wave-energy

name	wave-energy
description	Wave energy conversion — ocean wave power density (P = ρg²H²T/(32π)), point absorber (Froude-Krylov force, heave resonance, power take-off PTO), oscillating wave surge converter (OWSC), attenuator (Pelamis), oscillating water column (OWC, Wells turbine), resource assessment (significant wave height Hs, energy period Te), capture width ratio, WEC array design, mooring, and leading projects (Wave Dragon, Corpower, MARMOK-A-5).
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Wave Energy Conversion — Complete Skill

Ocean Wave Resource

Wave Power Density

Regular wave power (per unit crest width): P = ρ × g² × H² × T / (32π) [W/m; H = wave height; T = wave period; ρ = 1,025 kg/m³; g = 9.81 m/s²]

For irregular waves using Hs (significant wave height) and Te (energy period): P = ρ × g² × H_s² × T_e / (64π) [W/m; Hs = 4√m₀ = 4×standard deviation of surface elevation; Te = m₋₁/m₀] Simplified: P ≈ 0.5 × H_s² × T_e [kW/m; with ρg²/64π ≈ 0.491 kW·s/m³ ≈ 0.5]

Example: North Atlantic: H_s = 3 m; T_e = 10 s → P = 0.5 × 9 × 10 = 45 kW/m of wave crest Excellent resource sites:

• West Scotland, Orkney (EMEC): 40–70 kW/m annual mean
• West Ireland: 50–65 kW/m
• Peru/Chile (Humboldt Current): 25–40 kW/m
• Oregon Coast: 40–50 kW/m

Wave resource assessment: Scatter diagram: joint probability distribution of (H_s, T_z) pairs from measured data or ERA5 reanalysis Annual mean power P_annual = ΣΣ p(Hs, Tz) × P(Hs, Tz) [probability-weighted average]

Wave Spectrum

JONSWAP spectrum (North Sea and similar): S(ω) = (αg²/ω⁵) × exp(−5/4 × (ω_p/ω)⁴) × γ^(exp(−(ω−ω_p)²/(2σ²ω_p²))) ω_p = 2π/T_p (peak angular frequency); γ = peak enhancement factor (γ = 3.3 for North Sea; γ = 1 for Pierson-Moskowitz)

Pierson-Moskowitz (fully developed sea): S(ω) = (αg²/ω⁵) × exp(−0.74 × (g/(ω × U₁₉.₅))⁴) [U₁₉.₅ = wind speed at 19.5 m height]

WEC Types and Operating Principles

Point Absorber

Point absorber: small buoy (diameter << wavelength); heaves in response to wave Froude-Krylov force (exciting force for heave): F_FK = ρg × V_submerged × k_FK [k_FK = Froude-Krylov coefficient; depends on frequency and geometry] Or: F_exc = A_WL × ρg × η_surface [simple piston approximation; A_WL = waterplane area]

Equation of motion (heave): (m + m_a) × ẍ + (B_PTO + B_rad) × ẋ + (k_hydrostatic + k_PTO) × x = F_exc(t) m_a = added mass; B_rad = radiation damping; k_hydrostatic = ρg × A_WL

Resonance condition: ω_n = √(k_hydrostatic / (m + m_a)) [natural heave frequency] Maximum power at resonance: ω_exc = ω_n (wavelength matches buoy response) Typical T_n = 8–12 s → matches swell periods at good wave sites

Maximum capturable power per unit incident power (theoretical): P_max = |F_exc|² / (8 × B_rad) = P_incident × D_capture [capture width per unit wave energy] For 3D point absorber: D_capture_max = λ/(2π) = L/(2π) [capture width ratio CWR = D_capture / D_buoy]

CWR (Capture Width Ratio): CWR = P_absorbed / (P_incident × D_buoy) Point absorber: CWR = 0.15–0.30 (practical); maximum theoretical 1/(2π) × λ/D ≈ 2–10 for buoy

Power Take-Off (PTO)

Linear generator (direct drive): Buoy heave drives linear generator (magnet/coil assembly) Direct PTO: no mechanical conversion; efficient; high capital cost per kW; used by CorPower, AWS

Hydraulic PTO: Heave drives hydraulic cylinder → high-pressure fluid → hydraulic motor → generator Advantages: energy storage in accumulator (smooths power); flexible; established technology Disadvantages: sealing in marine environment; efficiency ~60–80%

Optimum PTO damping (matched to radiation damping for maximum power): B_PTO_opt = B_rad [at resonance; or more generally: B_PTO_opt = √(B_rad² + (ω(m+m_a) − k/ω)²)]

Oscillating Water Column (OWC)

Principle: fixed or floating structure with air pocket above internal water surface Waves cause water to rise/fall → air pumped through turbine → electricity Internal free surface acts as secondary oscillator

Air turbine options: Wells turbine: self-rectifying (works in both directions); constant rotation; poor efficiency at off-design; η ≈ 0.65–0.70 Impulse turbine (Dennis-Auld): self-rectifying; better off-design; η ≈ 0.75–0.80 Biradial turbine: rotating valve rectification; η ≈ 0.80–0.85

OWC power: P = η_turbine × ṁ_air × Δh [ṁ_air = air mass flow; Δh = enthalpy change through turbine] Or: P = η_turbine × p_chamber × Q_air [Q_air = volumetric flow through turbine]

Shore-based OWC: LIMPET (Scotland, 500 kW); Pico (Azores, 400 kW); Mutriku (Spain, 300 kW)

Oscillating Wave Surge Converter (OWSC)

Example: Oyster, BioWave: Hinged flap near seabed; surge (horizontal) waves drive flap rotation Amplified by proximity to seabed (shallow water compression of wave orbits) Drives hydraulic pumps to shore-based turbine (no moving parts offshore except flap)

Surge force: F_surge = C_m × ρ × π × D²/4 × ∂u/∂t + C_d × 0.5 × ρ × D × u|u| [Morison; u = horizontal velocity]

Attenuator (Pelamis-type)

Pelamis: long snake-like machine; segments connected by hydraulic joints Waves pass along length; segments articulate; relative motion drives hydraulic pumps Capture width proportional to length; length 150 m (Pelamis P2) Power: 750 kW rated; decommissioned 2014 (commercial challenges)

Array Design and Performance

Spacing: WECs absorb and radiate waves → wake effects on downstream devices Optimal spacing depends on wave direction spread; typically 3–10 × device diameter Array interaction can be constructive or destructive at specific frequencies

Capacity factor: Wave energy Cf_wave ≈ 25–40% (better than wind at same site; more consistent) Seasonal variation: higher in winter storms; lower in summer swells

Grid connection: Subsea cable to shore; voltage: 11–33 kV typical; HVDC for remote locations Power quality: smoothed by array diversity; storage (flywheel, battery) for further smoothing

Mooring Design

Key loading: Extreme survival storm: design wave Hs_100yr; mooring must withstand peak load without failure Fatigue: millions of wave cycles; chain and polyester fatigue data per DNV-ST-0437

Catenary mooring: Water depth > 50 m: catenary chain (steel or polyester tether) Water depth < 50 m: taut-leg mooring; stiffer; less horizontal excursion

Mooring horizontal restoring force: F_moor = (EA/L) × (ε − ε₀) + catenary weight term [ε = strain; EA = axial stiffness; L = mooring line length] Typical: 500–2,000 kN horizontal force in survival storm

Resource-to-Grid Efficiency Chain

η_total = η_WEC × η_PTO × η_generator × η_cable × η_transformer η_WEC: 20–35% (wave → mechanical) η_PTO: 70–85% (mechanical → hydraulic → mechanical) η_generator: 90–95% η_cable: 98–99% η_total: 15–30% overall wave-to-grid

Leading Projects and TRL

Project	Type	Rated Power	Status
CorPower C4	Point absorber	300 kW	TRL 6; wave trial completed 2023
MARMOK-A-5 (IDOM)	OWC buoy	30 kW	TRL 5; Biscay Marine Energy Platform
Wavepiston (Denmark)	Point absorber chain	Modular	TRL 5
AWS Ocean E1	Submerged point absorber	1 MW	TRL 5
Mutriku OWC	Shore-based OWC	300 kW	Operational commercial
Wave Dragon	Overtopping attenuator	4–11 MW	TRL 6; demonstrated

Standards and References

Standard	Scope
IEC 62600-100	Wave energy — WEC power performance assessment
IEC 62600-200	(Marine energy — resource assessment)
DNV-ST-0437	Loads and site conditions for WTGs (applicable to WEC mooring)
EMEC guidelines	Wave energy device testing protocol
Cruz "Ocean Wave Energy" (Springer)	Comprehensive reference

Output

Provide: site data (location; H_s [m] and T_e [s] at annual mean and 1-yr/50-yr return; water depth [m]; resource P [kW/m]; predominant direction; seabed type), WEC type selection (point absorber/OWC/OWSC/attenuator; basis: water depth, wave climate, technology maturity), device design (for point absorber: D [m]; m [t]; m_a [t]; ω_n [rad/s]; resonance T_n [s]; vs. T_e match), PTO (linear/hydraulic/pneumatic; rated force or pressure; rated power [kW]; PTO damping B_PTO [kN·s/m]; energy storage buffer [kWh]), capture width ratio (CWR = [value]; P_absorbed per device [kW] at annual mean sea state; vs. incident P × D), annual energy production (AEP per device [MWh]; capacity factor [%]; array of N devices; total AEP [GWh/year]), mooring (type: catenary/taut; water depth [m]; line material; survival tension at H_s_100yr [kN]; fatigue life [years]), grid connection (cable voltage [kV]; length [km]; array layout; cable losses [%]), economics (capex estimate [$/kW]; LCOE [$/MWh]; comparison to offshore wind [$/MWh]; path to cost reduction), and applicable standard (IEC 62600-100 for performance; EMEC protocol for testing; DNV-ST-0437 for mooring).