Data-driven modeling and extreme-event analysis of a floating power system with mooring dynamics in random waves

Ocean wave energy represents a promising renewable energy source with the potential to contribute significantly to the global energy mix.¹ As the world shifts toward sustainable energy solutions, the development of wave energy technologies, particularly various wave energy converters (WECs), has gained considerable momentum. They are typically classified according to their interaction with wave direction, into three main categories: attenuating, point-absorbing, and terminating devices.² The wave energy test site at Galway Bay in Ireland is set to become a valuable resource for the research and development of wave energy. Offering real-world testing within a relatively mild sea state, the site features a wave environment approximately one-fourth the intensity of those in the North Atlantic conditions.³ This location, which is currently in the development phase, will serve as a critical intermediary step for WECs, bridging the gap between wave tank testing and full-scale operation in the open ocean.⁴ In the course of this project, a floating power system (FPS) was engineered to facilitate data acquisition and deliver power. Enabling the integration of multiple WECs, this system also serves as a reliable local grid connection point.⁵ 

Accurate predictions of the motions and loads on WECs and the associated FPSs are crucial for ensuring their structural integrity and operational safety, particularly in the context of mooring dynamics. Mooring systems play a vital role in the overall design of WECs, maintaining the position of the device and managing wave-induced motions and loads in harsh and complex random waves.⁶ The design of mooring systems is challenging due to the nonlinear hydrodynamic phenomena encountered during severe sea conditions and the fatigue forces experienced under operational wave environments.⁷ Reliable assessment of floating offshore structures with mooring systems requires interdisciplinary knowledge encompassing hydrodynamics, mooring dynamics, and their interactions. Apart from physical model tests, analytical techniques are generally classified into middle-fidelity methods, such as potential solvers, and high-fidelity tools like computational fluid dynamics (CFD). Potential flow-based solvers, although unable to explicitly account for viscous effects, are widely used in marine hydrodynamics due to their robustness and computational efficiency.^8,9 Conversely, CFD approaches are capable of capturing the complex interactions between nonlinear waves and floating structures, providing comprehensive insights into the diverse flow phenomena associated with floating bodies. These approaches have been employed to analyze wave–structure interaction with mooring dynamics.^10–12 While traditional numerical methods are effective, they can be computationally intensive and unable to provide real-time predictions. This has led to the exploration of surrogate models, which offer faster predictions without significant loss of accuracy. Enhanced efficiency is particularly important for monitoring the tension response of mooring lines, which is crucial for preventing potential damage.

In terms of analyzing the dynamics of mechanical systems in marine applications, surrogate methods are generally classified into two main types, namely, the model-free data-driven approaches and the model-based techniques.¹³ Model-based methods rely on an in-depth understanding of marine hydrodynamics and wave mechanics, typically assuming the availability of model parameters and the measurability of state variables.¹⁴ Nevertheless, determining model parameters or acquiring these measurements may be costly, difficult, or even unfeasible for systems with high nonlinearity.¹⁵ In contrast, data-driven approaches are capable of directly extracting system features from labeled data without requiring a detailed understanding of the underlying physics.¹⁶ These methods, including sparse identification for nonlinear dynamics,¹⁷ dynamic mode decomposition,¹⁸ and deep neural networks,¹⁹ have been widely applied in the field of fluid mechanics.²⁰ Specifically, the long short-term memory (LSTM) algorithm, a type of recurrent neural network (RNN) designed for handling longer and more complex time series, has been extensively used.²¹ Applications of LSTM include short-term deterministic predictions of wave-induced ship motions^22–25 and dynamics of floating bodies in waves.^26–30 However, there has been limited research on applying these methods to addressing problems of nonlinear interactions between waves and floating bodies, especially when accounting for the nonlinear behavior of mooring systems across different sea states.³¹ 

In previous work, we developed a data-driven model to predict the nonlinear dynamics of a floating body moored with four catenary lines.³¹ This method treats the system's output, namely mooring loads and body motions, as a functional response to the ocean waves. The model's effectiveness underwent systematic validation through a sensitivity examination, which scrutinized parameters like the hidden layer configurations, backpropagation length, and number of epochs. This study extends our evaluation of the data-driven surrogate model's performance in the prediction of motions and loads for a FPS in irregular waves, utilizing valuable experimental datasets as the foundation. Specifically, we test the model on various irregular waves with different peak periods and wave steepness, assessing its ability to generalize to new sea states. Furthermore, we focus on the model's capability to predict nonlinear hydrodynamics, particularly the higher-order harmonic components of extreme events. Predictions of extreme events are analyzed in detail using a phase-manipulation harmonic extraction method, which is crucial for capturing the nonlinear dynamics essential for the safety and reliability of offshore structures. Following the introduction, the paper is structured as follows: Sec. II introduces the adopted methods, including those used in the experimental study, data-driven modeling, and harmonic assessment. Section III discusses the associated results for various sea states. Concluding remarks are provided in Sec. IV.

The section introduces the methodology employed for evaluating the nonlinear dynamics of a floating power system in random waves. It starts with an overview of the experimental investigation, and then proceeds with elucidating the data-driven model utilized to establish a surrogate model. Additionally, it outlines the harmonic approach employed for analyzing the extreme motions and loads of both measured and predicted results.

To assess the performance of a floating power system (FPS) for the Galway Bay wave energy test site, the first round of physical wave basin tests were performed at University College Cork, Ireland, using the National Ocean Energy Test Facility.⁴ Expected to be an invaluable resource for developers and researchers in the wave energy sector, a FPS was designed to enable data acquisition and supply power for the Galway Bay wave energy test site. Additionally, the FPS will serve as a local grid connection point, facilitating the integration of multiple WEC devices.

The FPS was systematically designed to a scale of 1:25, resulting in a draft of 30 mm, an overall height of 90 mm, and a diameter of 200 mm.³² The model was constructed using thermoplastic polycarbonate material, painted, and sealed, with a mass of 766.4 g excluding the mooring lines. Its center of mass is situated approximately 29 mm above the water line. Figure 1 gives a diagram of the experimental setup, along with a photograph of the FPS model in the wave tank with markers indicating the optical tracking system.

A traditional three-point catenary mooring system was adopted for these tests. Three mooring lines are layout in a symmetric arrangement with angle between the lines were 120°. Two lines were positioned pointing obliquely up-wave, and one line was positioned pointing down-wave. With a linear density of 45.5 g/m, each mooring chain has a length of 3 m. The wave basin, with a water depth of 1 m, measures 25 m in length and 17.2 m in width. A flap-type ocean wave generator features 40 individually controlled paddles, capable of generating a diverse array of waves, such as regular and irregular single-direction waves, as well as directional-spread waves. Long-duration irregular wave tests were conducted during the experimental campaign and they are analyzed in detail in this study. Here, only waves propagating in the x-direction (parallel to line 2) are presented, where all conditions are long-crested waves. Measurements of free-surface elevations as well as motions and loads on the FPS were recorded at the frequency of 32 Hz. Table I lists the considered sea states, where the duration runs are about 350 s (30 min in full scale).

Figure 2 gives a diagram structure of an LSTM block, illustrating how an LSTM unit operates within a module at current time t, connected to estimations at times $t−1$ and $t+1$⁠. To address the problem of vanishing gradient effectively, the reintroduction of past information at later times is allowed within the design of an LSTM cell. Notably, our LSTM network was configured with a tanh activation function and four hidden layers. The network architecture's impact on accurately capturing the dynamics of a floating body with mooring dynamics in regular waves has been systematically examined in our previous study,³¹ where the sensitivity of epoch lengths, input sequence sizes, and the number of neurons in the hidden layers was examined. For the present study, we retained the same architecture, utilizing an epoch size of $E=100$⁠, a number of neurons of $N=100$⁠, and an input sequence size of $S=50$⁠. This configuration resulted in a low-loss value approximately on the order of $10−4$⁠.

Figure 4 illustrates an example of the original total time series of $F(t)$⁠, alongside its linearized form $F1(t)$ and the Hilbert transform of the linearized signal $F1H(t)$ for the SS1 case over 50 s. The accuracy of these transformations may be evaluated based on the crest and trough statistics of the resulting time series.³⁷  Figure 5 presents the sorted crests and troughs for the original data, the linearized data, and its Hilbert transform. Regarding the original data, its nonlinearity is evident in the sharper wave crests and flatter wave troughs. For both the linearized data and its Hilbert transform, the linear signals of sufficient duration align with the unit slope line. These comparisons indicate that $F1$ and $F1H$ exhibit reasonable linearity, although the largest crests display greater variability due to sampling limitations.

A nonlinear signal's harmonic components can be extracted through the phase-manipulation approach, achieved by linearly superimposing signals with phase shifts. To capture the first four harmonics, namely, $F0$⁠, $F90$⁠, $F180$⁠, and $F270$⁠, four phase-shifted repeats are necessary. It is important to note that phase manipulation is approximated using a single time series of irregular waves, with equivalent waveforms determined by averaging the extreme events, for example, the largest 100 crests and troughs.

As depicted in Fig. 6, extreme events are isolated from the original time series about the index time of the crests identified in the linearized time series, and their average represents the phase signal of $F0$⁠. Similarly, the average time series of extreme events are extracted from the original time series about the index time of the troughs identified in the linearized time series, corresponding to the phase signal of $F180$⁠. Repeating these procedures using the time index from the linearized signal's Hilbert transform yields the phase signals of $F90$ and $F270$⁠, respectively.

The subscripts of component f denote the power in amplitude and the harmonic, respectively. As seen, the first three harmonic components are derived using Eqs. (9)–(11), while Eq. (12) includes the second-order difference and the fourth-order superharmonic, which can be separated by frequency filtering. Notably, only the superharmonic terms of $Fn=fnn$ are analyzed within the present study.

First, the performance of the designed data-driven approach for estimating the motions and forces of the FPS in irregular waves is examined. This involves verifying the accuracy of the surrogate model in terms of predicting the temporal responses of the moored FPS as well as the associated extreme events. Following this, the performance of the surrogate model for waves with varying peak periods and steepness is explored, respectively. Specifically, the capability of the surrogate model for modeling the nonlinear hydrodynamics of a floating body with mooring dynamics is studied based on harmonic analysis of the measured and predicted results.

The surrogate model is initially trained and tested with a dataset of irregular waves, which is represented by a JONSWAP spectrum with a peak period of $Tp=1.5$ s, a significant wave height of $Hs=0.12$ m, and a shape parameter of $γ=3.3$⁠, corresponding to condition SS1 as listed in Table I. In our study, the wave elevation serves as the sequential input, while the resulting body motions in pitch, heave and surge, alongside the tensile forces acting on the front and rear mooring lines, are the outputs. Figure 7 plots the results of training and validation of the model using the SS1 dataset. Here, the data-driven model was trained with time series from 0 to 250 s, and validated using the remaining 30% of datasets from 250 to 350 s. Although there are some minor discrepancies between the local crests and troughs of the actual and predicted results. Overall, it is observed that the trained model accurately and consistently predicts the movements of the floating body, as well as the mooring loads, for unseen wave sequences.

Although the overall accuracy of the surrogate model is crucial for modeling the temporal responses and mooring tensions of a floating body in waves, the model's performance in accurately capturing nonlinear dynamics, especially during extreme events, is particularly significant. These extreme body motions and loads can pose severe damage or danger to floating structures in high sea states. Therefore, detailed comparisons between the actual and predicted results in terms of their exceedance probability are plotted in Fig. 8.

The trained model demonstrates an excellent capability to predict the dynamics of the FPS for wave condition SS1, utilizing 70% of the datasets for its training process. Overall, good agreements are observed between the predicted results and their actual values. However, when focusing on the prediction of extreme events, the model tends to underpredict a few of the largest responses compared to the measured results. Specifically, these discrepancies are noticeable in the motions of heave and pitch. Additionally, it is observed that the tensile forces acting on the mooring lines are slightly overpredicted during extreme events. Generally, the model is able to capture the physical correlations between the input of wave elevations and the outputs of body motions as well as mooring loads in random waves.

To investigate the deep-learning model for waves under varied conditions, we assess the prediction performance of the trained network model on irregular waves with a different peak period than those used during training. Specifically, the model trained on SS1 datasets is applied to predict the nonlinear dynamics of the moored FPS in a new sea state of SS2. Figure 9 shows partial time-series prediction results for SS2 alongside their corresponding actual values. As shown by the wave elevations, two distinct sea states used for training and testing are compared. We see that, despite being trained on the irregular waves with a smaller peak period, the network demonstrates robust performance in approximating the hydrodynamics of the FPS under the new sea state with a larger peak period. While there are some discrepancies between the predicted and actual measurements, the predicted results generally exhibit favorable agreements with the measured results. Specifically, the model captures both the phasing and amplitudes of the responses well compared to the measurements.

Once more, short-term statistics are computed for the wave elevations of the two different sea states, as well as for the predicted and measured body responses. Figure 10 provides a summary of the associated exceedance rates of these results, calculated using the range-pair counting approach. Notably, the maximum wave amplitudes of the two sea states reach levels exceeding twice the amplitude of the significant wave height. The overall counted wave amplitudes from the two distinct sea states align well with each other, except for differences in their tail distributions.

In terms of the motion and load responses, the network model successfully replicates the low and moderate response levels. However, discrepancies at the tails for large response levels are observed, particularly for the responses in surge and pitch, as well as for the tensions acting on the front mooring line. Examining the time series examples reveals that responses in surge and pitch contain strong nonlinear components, especially for surge, which is dominated by low-frequency motions. The correlations between the nonlinearity of the wave elevations and surge motions differ from those of the training waves and are not well captured using the constructed model. Regarding extreme values of the tensions for the front mooring, as they are dominated by the body's surge motions, they are also underpredicted using the trained model, as evidenced by the tails of the amplitude distribution.

The performance of the surrogate model is further evaluated by testing it on a new sea state characterized by a different significant wave height compared to its training waves. Figure 11 displays the time histories of the corresponding wave elevations in irregular waves of SS1 and SS3, alongside the predicted and measured body motions and mooring loads for this new testing sea state. Generally, the two distinct sea states exhibit similar wave phases, while waves of SS3 have relatively larger amplitudes compared to those of SS1. Therefore, the focus of this section is to investigate the impact of wave steepness on the performance of the surrogate model. As seen, while some deviations between model predictions and their actual values are observed, particularly in relation to certain local crests and troughs of heave motion and front mooring loads, the overall agreement is satisfactory.

Figure 12 gives the short-term statistics of both the predictions and their true values. Comparing the wave elevations, we observe that the maximum wave amplitudes of the testing sea state are relatively similar to those of the training sea state, despite having different significant wave heights. The testing waves exhibit a similar trend for low response levels compared to the training waves, but with more pronounced moderate response levels. Moreover, its extreme wave amplitudes appear slightly larger than those of the training sea state, as indicated by their tail distributions.

When it comes to the prediction performance of the network model for the FPS's motions and mooring dynamics, favorable agreements are observed mainly for the low response levels in surge motion and in tensions of the rear mooring lines. Overall, the model tends to overpredict heave motions across all response levels and slightly overpredicts the tensions on the front mooring line. It underpredicts the moderate response levels of pitch motion, while the maximum pitch amplitudes are accurately captured by the surrogate model. Specifically, the predicted extreme events in surge motion show significant deviations from the measured values. This discrepancy likely contributes to the pronounced differences in the tensions of both the rear and front mooring lines, as the mooring dynamics of a floating body are predominantly influenced by its horizontal motions, particularly surge motion in this study. Trained with data from a sea state with smaller wave steepness, the surrogate model's performance in a new sea state with higher wave steepness is acceptable for estimating low and moderate response levels. However, it exhibits significant deviations from the measured values when predicting extreme response levels of surge motion, generally underpredicting these extreme events. A detailed analysis of these discrepancies, based on a harmonic analysis, is provided in Sec. III D.

As the conventional method of assessing, the accuracy of a data-driven model, such as using root mean square (RMS) error, does not distinguish between different types of errors. It assumes that all data points are equally important, which is not the case in many marine applications where extreme events are more critical than others. To further investigate the extreme events that may pose a risk to the FPS, their higher harmonics are extracted using the phase separation method described in Sec. II C. It is worth noting that for this analysis, only the largest ten crests and troughs of these extreme events are considered, and the associated results represent their response amplitudes rather than their absolute values. For instance, the tensile force acting on the front mooring line fluctuates around its mean value of about 0.5 N, a property not captured by harmonic analysis. The ten largest events are selected by summing all the harmonic components, and the exemplary wave elevation signals $F1$ are centered at their peaks and troughs. The four-phase signals are then derived by averaging these selected events. Again, the primary objective of this section is to evaluate the capability of the surrogate model, trained on a sea state with a smaller wave steepness, in predicting the nonlinear hydrodynamics of the FPS, particularly its effectiveness in capturing the associated high harmonic components.

Figure 13 initially shows the harmonic components of the free-surface elevation for both the training and testing sea states, along with the corresponding harmonic components of the predicted and measured surge motions. As observed, with the increase in significant wave height, all the first-four harmonic components of the testing waves exhibit larger amplitudes compared to those of the training waves. The first-harmonic components predominantly contribute to the total wave elevation, with much smaller magnitudes for the higher harmonic components. Regarding the model's prediction, the linear component of the surge motion is accurately captured by the network. However, the model's accuracy diminishes as the order of the components increases. Relatively larger discrepancies are observed in the third and fourth harmonic components when comparing predictions to actual values. Unlike the wave elevations, higher-order components significantly influence the surge motion, contributing up to approximately 20% from the second harmonic component. Nevertheless, the second harmonic component contributes about 30% of its linear component for the testing wave elevations.

Accordingly, Fig. 14 presents the harmonic structures of the PFS's heave and pitch motions. For the heave motion, the extracted responses show a strong correlation to the wave elevations, both exhibiting a typical Stokes-type harmonic series. Its harmonic structure closely mirrors that of the wave elevation, with each harmonic component's magnitude comparable to that of the wave elevation. Regarding the surrogate model's performance, it is observed that, with the exception of the second harmonic components, all other harmonic components are slightly underpredicted compared to their true values. This underprediction at the largest response levels is also observed in other testing cases, with discrepancies noticeable in the tail distributions of heave motions. In terms of pitch motions, more pronounced deviations are observed between the model predictions and their actual values for all four harmonic components. Specifically, higher-order components, such as the third and fourth harmonics, are significantly underpredicted by the trained network. These higher-order components are crucial for pitch motion, with the second-order harmonic contributing up to about 45% of its linear harmonic. Furthermore, the amplitudes of the third and fourth harmonics remain in the same order of magnitude.

Finally, the extracted harmonics of the tensions experienced by the mooring lines are shown in Fig. 15. Generally, the force amplitudes on the front line are four times larger than those on the rear line for the extreme events considered. As these amplitudes are strongly influenced by the body's surge motion, the harmonic structures of these tensile forces resemble those of the surge motion. Upon examining the contributions of the higher-order harmonics for the front mooring lines, it is evident that the second harmonic has an amplitude approximating 40% of the first harmonic. In these extreme events, the contributions from the third- and fourth-order harmonics reach up to 8% and 3% of the amplitude of the linear harmonic. Additionally, the third- and fourth-harmonics exhibit response amplitudes of the same order of magnitude. Regarding the surrogate model's performance, the predicted response amplitudes exceed the actual values, which is unexpected. Unlike the exceedance probability shown in Fig. 12, where only the extreme crest values are provided, the harmonic assessment also includes extreme trough values. The slight overprediction by the surrogate model indicates that the response troughs of the front mooring lines are overpredicted significantly, as illustrated in some exemplary time series in Fig. 11.

Upon inspecting the harmonics of tensions acting on the rear mooring line, a notable distinction emerges between the lower-order and higher-order harmonics. The higher-order components are well predicted, while the first and second harmonics are slightly underpredicted. As observed from the time series in Fig. 11, the rear tension primarily oscillates at the wave frequency but also contains important higher-frequency oscillations. This indicates that local high-frequency dynamics play a critical role in the mooring dynamics, posing a risk for fatigue damage to the mooring lines. In general, compared to the harmonic structure of the front line, the higher harmonics of the rear mooring line contribute similarly. For instance, the amplitude of the second harmonic is about 40% of that of the first harmonic. Overall, the surrogate model underpredicts the tensions acting on the rear mooring line, with this underprediction becoming less pronounced as the harmonic order increases. For example, the fourth harmonic is only slightly underpredicted, approximately 5% smaller than its actual value.

This study systematically examines the performance of a data-driven approach for modeling the motion responses and mooring loads of a FPS in irregular waves, representing a complicated challenge in marine hydrodynamics and generally requiring a comprehensive analysis of the nonlinear interactions between waves, structures, and mooring dynamics. To develop a dependable foundation and datasets for training the network model, experimental model tests are conducted. The model's accuracy in predicting temporal responses and extreme events is validated across various sea states, focusing on different peak periods and wave steepness. Through harmonic analysis, we also investigate the model's performance to capture nonlinear hydrodynamics and associated high harmonic components.

The surrogate model was trained with wave elevations as inputs, and the resulting outputs are the predicted body motions and mooring loads. Initially trained and tested on a dataset characterized by a JONSWAP spectrum, the model demonstrates consistent accuracy in predicting FPS motions and mooring loads for unseen wave sequences. Despite minor discrepancies at local crests and troughs, the overall performance is satisfactory, particularly for low and moderate response levels. However, extreme events pose challenges, with the model underpredicting the largest responses, notably in heave and pitch motions, while slightly overpredicting tensile forces on mooring lines.

When applied to sea states with different peak periods, the model maintains robust performance. It accurately captures both the phasing and amplitudes of responses despite the variations in wave periods. Short-term statistics reveal good alignment between predicted and actual values, though discrepancies at large response levels, especially in surge and pitch motions, still exist. These deviations underscore the complex, nonlinear correlations between wave elevations and body motions that are not fully captured by the trained model.

Subsequently, we examined the model's performance under different wave steepness, which reveals similar trends. While the model satisfactorily predicts low and moderate response levels, significant deviations occur at extreme response levels, particularly for surge motion. The model tends to underpredict these extremes, highlighting the limitations of training on data with lower wave steepness. These findings indicate that for the considered FPS, the mechanics of wave–structure–mooring interactions at low and moderate response levels are similar between different sea states. A model trained using one sea state may also perform well in predicting low and moderate responses in another sea state. However, this is not the case for very large-amplitude motions, as their physical correlations among waves, structure, and mooring dynamics become strongly nonlinear. Furthermore, these events occur very rarely, providing only limited data points for training the model. Consequently, the model's performance in such extreme events is poor.

Harmonic analysis is then used to further elucidate these discrepancies. Generally, the model accurately captures the linear components of surge motion but shows reduced accuracy for higher-order components. Higher-order harmonics significantly influence surge and pitch motions, with the surrogate model often underpredicting these components. This underprediction extends to heave and pitch motions, where the model fails to capture the full magnitude of higher-order harmonics, critical for accurate dynamic response prediction. For mooring line tensions, the harmonic structures reveal that higher-order harmonics contribute substantially to the overall tension dynamics. The surrogate model's performance varies, with notable underprediction of higher-order harmonics, particularly for rear mooring lines. This underprediction becomes less pronounced with increasing harmonic order, indicating a need for enhanced model training to better capture these complex dynamics.

Overall, the data-driven model shows great potential for modeling wave-induced motions and loads on moored floating structures. It consistently delivered accurate predictions, even when sea characteristics differed from its training dataset. Its ability to handle various sea states, combined with the computation efficiency, proves its worth for real-world offshore engineering applications. While our surrogate model shows promise for modeling FPS dynamics in random waves, its limitations in capturing extreme events and higher-order nonlinearities highlight areas for improvement. To address these limitations, it is crucial to ensure that training datasets are representative of the specific sea conditions at the application location. Since extreme events are relatively rare, the datasets must be sufficiently large to train the model to accurately predict such events. The physical correlations between waves and body motions during extreme events differ significantly from those at lower and moderate response levels, where most data points are concentrated. Thus, updating the dataset with data from the specific location of application will enhance the model's performance and accuracy.

The first author gratefully acknowledges the financial support provided by the Institute of Ship Technology, Ocean Engineering, and Transport Systems at the University of Duisburg-Essen. The German Research Foundation (No. DFG:448471847) is acknowledged. We acknowledge support by the Open Access Publication Fund of the University of Duisburg-Essen.

The authors have no conflicts to disclose.

Qi Zhang: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal). Changqing Jiang: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Ould el Moctar: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal). Bret Bosma: Conceptualization (equal); Investigation (equal); Methodology (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.