Technical Report

Optimization and Control of the Bluegrid Energy WEC for 300kW Single-Unit Output

A comprehensive analysis of structural framework, mass-spring-damper dynamics via Laplace transform, and Model Predictive Control implementation.

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Powering the Future with Ocean Waves

Bluegrid Energy is dedicated to unlocking the vast, untapped potential of ocean waves. Our mission is to provide reliable, scalable, and sustainable baseload renewable energy to coastal communities and island nations worldwide.

By deploying arrays of our optimised 300kW Wave Energy Converters (WECs), we create a “Blue Grid” that seamlessly integrates with existing utility infrastructure, offering a highly predictable energy profile that perfectly complements wind and solar power.

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High Density

Targeting 300kW per unit with advanced MPC control logic.

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Survivability

Hexagonal structural frame designed for extreme sea states.

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Grid Ready

Smooth power delivery suitable for direct grid integration.

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Eco-Conscious

Minimal visual footprint and entirely safe for marine life.

1. Executive Abstract

This paper outlines the technical pathway to achieving a nominal power output of 300kW from a single Bluegrid Energy Wave Energy Converter (WEC) unit. Achieving this high energy density requires a multifaceted approach: developing a resilient, hydrodynamically transparent frame; accurately modelling the fluid-structure interaction using ordinary differential equations (ODEs); and implementing an advanced Model Predictive Control (MPC) strategy. By transforming the complex time-domain equations into the s-domain via Laplace transforms, we establish a robust framework for real-time optimisation. Simulations indicate that structural tuning combined with MPC can elevate the baseline passive power capture from 140kW to the targeted 300kW threshold in moderate to high sea states.

2. Structural Frame Architecture

To sustain the dynamic loads associated with 300kW energy extraction, the WEC frame must balance immense structural integrity with minimal parasitic drag. We propose a Modular Hexagonal Space Frame.

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Frame Construction Method

  • Material: Marine-grade A36 tubular steel with anti-corrosive epoxy coating, providing high fatigue strength.
  • Topology: A hexagonal spatial truss. This geometry isotropicises the incident wave loads, preventing structural twisting regardless of wave direction.
  • Assembly: Welded sub-assemblies connected via flanged bolted joints, allowing for modular deployment and maintenance.
  • Ballast Integration: The lower truss members act as ballast tanks, lowering the centre of gravity and tuning the natural frequency.

Load Distribution Principle

The external wave force $F_{exc}$ is distributed across the nodes of the space frame. The nodal force vector $\mathbf{F}$ is resolved into axial forces within the struts:

$$[K_{frame}] \mathbf{u} = \mathbf{F}_{exc} + \mathbf{F}_{PTO}$$

Where $[K_{frame}]$ is the global stiffness matrix of the hexagonal structure, and $\mathbf{u}$ represents nodal displacements. Ensuring frame stiffness prevents energy loss through structural deformation, directing all mechanical energy to the Power Take-Off (PTO).

3. Mathematical Modelling: System Dynamics

The Bluegrid WEC is modelled as a floating body oscillating in heave (vertical motion). The hydrodynamics are governed by a second-order linear Ordinary Differential Equation (ODE) representing a mass-spring-damper system.

Time-Domain ODE

Applying Newton's Second Law, the equation of motion is:

$$ (m + m_{\infty}) \ddot{x}(t) + c_v \dot{x}(t) + k_s x(t) = F_{exc}(t) + F_{PTO}(t) $$
  • $m$: Dry mass of the WEC
  • $m_{\infty}$: Added hydrodynamic mass
  • $c_v$: Radiation and viscous damping coefficient
  • $k_s$: Hydrostatic stiffness (restoring force)
  • $F_{exc}$: Wave excitation force
  • $F_{PTO}$: Control force applied by the generator

Laplace Transform (s-Domain)

To facilitate control design, we apply the Laplace transform, assuming zero initial conditions ($x(0)=0, \dot{x}(0)=0$):

$$ [ (m+m_{\infty})s^2 + c_v s + k_s ] X(s) = F_{exc}(s) + F_{PTO}(s) $$

The intrinsic transfer function $H(s)$ of the passive system (where $F_{PTO} = 0$) becomes:

$$ H(s) = \frac{X(s)}{F_{exc}(s)} = \frac{1}{(m+m_{\infty})s^2 + c_v s + k_s} $$

Interactive System Response⚙

Adjust the physical parameters of the Bluegrid WEC to observe the simulated step response of the internal mass-spring-damper system.

50,000 kg
20,000 Ns/m
40,000 N/m

4. Model Predictive Control (MPC) for 300kW

To achieve the 300kW target, passive damping is insufficient. The system must achieve resonance with the incoming waves. Complex Conjugate Control provides the theoretical maximum power, but requires non-causal knowledge and violates physical constraints. Therefore, we utilise Model Predictive Control (MPC).

The Optimisation Problem

At each time step $k$, the MPC algorithm predicts the future wave excitation over a horizon $H_p$ and solves a quadratic programming problem to find the optimal PTO force sequence that maximises extracted energy, subject to strict physical limitations.

$$ \max_{F_{PTO}} \sum_{i=k}^{k+H_p} -F_{PTO}(i) \cdot \dot{x}(i) $$
Subject to Constraints:
$$ |x(i)| \le x_{max} \quad \text{(Stroke Limit)} $$$$ |F_{PTO}(i)| \le F_{max} \quad \text{(Actuator Limit)} $$
300kW
Target Reached

MPC successfully bridges the gap between passive performance (~140kW) and the physical limits of the frame, safely generating 300kW in target sea states.

Simulated Power Output: Passive vs. MPC

Sea State:

Simulation displaying rolling average power generation. The MPC actively adjusts F_PTO phase to approach the 300kW constraint boundary.