

Time‐Domain ModelsFrequency‐Domain Models

•Applicable to non‐linear systems

•Capable of transitory modelling

•Time‐consuming and computationally exigent

Theoretically, when both applicable, time‐domain and frequency‐domain models are equivalent.

Due to the random nature of the waves, however, the output from a time‐domain model is very changing and dependent on the parameters and the spectral model used.



The dynamics can be described by the Cummins Equation:

),,()()()()()()( txxFdxtKtgSxtFtxAm ext

t

e +−−−=+ ∫∞−

∞ τττρ

Where Fext can represent every non-linear term, including PTO forces, moorings, drag and other terms.

∫∞

=0

)cos()(2)( ωωωπ

dtBtK

The convolution term represents a radiation memory effect, described by the radiation impulse response function:

There exist several methods to get rid of the convolution term:

•Identification in the frequency-domain

•Realization theory (Hankel SVD decomposition)

•Prony method

•And others …



•Prony Method and 5° Order Runge‐Kutta variable step integration

•Hankel SVD and 5° Order Runge‐Kutta variable step integration

•Frequency Identification Method

•Direct numerical integration

The convolution term is approximated by a sum of exponential terms. This allows the representation of the dynamics with a system of ODE and the subsequent solution with standard numerical routine (Dormand and Price for MATLAB)

The convolution term is approximated by a state‐space realization via a HankelSingular Value Decomposition. The solution of the resulting ODE system is achieved with the same numerical algorithm (Dormand and Price for MATLAB)

The frequency‐domain radiation transfer function is represented as fraction ofrational polynomials, by fitting it through a LS approach. It is then converted into a state‐space representation

Direct numerical integration of the convolution term and of the whole system of equations. Either implicit or semi‐implicit methods


Linear PTO case Hydraulic PTO case

DYNAMIC SIMULATION TOOLS: EXAMPLE OF TIME-DOMAIN SOLUTIONS



Challenges:

Commercial time-domain codes available for offshore industry but need purposely developed routines for marine renewables (often involving coding and benchmarking)Cummins’ equation widely applied for dynamic simulation of marine systems. Doubts arise on the integration of fluid losses terms (normally with drag terms). Possibility of a different approach based on fluid-related quantities?Time-domain simulations require either a large number of simulations or a long duration. Possibility of quicker methods?


INTEGRATION OF ENERGY CONVERSION AND MOORINGS

The development of dynamic simulation tools involves the proper mathematical modelling of all the components of the dynamic system:

Energy conversion system• Hydraulic system for wave energy extraction• Linear generators• Wind turbines

Mooring system• Catenary configurations• Taut configurations• Tension-leg design (positive buoyancy)

Dynamic cables for grid connection



Linear Power Take Off Modelling: PM Synchronous Linear Generator

Excitation field provided by Permanent Magnets:high force density (compact machine design),translator part free from electrical equipment

a

b

c

a’

b’

c’

a

b

c

a’

b’

c’

a

b

c

a’

b’

c’

a

b

c

a’

b’

c’IRON

MAGNET

COPPER

=>Can be approximated by a linear damping coefficient in the Equation of Motion


Example: Design and model of a hydraulic PTO

HP acc

LP accCompressibility of

the oilControl

accumulators

Power conversion

Offshore grid

Hydraulic PTO

Hydraulic motor Electrical generator

BuoyHP acc

LP accCompressibility of

the oilControl

accumulators

Power conversion

Offshore grid

Hydraulic PTO

Hydraulic motor Electrical generator

Buoy




The torque applied by the generator can be considered as a control variableLet us consider a 1-meter wave of a period of 10 seconds

The torque of the generator represents a kind of resistive load on the system by means of which it is possible to control the flow rate across the motor and eventually perform a rough phase control

Under-loaded

T=80 Nm

Optimal

T=200 Nm

Overloaded

T=300 Nm




The optimal result of the opening of the extra-accumulator would be to generate an acceleration increase on the buoy in such a way that the

velocity peak occurs in phase with the peak of the excitation force

Period=10 s

Torque=300 Nm

Thres=300 kN

Bretschneider Hs=1 m, Te=7 s

Threshold=100 kN

Torque=60 Nm




The problem of control:

To improve power absorption, a control system should be designed and implemented. The concept of the control could indeed influence the whole design of the device. Without entering into details we may distinguish between two strategies:

•Real-time control (latching or phase control): Feedback strategy aimed to continously change the phase of the motion of the body. To do so, one or more parameters have to be changed second by second.

•Sea-state control: The control law is much simpler and is changed only in correspondence of changes in the sea state.



One of the largest research subject in offshore wind is the development of coupled codes that include aerodynamics, aero-elasticity and hydrodynamics

The basic approach is the introduction of separate equations in the time-domain model:

Constant force for a preliminary modelling of the thrustThrust and torque curves depending on the position of the platform based on Blade Momentum TheoryAdvanced aerodynamic models that describe better the aerodynamics of the blades under severe oscillatins

Coupling with wind turbine aerodynamics



For the analysis and design of the moorings of a floating system, three separate methods can be identified:Quasi-static method (application of the steady loads, computation of the offset using static equilibrium equations and subsequent application of offsets due to wave loads) Frequency-domain method (from an initial position defined based on static equilibrium of the configuration, all contributions from moorings and other systems are linearised to build a frequency-domain model)Time-domain method (the dynamic equations are fully solved including all the non-linear effects and possibly low-frequency drift forces. Lines can be also modelled by using FE analysis)

Design of Mooring systems


Simple assessment involve the use of the catenary equations

Load curves that specify the tension in function of the position of the cable end can be introduced in any of the models described before




The line is divided into a series of line segments which are then modelled by straight masslessmodel segments with a node at

each end

The model segments only model the axial and torsional properties of the line. The other properties (mass, weight, buoyancy etc.)

are all lumped to the nodes

FE Modelling of Mooring lines and cables





Challenges:

Integration of PTO models can be complicated when considering real systemsLosses and efficiency of the conversion needs to be modelled properly (convergence issues might arise when integrating complex physical processes)Time scales might be different for separate systems (e.g. power electronics)Coupled aero-hydrodynamic models required to estimate the effect of the motion of the platform on the turbineMooring lines software available but not easy to handle and not fit to preliminary design purposes (time-consuming and very sensitive to any parameter)


OTHER ASPECTS

Response of offshore wind turbines frequently involves structural modes of vibration

Possibility of building mathematical models accounting for structural response:

Basic method involves beam theory applied to sub-elements (i.e. tower)Integration of structural modes into a multi-body approachSome codes under development (FAST) to tackle this issue. The coupling with hydrodynamics, aerodynamics and moorings makes thewhole model complicated and time-consuming

Aero/Hydro‐elasticity


OTHER ASPECTS

Structural, Electrical and Thermal analysis of specific components

Some components might require detailed analysis for their

development

Thermal, mechanical and electrical behaviour can be

modelled through FE analysis. Boundary conditions and Load

regimes?


OTHER ASPECTS

Grid Integration and Power Quality Issues

The impact of wave farms on power quality depends on the strength of the grid

High impedance Large voltage variations Weak grid (DS)

Renewable energy systems convert the energy flux from natural sources into useful forms

Stochastic and periodic nature affects operation, output and availability

To transport and control with relative ease and with a high degree of efficiency and reliability

Power quality issues


OTHER ASPECTS

172.5 173 173.5 174 174.5 175 175.5 176 176.5 1770

0.2

0.4

0.6

0.8

1

PCCGEN 1GEN 2GEN 3GEN 4

Voltage (pu) for Attenuator with SC

7MW

Voltage (pu) for Point absorber with SC

1.1MW

12MW 6MW


CONCLUSIONS

Mathematical modelling is fundamental at every stage of the development of a technology and is particularly crucial for marine energy devices

Some ideas for improving and extending the current state‐of‐the‐art models:

Non‐linear hydrodynamic effects are poorly understood and rarely included in advanced models. Extensive improvement in the existing codes could provide some reliable tools that avoid the recurrence to experimental testing

Future operations might benefit of advanced time‐domain solvers capable of short‐term wave forecasting

Coupled aero‐hydrodynamic models are still at a development stage and are absolutely needed for the development of the offshore wind industry

In the whole, confidence in the numerical modelling has yet to be built. A thorough perspective from applied mathematicians is often welcomed with respect to the well‐posedness and solvability of the numerical problems

62

Thank you!

Pierpaolo Ricci

TECNALIA – Marine Energy

[email protected]

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Mathematical modelling and analysis in marine renewable energy · 2012-05-08 · BCAM Seminar...

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