Machine Learning Based Control Co-Optimization for Geometry and Power Take-off of a Point Absorber Wave Energy Converter

16th International Symposium on Practical Design of Ships and Other Floating Structures PRADS 2025 Ann Arbor, MI, USA, October 19th-23rd 2025 Machine Learning Based Control Co-Optimization for Geometry and Power Take-off of a Point Absorber Wave Energy Converter Weihan Lin1, Andrew Fang2, Lisheng Yang1and Lei Zuo1,* 1University of Michigan, Ann Arbor, MI, USA 2Virginia Tech, Blacksburg, VA, USA Abstract. The floating buoy geometry and the power take-off (PTO) system of a point absorber wave energy converter (WEC) were the two main factors that influence the total power extraction performance. This paper aims to present a machine learning based control co-optimization method for geometry and PTO of a point absorber, with a discussion of performance by using various training models including Support Vector Machines (SVM) and Neural Networks (NN). The machine learning model in this study is developed to replace the conventional frequency domain boundary element method (BEM) hydrodynamics software with thousands times faster simulation speed and with error margin from 0.5% to 2%. The trade-off between the number of training samples and the accuracy of the machine learning model is discussed to further improve the collection speed of training samples. The advantages and disadvantages of various types of machine learning models, SVM and NN, are discussed and identified. Finally, the control co design for the PTO of the point absorber is considered during geometry optimization stage, which allows the estimation of the electrical power to become more realistic to be used in the main objective function. In this process, PTO parameters are optimized at the same time as the floating buoy geometry, achieving global optimal performance. Key words: Machine learning, Renewable energy, Point absorber, Support vector machines, Neural networks, Power take-off 1. INTRODUCTION As global demand grows for clean and sustainable energy sources, interest in renewable energy, particularly from solar, wind, and ocean waves, has significantly increased. Renewable energy is expected to account for over 40% of global electricity demand within the next 15 years, rising from the current 25%, as total electricity usage expands from 22,200 TWh/year to 35,500 TWh/year [1]. Among renewable resources, ocean wave energy stands out due to its high energy density (10-100 kW/m [2]), notably surpassing that of solar (100-150 W/m² [3]) and wind (200-400 W/m² [4]). Theoretical estimates for ocean wave energy potential range from 8,000 to 80,000 TWh/year [5], with the broader potential for all ocean energy forms estimated at around 151,300 TWh/year [6]. Although these theoretical potentials exceed current technical capabilities, ongoing scientific and technological advancements promise improved efficiencies. Wave Energy Converters (WECs) are devices designed to transform ocean wave motion into electrical energy, suitable for residential, community, and industrial applications. The point absorber type of WEC captures energy through the relative movement between a buoy—either floating or submerged—and a stationary or oscillating reference [7]. This concept, initially presented by Budal and Falnes [8][9], has spurred extensive research into hydrodynamics [10][11], structural integrity [12], and power take-off (PTO) system design [13][14]. Additionally, significant efforts have gone into optimizing the geometry of point absorber buoys to enhance their energy capture capabilities [15]. Gilloteaux and Ringwood optimized cylindrical buoy dimensions by adjusting diameter and draft for single-body heaving absorbers [16]. De Backer explored the hydrodynamic performance of cylindrical buoys with various bottom shapes, such as conical and spherical designs, identifying optimal geometries for specific sea conditions [17]. Moving beyond simple geometric shapes, McCabe introduced bi-cubic B-spline parametric modeling combined with genetic algorithm *Correspondence to: [email protected] 1 (GA) optimization to maximize power capture of surge-and-pitch WECs [18]. Garcia-Teruel et al. further analyzed various optimization strategies, applying them to multiple buoy shapes, including barge, hemisphere, vertical cylinder, and bi-cubic B-spline-defined surfaces [19]. To reduce computational complexity, Lin et al. used 12 non-uniform rational B-spline (NURBS) control points and employed neural networks to significantly speed up hydrodynamic evaluations within GA optimizations, cutting computation times dramatically [20]. Previous WEC shape optimization studies often simplified power calculations by using an idealized PTO model as a passive damper, neglecting real-world losses and dynamic adaptability to changing wave conditions. This approach leads to two primary inaccuracies: first, it assumes an ideal PTO with no power losses, and second, passive damping only achieves maximum efficiency at resonance. Recent studies have highlighted the benefits of incorporating PTO losses and active control into optimization frameworks, providing a more realistic estimation of electrical power output from WECs [21, 22]. Accurate power assessment is critical, as optimistic estimations based on simplified models can lead to ineffective device designs in later development stages. Addressing these limitations, this paper introduces a comprehensive and realistic approach to WEC optimization through machine learning-based control co-optimization. Two main contributions set this work apart from earlier studies. First, PTO characteristics and active control strategies are integrated into the shape optimization loop, enabling rapid and realistic evaluations of electrical power output in the frequency domain. Second, a machine learning approach, using SVM or Neural Networks, quickly predicts hydrodynamic parameters based on geometric inputs, significantly accelerating the genetic algorithm-based optimization process. As generating training data through boundary element method (BEM) simulations can be computationally intensive, this paper also investigates how the accuracy of SVM and NN models varies with the number of training samples. 2. MODELING In this section, we present the modeling methods for the floating buoy used in the WEC design. A parametric geometry based on Non-Uniform Rational B-Splines (NURBS) control points is adopted, as outlined by Lin et al. [20]. Dynamic models of the two-body point absorber under wave and current conditions are developed to evaluate both power performance and hydrodynamic drag. Additionally, we detail the modeling of the PTO system, highlighting its mechanical performance. Finally, the machine learning approaches, including SVM and neural networks, are introduced. 2.1. Parametric Modeling The buoy geometry is defined using a parametric NURBS model introduced by Lin et al. [20], as shown in Fig. 1.. This model uses 12 parameters, organized into the shape vector s: s= [sx, sy, sr](1) and, sx= [x1, x2, x3, x4] sy= [y1, y2, y3, y4] sr= [rx41, rx42, ry41, ry42] (2) By adjusting these parameters, complex buoy geometries can be created. However, certain constraints must be met to avoid infeasible deck lines, examples of which are shown in Fig. 2.. 2.2. Power Take-off Modeling In this study, we restricted the analysis to the heave DOF of the two bodies and and used a linear, frequency domain BEM to get the added mass, radiation damping, wave excitation, and hydrostatics for each candidate shape. Viscous effects in waves and second-order drift are not solved in the BEM. The viscous current drag is handled separately through the objective in Sec. 3. The device is self–reacting, so 2 Figure 1.: Parametric modeling of the floating buoy using NURBS control points [20] Figure 2.: Examples of (a) a feasible deck line (b) an infeasible deck line hydrodynamic coupling to a separate reference structure is not modeled, and the only interaction of the two bodies is through the PTO. Mooring dynamics is also not considered for the frequency domain solution. The dynamics of this two-body system can be described as: (m1+A1)¨x1=k1x1+c1˙x1+fe1+fPTO (3) (m2+A2)¨x2=k2x2+c2˙x2+fe2−fPTO (4) Here, m1,2are body masses, A1,2are added masses, k1,2are hydrostatic stiffnesses, c1,2are radiation damping coefficients, and fe1,2are wave excitation forces. fPTO represents the PTO force, and x1,2are displacements from equilibrium positions. Figure 3.: The free-body diagram of the point absorber proposed in this study 3 The PTO consists of a ball screw coupled to a generator, modeled as equivalent mass and damping with a linear electromagnetic torque relationship. The dynamics are described by: mb¨x=cb˙x+NKtiL+fPTO (5) Here, x=x1−x2represents relative displacement, mband cbrepresent the ball screw properties, Ktis the generator torque constant, Nis the gear ratio, and iLis the load current. Positive PTO force is defined as upward thrust. Transforming the equations into the frequency domain, we define intrinsic impedance Zifor the WEC: Zi=−c1+jω(m1+A1−k1/ω2),0 0,−c2+jω(m2+A2−k2/ω2)(6) The ball screw impedance is defined as Zb=−cb+jωmb, and the generator impedance is defined as Zw=Rs +jωLq, where Rsis the stator resistance and Lqis the synchronous inductance. A PI controller is assumed to control the PTO force such that fPTO =Ki(x1−x2) + Kp( ˙x1−˙x2). The controller impedance is thus defined as Zc=Kp+Ki/jω. Transform Eq. 5 to the frequency domain yields: Zb(V1−V2)−NKtIL=FPTO (7) Upper cases are used to represent values in the frequency domain. V1and V2are equal to jωX1and jωX2. From this PTO dynamics we can derive the required generator load impedance ZLin order to achieve the desired controller impedance. Notice that Zc=FPTO/(V1−V2), we then have: Zc=Zb−NKtIL V1−V2 (8) Another equation can be established using the back EMF of the generator. Specifically, torque constant Kt is also the linear coefficient between generator rotational velocity and the back EMF. Thus, we have the electrical part and mechanical part coupling equation: IL(ZL+Zw) = NKt(V1−V2)(9) Plug Eq. 9 into Eq. 8 and rearrange the equation, we get the corresponding load impedance to the controller: ZL=N2K2 t Zb−Zc −Zw(10) Finally, the electrical power absorbed by the generator load is expressed as the average active power of the load: Pe=1 2R(ZL)|iL|2(11) This average power is used in the optimization objective in the following section. 2.3. Building the Dataset Given the novelty of this work, we created a dataset comprising 120,000 distinct buoy hull shapes for machine learning training. Each sample includes 13 parameters—12 defining the buoy geometry and one specifying the wave period. To manage computational resources, we divided the dataset into smaller subsets suitable for efficient training. The dataset was generated in batches, each containing 100 random shapes adhering to predefined geometric constraints from Lin et al. [20]. To reduce the sample count while maintaining prediction accuracy, validation testing determines the optimal number of samples per batch. 4 2.4. Support Vector Machines To select the best model for predicting hydrodynamic parameters, two machine learning methods are considered: SVM and neural networks. Support Vector Regression (SVR), a variant of SVM customized for continuous value predictions, is used [23]. SVR operates similarly to linear regression by fitting a line to data points, but with two main differences: its unique optimization procedure and the application of kernel functions. In SVR optimization, consider a dataset with input vectors xNand corresponding continuous outputs yN. The SVR aims to find an optimal line defined as f(x) = wx +bby performing two optimizations. First, it minimizes the loss function, defined in Eq. (12), with an allowable margin ϵ, which for this problem is set at 0.0554 based on the interquartile range of the dataset. Second, it enforces flatness of the regression line by minimizing the norm of the weight vector, as shown in Eq. (13). L=(0|y−f(x)|< ϵ |y−f(x)| − ϵ otherwise (12) minimize :J(w) = 1 2||w||2(13) Subject to: ∀n:|y−f(x)|< ϵ (14) Here, wrepresents model weights, ||w|| is their Euclidean norm, and bis a bias term. This optimization focuses on data points outside the margin ϵ, known as support vectors, and utilizes Sequential Minimal Optimization (SMO). SVRs also employ kernel functions to handle data not suitable for linear fitting. These functions map inputs into higher-dimensional spaces to allow linear separation. Three kernels tested in this study are linear, polynomial, and Gaussian, as described in Eq. (15): linear :K(xi, xj) = xi·xj polynomial :K(xi, xj) = (1 + xi·xj)q gaussian :K(xi, xj) = e−||xi−xj||2 (15) Where: xiand xjare two different sample vectors from the input xN.Kis the kernel function. qis some exponent such that it is an integer and q≥2. For optimal kernel selection, 10-fold cross-validation is applied, dividing the data into ten subsets. Each subset is sequentially used for validation while training the model with the remaining subsets. Results showed that a linear kernel best predicts added mass and excitation force, while a polynomial kernel most effectively predicts damping. Consequently, three separate SVRs are established, each with 13 input parameters corresponding to the buoy shape and wave period, to predict added mass, damping, and excitation force separately. 2.5. Neural Networks The second method considered is a feedforward neural network (NN). This NN includes an input layer, two hidden layers with 15 neurons each, and an output layer. Unlike SVRs, a single neural network predicts all three hydrodynamic parameters simultaneously. The NN structure and neuron counts are based on previous optimization studies by Lin et al. [20]. Calculations for hidden layers are performed according to Eq. (16), and the output layer employs the tan-sigmoid transfer function as shown in Eq. (17): h(a) j=σ(Xw(a) ij h(a−1) i+b(a) j)(16) σ(θ) = tansig(θ) = eθ−e−θ e−θ+eθ(17) 5 Where, h(a) jis the j-th neuron at the a-th layer. w(a) ij is the weight of the j-th neuron at the a-th layer from the i-th neuron value at the (a−1)-th layer. h(a−1) iis the i-th neuron at the (a−1)-th layer. b(a) jis the bias for the j-th neuron at the a-th layer. σis the tan-sigmoid transfer function [24]. The NN training employs a backpropagation algorithm, adjusting weights and biases to minimize prediction errors. The neural network’s detailed structure is illustrated in Fig. 4.. Figure 4.: The neural network structure 3. RESULTS AND DISCUSSIONS 3.1. Validation Tests To evaluate the effectiveness of support vector machines (SVMs) and neural networks (NN) in predicting hydrodynamic parameters, validation tests were conducted using a dataset of 18,240 samples not used in the initial training. The performance of these two models was compared across varying numbers of training samples per batch (2, 10, 50, and 100), corresponding to total training inputs of 2,400, 12,000, 60,000, and 120,000 respectively. These validation tests help verify optimal model performance and identify any overfitting or underfitting issues. Prediction accuracy is measured using the percentage error formula: Error =BEMResults −MLResults BEMResults ∗100% (18) The validation results reveal substantial differences in performance between the two methods. Initially, when using only 2 samples per batch, SVMs outperform neural networks in general. However, while SVM performance remains relatively constant with increased training samples, neural networks demonstrate significantly improved accuracy as training sample size grows. By the time the batch size reaches 50 samples, neural networks surpass SVMs for predicting all three parameters: added mass, damping, and excitation force. Figures 5.–7. illustrate the error distribution through boxplots for each parameter. Boxplots depict the median as the central line, with the top and bottom edges representing the 75th and 25th percentiles respectively. Outliers, indicated by red crosses, fall outside 1.5 times the interquartile range from the box edges, and whiskers extend to the maximum and minimum non-outlier values. A critical limitation of SVMs is their difficulty predicting radiation damping, particularly with smaller training samples. Errors can reach as high as 150% with just two samples per batch. Although increasing the training data size reduces this error to approximately 25%, the damping data’s complexity prevents accurate modeling using linear, polynomial, or Gaussian kernels. Thus, despite performing adequately for added mass and excitation force, SVMs consistently struggle with damping predictions. In contrast, neural networks achieve significantly lower error rates across all three parameters, even with small sample sizes. With only two samples per batch, neural networks maintain errors under 6%, improving dramatically to under 2% with increased training samples. At 50 samples per batch, neural network predictions exhibit errors of less than 0.5%, demonstrating high accuracy and suitability for this application. 6 Figure 5.: Added mass error box plot of SVM and NN model trained using 2, 10, 50 and 100 samples per batch Figure 6.: Radiation damping error box plot of NN model (left) and SVM model (right) trained using 2, 10, 50 and 100 samples per batch. Figure 7.: Excitation force error box plot of SVM and NN model trained using 2, 10, 50 and 100 samples per batch Based on these validation results, neural networks clearly offer superior performance compared to SVMs, especially for complex hydrodynamic predictions. For the most accurate predictions, training neural networks with 50 samples per batch (60,000 total samples) is recommended, achieving errors below 0.5%. For faster training while maintaining acceptable accuracy, reducing the training size to 10 samples per batch 7 (12,000 total samples) is feasible, with predicted errors remaining below approximately 2%. 3.2. Optimal Shapes and Performance Two distinct objective functions were analyzed to optimize the WEC: one maximizing average power output, and the other balancing normalized power output and current drag performance. The objective functions, with the goal of max f(x), are defined as: f1=P=f{x1, x2, ...ry41, ry42, T, N, mb, Lq} f2=P Pmax +Fmax −F Fmax =f{x1, x2, ...ry41, ry42, T, N, mb, Lq}(19) These functions are evaluated with fixed equality constraints: m= 80kg;H= 0.5m;v= 2m/s;(20) where Prepresents the average power output (W), Fthe drag force (N), mbuoy mass, Hwave height, and vcurrent speed. Additionally, the following inequality constraints guide the buoy shape optimization: case1 : si(j1)≤si(j1+ 1) where, i ={x, y}, j1={1,2,3} case2 : si(j1)≤si(j1+ 1) si(j2)≥si(j2+ 1) where, i ={x, y}, j1={1,2}, j2={3} case3 : si(j1)≤si(j1+ 1) si(j2)≥si(j2+ 1) where, i ={x, y}, j1={1}, j2={2,3} case4 : si(j1)≥si(j1+ 1) where, i ={x, y}, j1={1,2,3} (21) plus: 0.05 ≤sx≤0.5 0.05 ≤sy≤0.3 0.01 ≤sr(1,2) ≤0.5 0.01 ≤sr(3,4) ≤0.3 0.5≤area ≤0.6 5≤T≤10 (22) where with shape vector [sx, sy, sr]unit of m, and area unit of m2, and wave period Tunit of sec. The optimal PTO parameters and corresponding performance are presented in Table 1., while Fig. 8. illustrates the optimized buoy shapes under the different objective functions. The results indicate that optimal PTO parameters remain consistent across objectives due to their direct influence on power output alone. However, significant shape differences occur because drag performance is considered only in one objective. Maximizing average power only (f1) pushes the search toward fuller sections that raise excitation near the target period. When we add the normalized drag term (f2), the shape shifts to a slimmer underwater profile with smoother shoulders. That cuts projected area and cross-flow length, so current drag drops, while waterplane and volume stay high enough for good heave response. This trade-off explains why the right figure in Fig. 8 looks more “planing hull” and gives a better value of f2even though the PTO settings in Table 1. barely change. In short, geometry carries most of the drag reduction, while the PTO still targets electrical power. Introducing cost considerations, including buoy and PTO system costs, could further refine 8 the optimization, but assigning cost weights is arbitrary and dependent on project-specific needs, and thus was excluded from this study. Compared to Lin et al.’s optimal boat-shaped buoy [20], the optimized shape using the combined power and drag objective closely matches their result, but offers greater realism due to the inclusion of detailed PTO component modeling and control optimization. Table 1.: Optimization performance of objective function f1(power, W) and f2(normalized power plus drag) at H= 0.5m, T∈[5,10] s, v= 2 m/s Objective function Optimal PTO parameters Performance f1N= 80, mb= 0.001, Lq= 0.0588 167 f2N= 80, mb= 0.001, Lq= 0.0494 1.49 Figure 8.: Optimal shape solved using averaged power as objective function (left) and using normalized power plus current drag performance as objective function (right). 4. CONCLUSION In this study, we introduced a practical approach using machine learning methods to optimize both the shape and PTO control of a point absorber WEC. Two popular machine learning models, SVMs and neural networks, were tested to predict key hydrodynamic parameters, including added mass, radiation damping, and excitation force. Our validation showed that neural networks provided better overall performance, especially for radiation damping. Although SVMs handled simpler predictions like added mass and excitation force well when fewer training samples were used, they struggled significantly with the more complex damping data. By integrating detailed PTO component modeling and control into the optimization process, we were able to create more realistic and effective WEC designs than traditional methods. Additionally, considering both drag forces and power output during optimization resulted in notably different buoy shapes, highlighting the importance of balancing multiple objectives for practical designs. Our results indicated that neural networks trained on larger datasets (50 samples per batch, totaling 60,000 samples) achieved the highest accuracy, keeping errors below 0.5%. However, when training speed is a priority, neural networks trained on smaller datasets (10 samples per batch, totaling 12,000 samples) still provided reasonably accurate results, with errors around 2%. In conclusion, this research demonstrates that machine learning methods can significantly improve the efficiency and realism of WEC optimization, offering engineers a valuable tool to design more effective marine energy technologies. Acknowledgments The authors express thanks for the funding support provided by the Office of Naval Research (ONR) through grants # N00014-23-1-2100 and # N00014-21-1-2152. 9