Specialization in Ocean Energy MODELLING OF WAVE ENERGY CONVERSION António F.O. Falcão Instituto Superior Técnico, Universidade de Lisboa 2017

STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION PART 5 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.

Frequency domain model Introduction Frequency domain model Basic assumptions: Monochromatic (sinusoidal) waves The system (input  output) is linear Historically the first model The starting point for the other models Advantages: Easy to model and to run First step in optimization process Provides insight into device’s behaviour Disadvantages: 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: Advantages: In a given sea state, the waves are represented by a spectral distribution Advantages: Fairly good representation of real waves Applicable to all systems (linear and non-linear) Yields time-series of variables Adequate for control studies Disadvantages: Computationally demanding and slow to run Essential at an advanced stage of theoretical modelling

Gaussian process Physical random variables that are expected to be the sum of many independent processes have distributions that are nearly Gaussian. This is the case of the free surface elevation of real irregular waves

Introduction Stochastic model Basic assumptions: Advantages: In a given sea state, the waves are represented by a spectral distribution The waves are a Gaussian process The system is linear Advantages: Fairly good representation of real waves Very fast to run in computer Yields directly probability density distributions Disadvantages: Restricted to approximately linear systems (e.g. OWCs with Wells turbines) Does not yield time-series of variables

LINEAR SYSTEM Input signal Ouput signal Random Gaussian Given spectral distribution Root-mean-square (rms) Spectral distribution

Ouput signal Input signal LINEAR SYSTEM

Input signal Ouput signal LINEAR SYSTEM

Input signal Ouput signal LINEAR SYSTEM

Linear air turbine (Wells turbine)

Linear air turbine (Wells turbine) Average power output

Linear air turbine (Wells turbine) Average turbine efficiency

AIR TURBINE AND ELECTRICAL EQUIPMENT FOR THE PICO OWC PLANT

The Pico plant

9 sea states, and their frequency of occurrence Simplified version of the wave climate (in deep water) 9 sea states, and their frequency of occurrence

How to model the energy conversion chain ELECTRICAL POWER OUTPUT 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. OWC AIR PRESSURE WAVES TURBINE Linear system. Known hydrodynamic coefficients Random, Gaussian Known performance curves Random, Gaussian rms: p TURBINE SHAFT POWER ELECTRICAL POWER OUTPUT GENERATOR Electrical efficiency Time-averaged Time-averaged

Pico plant

Compare two types of air turbines Biradial turbine Wells turbine

Turbine performance curves versus pressure head (dimensionless) Wells Pressure head

Turbine performance curves versus pressure head (dimensionless) Biradial Pressure head linearize

Constraint: blade tip velocity of the turbine rotor should not exceed 180 m/s WD/2 < 180 m/s Why? Centrifugal stresses, shock waves

Wells turbine: single stage and two-stages were considered For each turbine size D and each sea state (Hs,Te), the rotational speed W was numerically optimized for maximum averaged power output of the turbine. The annual-averaged power output was computed.

Turbine type and size optimization Rotor diameter

Turbine efficiency Incident wave power

Rotational speed control Points fairly well aligned along lines BIRADIAL TURBINE But, in average values Control algorithm (instantaneous values): Electromagnetic torque: Rotational speed

This affects the plant performance and the rotational speed control Effect efficiency and rated power of electrical equipment The efficiency of the electrical equipment (generator and power electronics) decays markedly for load factor less than about 30 – 35% The electrical rated power should not be exceeded at any time. The power electronics is very sensitive to overheating. This affects the plant performance and the rotational speed control

A B C D E F G H Turbine size D = 2 m

Average rotational speed W versus electrical rated power Prated in the most energetic sea state i = 9, for different turbine sizes D = 2.0 to 3.0 m

The annual production of electrical energy depends on turbine size and electrical rated power

The final choice of turbine size and electrical rated power will also depend energy tariff and equipment costs. Maximum profit versus maximum energy production.

Maximum energy production and maximum profit as alternative criteria for wave power equipment optimization

Which criterion to adopt for optimization? The problem When designing the power equipment for a wave energy plant, a decision has to be made about the 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?

Annual profit The costs Capital costs Annual repayment other elec mech struc C + = Annual repayment Operation & maintenance annual costs Income Annual profit

Pico OWC plant Calculation example OWC cross section: 12m 12m Computed hydrodynamic coefficients

Wells turbine Calculation example Turbine geometric shape: fixed Dimensionless performance curves Turbine geometric shape: fixed Turbine size (D): 1.6 m < D < 3.8 m Equipped with relief valve

Wave climate: set of sea states Each sea state: 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

Turbine size range 1.6m < D < 3.8m Calculation example Turbine size range 1.6m < D < 3.8m 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 100 150 200 250 300 350 WD (m/s) Dimensionless power output D =1.6m =2.3m =3.8m Turbine rotational speed W optimally controlled. Max tip speed = 170 m/s Plant rated power (for Hs = 5m, Te=14s)

Wave climates Calculation example 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

Annual averaged net power (electrical) Calculation example Annual averaged net power (electrical)

Costs Capital costs Operation & maintenance Availability Calculation example Costs Capital costs Operation & maintenance Availability

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 energy price

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 and discount rate r

Influence of wave climate & mech. equip. cost 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

Influence of wave climate and lifetime n Calculation example 29 kW/m 14.5 kW/m 7.3 kW/m Influence of wave climate and lifetime n

CONCLUSIONS

Example: Optimization of an OWC sparbuoy for the wave climate off the western coast of Portugal (31.4 kW/m) Optimization involved several geometric parameters

Size and rotational speed of air turbine were optimized R.P.F. Gomes, J.C.C. Henriques, L.M.C. Gato, A.F.O. Falcão. "Hydrodynamic optimization of an axisymmetric floating oscillating water column for wave energy conversion", Renewable Energy, vol. 44, pp. 328-339, 2012.

STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION END OF PART 5 STOCHASTIC MODELLING OF WAVE ENERGY CONVERSION