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Short Course on Wave Energy Technology Lisbon, 14-18 July 2014. MODELLING OF OWC WAVE ENERGY CONVERTERS. António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2014. Basic approaches to OWC modelling. will be analized here. Basic equations.

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MODELLING OF OWC WAVE ENERGY CONVERTERS

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Short Course on Wave Energy Technology Lisbon, 14-18 July 2014

MODELLING OF OWC WAVE ENERGY CONVERTERS

António F.O. Falcão

Instituto Superior Técnico,

2014

Basic approaches to OWC modelling

will be analized here

Basic equations

Volume flow rate of air displaced by OWC motion

• Decomposeinto

• excitationflow rate

Basic equations

Thermodynamics of air chamber

Assume compression/decompression process in air chamber to be isentropic (adiabatic + reversible)

Aerodynamics of air turbine

X

X

Dependence on Mach number Ma in general neglected, because of scarce information from model testing.

Frequency domain analysis

• Linear turbine

• Linear relationship air density versus pressure

Linearize:

Wells turbine

Frequency domain analysis

The system is linear

Decompose

Note: radiation conductance G cannot be negative

PICO OWC PLANT, AZORES, PORTUGAL

Frequency domain analysis

(deep water)

Axisymmetric body

(deep water)

Frequency domain analysis

Power

Power available to turbine =

pressure head x volume flow rate

Regular waves

Time average

Frequency domain analysis

Power

Turbine power output

Wells turbine

Exercise

Compute the turbine power ouput of the Pico OWC plant, for regular waves of period 10 s and amplitude 1.0 m.

The diameter of the turbine rotor is 2.3 m. The maximum alowable rotational speed is about 1500 rpm.

Wells turbine of Pico plant

Frequency domain analysis

Dimensional analysis

Model testing: similarity laws for air chamber and air turbine

Correct dynamic similarity requires all terms in equation to take equal values in similar conditions at model size 1 and full size 2 .

1

2

air chamber

Turbine dimensionless parameters (representing the turbine aerodynamic performance) take equal values for similar conditions of the air pressure cycle in the chamber of the model and the full-sized converter. We take such conditions as those of maximum air pressure .

Turbine size

Turbine rotational speed

The two turbines are geometrically similar

Time-domain analysis of OWCs

The Wells turbine is approximately linear. So frequency-domain analysis is a good approximation.

Other turbines (e.g. impulse turbines) are far from linear. So, time-domain analysis must be used, even in regular waves.

This affects specially the radiation flow rate, with memory effects.

The theoretical approach is similar to time-domain analysis of oscillating bodies.

memory function

STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION

Introduction

Theoretical/numerical hydrodynamic modelling

• Frequency-domain

• Time-domain

• Stochastic

In all cases, linear water wave theory is assumed:

• small amplitude waves and small body-motions

• real viscous fluid effects neglected

Non-linear water wave theory and CFD may be used at a later stage to investigate some water flow details.

Introduction

Frequency domain model

Basic assumptions:

• Monochromatic (sinusoidal) waves

• The system (input  output) is linear (e.g. a linear damper and a linear spring)

• Historically the first model

• The starting point for the other models

• Easy to model and to run

• First step in optimization process

• Provides insight into device’s behaviour

• Poor representation of real waves (may be overcome by superposition)

• Only a few WECs are approximately linear systems (OWC with Wells turbine)

Introduction

Time-domain model

Basic assumptions:

• In a given sea state, the waves are represented by a spectral distribution

• Fairly good representation of real waves

• Applicable to all systems (linear and non-linear)

• Yields time-series of variables

• Computationally demanding and slow to run

Essential at an advanced stage of theoretical modelling

Introduction

Stochastic model

Basic assumptions:

• In a given sea state, the waves are represented by a spectral distribution

• The waves are a Gaussian process

• The system is linear

• Fairly good representation of real waves

• Very fast to run in computer

• Yields directly probability density distributions

• Restricted to approximately linear systems (e.g. OWCs with Wells turbines)

• Does not yield time-series of variables

Many processes in Nature behave in such a way that the Gaussian probability density function applies.

The sum of a large number of independ random variables (without any one being dominat) is Gaussian distributed.

The surface elevation at a given point in real ocean waves is approximately a Gaussian random process.

Ouput signal

Input signal

LINEAR

SYSTEM

• Random

• Gaussian

• Given spectral distribution

• Root-mean-square (rms)

• Random

• Gaussian

• Spectral distribution

• Root-mean-square (rms)

Ouput signal

Input signal

Ouput signal

Input signal

Ouput signal

Input signal

Linear air turbine (Wells turbine)

Linear air turbine (Wells turbine)

Average power output

Linear air turbine (Wells turbine)

Average turbine efficiency

Application of stochastic modelling

Maximum energy production

and maximum profit

as alternative criteria for

wave power equipment optimization

The problem

When designing the power equipment for a wave energy

size and rated power capacity of the equipment.

Which criterion to adopt for optimization?

Maximum annual production of energy,

leading to larger, more powerful, more costly equipment

or

Maximum annual profit,

leading to smaller, less powerful, cheaper equipment

How to optimize? How different are the results from these two optimization criteria?

How to model the energy conversion chain

Wave climate represented by a set of sea states

• For each sea state: Hs, Te, freq. of occurrence .

• Incident wave is random, Gaussian, with

known frequency spectrum.

AIR

PRESSURE

OWC

WAVES

TURBINE

Linear system.

Known hydrodynamic

coefficients

Known

performance

curves

Random,

Gaussian

Random,

Gaussian

rms: p

TURBINE SHAFT POWER

ELECTRICALPOWER OUTPUT

GENERATOR

Electrical

efficiency

Time-averaged

Time-averaged

=

+

+

+

C

C

C

C

C

struc

mech

elec

other

The costs

Capital costs

Annual repayment

Operation & maintenance

annual costs

Income

Annual profit

Calculation example

Pico OWC plant

OWC cross section:

12m 12m

Computed hydrodynamic coefficients

Calculation example

Wells turbine

Dimensionless performance curves

Turbine geometric shape: fixed

Turbine size (D): 1.6 m < D < 3.8 m

Equipped with relief valve

Inter

Calculation example

Wave climate: set of sea states

Each sea state:

• random Gaussian process, with given spectrum

• Hs, Te, frequency of occurrence

Calculation method:

• Stochastic modelling of energy conversion process

• 720 combinations 

Three-dimensional interpolation for given wave climate and turbine size

0.6

0.55

0.5

D

=1.6m

0.45

Dimensionless power output

D

=2.3m

0.4

0.35

D

=3.8m

0.3

0.25

100

150

200

250

300

350

WD (m/s)

Calculation example

Turbine size range 1.6m < D < 3.8m

Turbine rotational speed W optimally controlled.

Max tip speed = 170 m/s

Plant rated power

(for Hs = 5m, Te=14s)

Calculation example

Wave climates

Wave climate 3: 29 kW/m

Reference climate:

• measurements at Pico site

• 44 sea states

• 14.5 kW/m

Wave climate 2: 14.5 kW/m

Wave climate 1: 7.3 kW/m

Calculation example

Wind plant

average

Utilization factor

Calculation example

Annual averaged net power (electrical)

Calculation example

Costs

Capital costs

Operation & maintenance

Availability

Calculation example

wave climate 3: 29 kW/m

wave climate 2: 14.5 kW/m

wave climate 1: 7.3 kW/m

Influence of

wave climate

and energy price

Calculation example

wave climate 3: 29 kW/m

wave climate 2: 14.5 kW/m

wave climate 1: 7.3 kW/m

Influence of wave climate and discount rate r

Calculation example

wave climate 3: 29 kW/m

wave climate 2: 14.5 kW/m

wave climate 1: 7.3 kW/m

Influence of wave climate & mech. equip. cost

Calculation example

29 kW/m

14.5 kW/m

7.3 kW/m

Influence of wave climate and lifetime n

CONCLUSIONS

END OF

MODELLING OF OWC WAVE ENERGY CONVERTERS