Multidisciplinary Design Optimization of Wave Energy Converter Farms Considering Uncertainty through Polynomial Chaos Expansion

Technical Summary: Multidisciplinary Design Optimization of Wave Energy Converter Farms Considering Uncertainty through Polynomial Chaos Expansion

Problem Statement
The design of Wave Energy Converter (WEC) farms faces significant challenges due to the coupled nature of hydrodynamic interactions, geometric constraints, and control strategies, often referred to as the "park effect." Traditional optimization approaches frequently treat these disciplines sequentially or deterministically, neglecting the inherent stochasticity of ocean wave conditions (height, period, and direction) and control responses. This oversight leads to designs that may perform suboptimally in real-world environments. Furthermore, existing studies often decouple control optimization from geometry and layout, limiting potential performance gains. The core problem addressed is the need for a robust, multidisciplinary design optimization (MDO) framework that simultaneously optimizes WEC geometry, array layout, and control parameters under uncertain sea states to maximize energy production while ensuring reliability.

Methodology
The authors formulate a control co-design (CCD) problem within a Multidisciplinary Design Optimization (MDO) framework using a Multidisciplinary Feasible (MDF) architecture. The system couples four primary disciplines: geometry, hydrodynamics, dynamics/control, and uncertainty quantification.

1. Design Variables and Objective:

   • The optimization seeks to maximize the expected electrical power per unit displaced volume (a proxy for cost and Levelized Cost of Electricity).
   • Design variables include geometric parameters (buoy radius $r$ and height $h$), array layout coordinates ($x_i, y_i$), and Power Take-Off (PTO) control parameters (drivetrain inertia $I_i$ and stiffness $K_i$).
   • The study considers a six-body array ($n=6$) with a total of 144 design variables, including controller action variables for 10 frequency components.

2. Disciplinary Models:

   • Hydrodynamics: Linear potential flow theory is employed using the open-source Boundary Element Method (BEM) solver, Capytaine. This calculates added mass, radiation damping, and excitation forces, explicitly capturing pairwise wave interference between devices. To manage computational cost, low-rank matrix compression (Adaptive Cross Approximation) and mesh convergence strategies are utilized.
   • Dynamics and Control: The system uses a pseudo-spectral optimal control solver (WecOptTool) to jointly optimize the WEC motion trajectory and control actions. An unstructured, idealized Model Predictive Control (MPC) scheme is assumed to establish an upper bound on performance, subject to physical constraints on maximum PTO force.
   • Uncertainty Quantification (UQ): Aleatoric uncertainty in wave height ($H_s$), peak period ($T_p$), and wave direction ($\beta$) is modeled using probability distributions fitted to NOAA buoy data (Station 46042). A Nataf transformation generates correlated random samples.

3. Optimization Strategy:

   • The problem is solved using the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), a gradient-free algorithm suitable for black-box, noisy problems.
   • Polynomial Chaos Expansion (PCE): To avoid the prohibitive cost of Monte Carlo simulations, a non-intrusive, regression-based PCE surrogate model is constructed at each design iteration. This surrogate approximates the expected power output over the uncertain wave parameters. The PCE coefficients are determined via least-squares regression using samples generated via Gaussian quadrature.
   • The workflow operates in a "double-loop" structure: the outer loop (CMA-ES) proposes design vectors, while the inner loop trains the PCE surrogate to evaluate the expected objective function.

Key Contributions

• Integrated Control Co-Design: The paper presents a unified framework that simultaneously optimizes WEC geometry, array layout, and control gains, addressing the limitations of decoupled design approaches.
• Uncertainty-Aware Optimization: Unlike traditional deterministic methods, this work directly incorporates environmental uncertainty into the design process using PCE. This allows for the quantification of performance variability and the derivation of robust designs.
• Efficient Surrogate Modeling: The application of regression-based PCE within a high-fidelity MDO loop for WEC farms enables the efficient propagation of uncertainty without requiring thousands of expensive hydrodynamic-control simulations.
• Demonstration of Robustness: The study demonstrates that optimizing a WEC farm under uncertainty yields designs that are significantly less sensitive to input variations compared to optimizing a single device and placing it in an array.

Results and Significance
The paper establishes a theoretical and computational framework for robust WEC farm design. While the specific numerical results of the optimization experiments are noted as forthcoming in the manuscript (Section 5), the authors claim the following significance based on their formulation and preliminary analysis:

• Superiority of Array Optimization: The study posits that optimizing a WEC farm as an integrated array under uncertainty is significantly more effective than optimizing a single WEC and subsequently arranging it in an array.
• Robustness to Variability: The proposed framework produces designs that maintain high performance across a wide range of environmental conditions, mitigating the risk of underperformance in real-world stochastic wave environments.
• Methodological Advancement: By integrating PCE into the MDO loop, the work provides a computationally tractable method for handling the high-dimensional, coupled nature of wave-to-wire systems, which are otherwise intractable for local sensitivity or adjoint-based methods due to computational expense.

The authors emphasize that this approach moves beyond simplified hydrodynamic models and fixed control strategies, offering a pathway toward economically viable and reliable grid-scale wave energy deployment. The code and scripts for this work are made open-source to support reproducibility and further research in marine energy system optimization.