Long-term Extreme Response Research Articles

The SPAR Model: A New Paradigm for Multivariate Extremes: Application to Joint Distributions of Metocean Variables

Abstract This paper presents the application of a new multivariate extreme value model for the estimation of metocean variables. The model requires fewer assumptions about the forms of the marginal distributions and dependence structure compared to existing approaches, and provides a flexible and rigorous framework for modeling multivariate extremes. The method involves a transformation of variables to polar coordinates. The tail of the radial variable is then modeled using the generalized Pareto distribution, with parameters conditional on angle, providing a natural extension of univariate theory to multivariate problems. The resulting model is referred to as the semi-parametric angular-radial (SPAR) model. We consider the estimation of the joint distributions of (1) wave height and wave period, and (2) wave height and wind speed. We show that the SPAR model provides a good fit to the observations in terms of both the marginal distributions and dependence structures. The use of the SPAR model for estimating long-term extreme responses of offshore structures is discussed, using some simple response functions for floating structures and an offshore wind turbine with monopile foundation. We show that the SPAR model is able to accurately reproduce response distributions, and provides a realistic quantification of uncertainty.

Statistical Modeling and Applications of Joint Distributions for Significant Wave Height, Spectral Peak Period, and Peak Direction of Propagation: A Case Study in the Norwegian Sea

The estimation of long-term extreme response is a crucial task in the design of marine structures. The target extreme responses are typically defined by annual exceedance probabilities of 10−2 and 10−4. Various approaches can be employed for this purpose, with preference given to statistical long-term analysis, which involves aggregating the exceedance probabilities of all potential sea states contributing to the exceedance of the target extremes. A joint model encompassing important metocean parameters such as wind, waves, and current is often necessary. This study specifically focuses on waves and wave-induced responses. In characterizing short-term sea state conditions, significant wave height (Hs), spectral peak period (Tp) and peak direction of propagation (Φp) are identified as the most important sea state characteristics. The objective of this work is to present the results of the joint model for the three sea state parameters, i.e., Hs, Tp and Φp, at an offshore site in the Norwegian Sea. The conditional modeling approach is applied using long-term hindcast data, and different statistical models are discussed for fitting the marginal and conditional distributions. The fitted parameters for all directional sectors are provided, offering a comprehensive representation of the joint model for direct use in long-term response analysis. Two case studies are included to illustrate the application of the fitted joint model in long-term response analyses. The case studies identify the governing wave directions and the most important combinations of short-term sea state characteristics regarding the estimation of long-term extreme responses.

Long-term extreme responses of torsional moments at two-directional hinges for moored very large floating structures

This paper presents an analysis procedure for estimating long-term extreme torsional moments at connections of very large floating structures (VLFS) with two-directional hinges and mooring lines under random waves. This procedure, compared to the common short-term estimation method, can accurately and quickly calculate the extreme response at a certain return period. Applying a coupled frequency-time domain approach, we simulate the hydroelastic response of VLFS with two-directional hinges and vertical mooring lines. In order to validate the proposed approach, numerical simulations are firstly carried out under the action of regular waves. The present approach demonstrates a high level of consistency with experimental data and numerical results from existing studies in terms of vertical displacement, bending moment and shear force. Then, on the basis of the short-term extreme response distribution and environmental model, a procedure for solving long-term extreme responses of the structure is introduced. The long-term extreme response characteristics of the torsional moment at hinges are evaluated and studied in depth using the procedure, providing a reference for the structural design of moored VLFS with two-directional hinges.

Robust adaptive analysis of dynamic responses of wave energy converters

PurposeThe purpose of this paper is to exploit a new and robust method to forecast the long-term extreme dynamic responses for wave energy converters (WECs).Design/methodology/approachA new adaptive binned kernel density estimation (KDE) methodology is first proposed in this paper.FindingsBy examining the calculation results the authors has found that in the tail region the proposed new adaptive binned KDE distribution curve becomes very smooth and fits quite well with the histogram of the measured ocean wave dataset at the National Data Buoy Center (NDBC) station 46,059. Carefully studying the calculation results also reveals that the 50-year extreme power-take-off heaving force value forecasted based on the environmental contour derived using the new method is 3572600N, which is much larger than the value 2709100N forecasted via the Rosenblatt-inverse second-order reliability method (ISORM) contour method.Research limitations/implicationsThe proposed method overcomes the disadvantages of all the existing nonparametric and parametric methods for predicting the tail region probability density values of the sea state parameters.Originality/valueIt is concluded that the proposed new adaptive binned KDE method is robust and can forecast well the 50-year extreme dynamic responses for WECs.

Uncertainty quantification in low-probability response estimation using sliced inverse regression and polynomial chaos expansion

For wave energy converters (WECs), wind turbines, etc., estimation of response extremes over a selected exposure time is important during design. Sources of uncertainty arising from background slowly-varying environmental conditions and from shorter time-scale fluctuations in ocean winds, turbulence, etc. must all be considered. Together, these different sources can comprise a high-dimensional vector of stochastic variables (often on the order of hundreds or thousands). To accurately propagate the influence of these uncertainty sources to model outputs, conventional surrogate model building approaches such as polynomial chaos expansion (PCE), stochastic collocation, low-rank tensor approximations, etc. must consider dimension reduction. We explore the use of sliced inverse regression (SIR) combined with polynomial chaos expansion. SIR first reduces the original high-dimensional problem to a low-dimensional one; then, an optimal polynomial PCE model is proposed and applied on “effective” components in the low-dimensional space. SIR-PCE can mitigate the curse of dimensionality. It is employed here in the prediction of the long-term extreme response of offshore structures; it is demonstrated using classical benchmark analytical functions as well as offshore applications including extreme waves and the response of a wave energy converter. Efficiency and accuracy gains over Monte Carlo simulation and other methods in literature are found.

Artificial Neural Network-Based Prediction of the Extreme Response of Floating Offshore Wind Turbines under Operating Conditions

The development of floating offshore wind turbines (FOWTs) is gradually moving into deeper offshore areas with more harsh environmental loads, and the corresponding structure response should be paid attention to. Safety assessments need to be conducted based on the evaluation of the long-term extreme response under operating conditions. However, the full long-term analysis method (FLTA) recommended by the design code for evaluating extreme response statistics requires significant computational costs. In the present study, a power response prediction method for FOWT based on an artificial neural network algorithm is proposed. FOWT size, structure, and training algorithms from various artificial neural network models to determine optimal network parameters are investigated. A publicly available, high-quality operational dataset is used and processed by the Inverse First Order Reliability Method (IFORM), which significantly reduces simulation time by selecting operating conditions and directly yielding extreme response statistics. Then sensitivity analysis is done regarding the number of neurons and validation check values. Finally, the alternative dataset is used to validate the model. Results show that the proposed neural network model is able to accurately predict the extreme response statistics of FOWT under realistic in situ operating conditions. A proper balance was achieved between prediction accuracy, computational costs, and the robustness of the model.

Investigation on the extreme peak mooring force distribution of a point absorber wave energy converter with and without a survivability control system

To determine the optimal design of the wave energy converter (WEC) that can withstand extreme waveconditions, the short- and long-term extreme responses of the system need to be determined. This paper focuses on the extreme peak force distribution of the mooring force for a 1:30 scaled point absorber WEC. The basis of this analysis is the mooring force response obtained from a WEC-Sim model calibrated by wave tank experimental data. The extreme sea states have been chosen from a50-year environmental contour. Here, first, the long-term extreme response using the full sea state approach is obtained for three constant damping cases of the power take-off (PTO) system. Then, using a contour approach, the expected value of the extreme peak line (mooring) force distribution is computed for the sea states along an environmental contour. Further, for the most extremesea state, the extreme peak line force distribution is also computed where a survivability control system, based on a deep neural network (DNN), changes the PTO damping to minimize the peak mooring force in each zero up-crossing episode of surface elevation. The results show that in the absence of a control system, the zero PTO damping case is a conservative choice in the analysis of the long-term response and the design load. For the most extreme sea state along the environmental contour, the survivability control system slightly reduces the expected value of the extreme peak force distribution when compared with lower constant PTO damping configurations.

Full long-term extreme buffeting response calculations using sequential Gaussian process surrogate modeling

The full long-term method (FLM) provides the most accurate estimates of the long-term (i.e., 10–100 years) extreme buffeting responses in the design of wind-sensitive structures. However, it suffers from high computational demand if the number of uncertain turbulence parameters becomes high. In this paper, we propose a new algorithm for efficient solution of the full long-term problem through sequential Gaussian Process (GP) surrogate modeling. The algorithm efficiently trains a GP surrogate model that replaces the buffeting response calculation while exploiting the specific properties of the long-term problem. The performance of the algorithm is demonstrated on a practical design problem of estimating extreme buffeting-induced stresses of a long-span suspension bridge, where the results are compared with the ones obtained from other viable methods. The results show that the algorithm achieves computational efficiency comparable to the highly efficient IFORM, while also significantly enhancing the accuracy of the response estimate. The algorithm also has the added benefit of converging to the true solution of the full long-term problem, in contrast to the IFORM solution, which converges to an approximate solution.

Effects of inhomogeneous wave modeling on extreme responses of a very long floating bridge

Floating bridges are usually deployed in fjords where the wave fields are inhomogeneous. The wave inhomogeneity has been found to induce significant effects on dynamic responses, however, it was rarely considered during the extreme response analyses of floating bridges. The present study deals with the effects of inhomogeneous wave modeling on extreme responses of a very long end-anchored floating bridge. The inhomogeneous wave field modeling accounts for the spatial variation of wave spectral parameters as well as the coherence of wave elevations between different positions. Structural responses under various wave conditions are simulated through fully-coupled hydro-elastic simulations. The long-term extreme responses are investigated by a simplified environmental contour method, where the long-term extremes are approximated by extremes at a high quantile in carefully selected short-term sea states. The characteristics of simulated waves and responses are first analyzed to serve as a basis for extreme prediction. Uncertainties associated with extreme analyses are also studied. It is found that the coherence of wave elevations has significant influences on the standard deviation and the spatial correlation of the responses. The extreme values of the axial force vary between different modeling of inhomogeneous wave loads and become significantly larger considering partially correlated wave fields.

Extreme response analysis of a floating vertical axis wind turbine based on modified environmental contour method

Floating vertical axis wind turbine (FVAWT) is a potential concept that could compete with the floating horizontal axis wind turbine (FHAWT) in the offshore wind industry. For a reliable FVAWT design, accurate prediction of extreme structural responses is crucial for ultimate limit state (ULS) design evaluation. Extreme structural responses of offshore wind turbines depend on operational modes and could present non-monotonic behavior with respect to environmental parameters. In this study, the extreme responses of a 3-bladed H-rotor semi-submersible FVAWT are evaluated using a simplified engineering approach, i.e., the modified environmental contour method (MECM). Judging from the response characteristics of the FVAWT with respect to increasing wind speed, the environmental contour with a return period of the cut-out wind speed is used. Along the selected environmental contour, the cut-out wind speed and the corresponding most probable sea state is chosen as the environmental design point. To generate statistical samples of structural responses, the Monte Carlo (MC) method is applied by carrying out fully-coupled time domain simulations under the selected environmental loadings. The Gumbel method, the Weibull tail method and the mean upcrossing rate (MUR) method are used to extrapolate the 50-yr long-term extreme responses. With a sufficiently large sample size, e.g. 100 MC simulations, the estimated extremes using different extrapolation methods are in close agreement with each other. However, the Weibull tail and MUR methods reduce the statistical uncertainty of the estimates due to higher data utilization. For practical engineering applications, massive amount of MC simulations is often not desirable. Therefore, the accuracy and statistical uncertainty of the estimated extreme responses associated with a reduced sample size is also investigated. For a sample of 10 simulations, the Weibull tail and MUR methods are able to predict extreme responses that are closer to the estimates using 100 simulations. The MUR method results in the lowest statistical uncertainty of the estimates.

Prediction of long-term extreme response due to non-Gaussian wind on a HSR cable-stayed bridge by a hybrid approach

The effect of non-Gaussian inflows on structural long-term extreme buffeting responses has been little investigated. In this study, the sensitivity of long-term extreme value distribution (EVD) of a high-speed railway cable-stayed bridge to the non-Gaussian intensity is studied first. The turbulence skewness and kurtosis are then taken as the environmental variables to investigate their single and combined effects on bridge's long-term EVDs based on a proposed hybrid approach that combines the machine learning algorithm and virtual process method. The 2.5-year measured turbulence wind and 40-year annual extreme wind speed recorded near the bridge site are utilized to describe the probability distributions of the skewness and kurtosis of turbulence wind and 10-min mean wind speed. The research results reveal that: (1) the long-term EVD of torsional angle is more sensitive to non-Gaussian turbulence wind than vertical and lateral extreme responses; (2) the single effect of turbulence skewness is detrimental but limited, and the combined effect of skewness and kurtosis of turbulence u (w) is also weak within the considered MRIs (1–100 years). Lastly, the virtual process method is shown to be applicable to predict structural long-term EVDs; and it is efficient without losing significant prediction accuracy.

Investigations of the long-term extreme buffeting response of long-span bridges using importance sampling Monte Carlo simulations

Several studies have shown that site measurements of the buffeting responses of long-span bridges do not always agree with predictions from analytical models. The observed discrepancy is believed to be explained by the variability in the wind’s turbulence field, which is ignored in the current design guidelines. Additionally, the guidelines neglect the true variability in the buffeting response, as this parameter is considered to be deterministically dependent on the mean wind speed only, and a single short-term-averaged wind condition is assumed to be representative of the bridge’s behaviour over its lifetime. Such assumptions underestimate the design buffeting response. Alternatively, a full long-term analysis is regarded as the most accurate approach to handling extreme response estimations. Applications of this analysis in the wind engineering literature are scarce due to its large computational demand. With the objective of alleviating such demand, this paper proposes a full long-term analysis framework based on importance sampling Monte Carlo (ISMC) simulations and investigates, for the first time, the extreme buffeting response of a suspension bridge in its design phase with this framework. The results were compared with available formulations based on reliability methods and numerical integration. The proposed ISMC framework requires fewer simulations than the other formulations while maintaining a good degree of accuracy. The results show that the estimated extreme response is 30% larger than the current design guidelines suggest, showing the importance of considering the turbulence parameters' and loading effects' variabilities when calculating extreme responses.

Environmental design load for the line force of a point-absorber wave energy converter

To ensure a reliable operation over the life time of wave energy converters (WECs), a number of load cases need to be considered according to international standards for marine structures to determine an optimal design. This paper outlines the procedure of obtaining an environmental design load for the line force of a 1:30 scaled point-absorber WEC using an environmental contour with a 50-year return period for the Dowsing site in the North Sea. To obtain the response of the WEC during extreme conditions, a numerical WEC-Sim model is developed and calibrated with experimental wave tank tests to augment the data required for such design load analysis. The design load for the line force is estimated based on the full long-term extreme response computed from the full sea state approach by considering 180, 360, and 720 sea state samples as well as the contour approach for the sea state that gives the most extreme response. Further, a probabilistic approach is used to quantify the probability of failure for a critical mechanical component of the system such as shackle. The result shows that the numerical WEC-Sim model is able to sufficiently replicate the real response of the system during extreme irregular waves. The Bayesian theory with Monte-Carlo algorithm is found to be an excellent tool for identifying the degree of belief in the statistical models used for the short-term extreme response analysis. Considering the ultimate limit state, the design load for the 1:30 scaled system is calculated as 670.95 N (i.e. 18.11 MN for a full-scale system) after applying the partial load safety factor of 1.35 on the full long-term extreme response of the system for the 9.1 years return period (i.e. 50 years in a full-scale model).

Prediction of long-term extreme response of two-rotor floating wind turbine concept using the modified environmental contour method

The modified environmental contour method (MECM) is assessed for the prediction of 50-year extreme response of a two-rotor floating wind turbine concept (2WT) deployed in two offshore sites in the northern North Sea (Norway 5) and the North Atlantic Ocean (Buoy Cabo Silleiro). The sites considered are in areas known for their floating wind development potential. The environmental contour method (ECM) is used to reduce the computational effort of full long-term analysis (FLTA) by only considering environmental conditions associated with a given return period. MECM is a modification of the ECM where additional environmental contours are included to account for discontinuous operation modes of dynamic structures. The results obtained in MECM are benchmarked against FLTA results and compared to ECM results. ECM leads to large underpredictions of responses governed by wind loads if compared to FLTA, as it is not capable of taking into account important operational modes of the 2WT. It is found that MECM, which includes the wind turbines cut-off contour, is able to reduce most response underpredictions within 15% difference compared to FLTA results. MECM may thus be considered as a sufficiently accurate and computationally efficient method for the long-term extreme analysis of 2WT concepts.

Quantitative comparison of environmental contour approaches

Environmental contours are a pragmatic and widespread method to estimate the long-term extreme response of marine structures. Over the years, a range of approaches have been proposed. A benchmarking study was recently conducted to compare the various methods using a common set of data. The current work extends this benchmark study by providing a quantitative assessment of the contours submitted to the exercise. The estimates of long-term responses from the contours were compared against a response-based analysis (RBA) for a wide range of responses. While some contour methods agreed well with estimates from the RBA (relative errors less than 10%), most methods were found to give large errors relative to the RBA. For the 1-year responses most methods showed a large positive bias, whilst both positive and negative biases were found for the 20-year responses. The reasons for the differences between the contours and RBA were explored. It was shown that the fitted statistical models accounted for a large portion of the error in some approaches, with both positive and negative biases of the order of 50% for some contributions, depending on the response type. Whilst for other methods, the statistical model gave accurate predictions for most responses, no models were able to capture all response behaviours for all locations. Secondly, most contour methods do not account for serial correlation in the data. It is shown that this introduces a significant positive bias into long-term response estimates, especially for lower return periods. The level of error introduced by the type of contour method is dependent on the assumption made about the shape of the failure region in the contour definition. For the predominantly unimodal response types considered, contours which approximate the failure region as having a linear boundary (IFORM and direct sampling contours), introduce relatively little error for most responses. However, for some responses, the direct sampling contours were found to introduce errors in the range 20%–40%, depending on the variable space in which they are constructed. The ISORM and highest density contours were found to have a significant over-conservatism bias, which would be expected for the response types considered.

The effect of serial correlation in environmental conditions on estimates of extreme events

In offshore engineering, it is common practice to estimate long-term extremes under the assumption that environmental conditions are independent. However, many environmental variables, such as winds and waves, exhibit correlation over several days. In this work, we consider the impact that this has on estimates of return values of metocean variables, environmental contours and long-term extreme responses. It is shown that methods which neglect serial correlation over-estimate the size of extreme events at a given return period. We introduce a new definition of a sub-asymptotic extremal index, and show how this can be used to quantify the effect of neglecting serial correlation. Simple examples are presented to illustrate why neglecting serial correlation leads to positive bias. We show how the size of the bias is related to the average shape of storm events and the shape of the tail of the distribution of storm peak values, with the latter having the dominant effect. Storm peak distributions with longer tails lead to larger biases when serial correlation is neglected. In the examples presented, neglecting serial correlation resulted in relative errors of over 50% in the 25-year extreme response estimates in some cases. The examples presented show that accounting for serial correlation in estimates of environmental contours and long-term extreme responses can reduce over-conservatism and result in more efficient designs.

Long-term extreme response of an offshore turbine: How accurate are contour-based estimates?

According to design standards, offshore wind turbines need to withstand environmental loads with a return period of 50 years. This work compares the extreme response along the 50-year environmental contour with the true 50-year wind turbine response. It was found that the environmental contour method that is currently described in the IEC design standard for offshore wind turbines can strongly under-predict the 50-year return value of response variables whose annual maxima typically occur during power production. The bias in the contour-based estimate of the 50-year response can be attributed to three sources: (1) the method used to construct the contour; (2) neglecting serial correlation in environmental conditions; and (3) neglecting the short-term variability in the response. In our analysis the 50-year maximum mudline overturning moment was underestimated by 4–8% by the contour-based approach that is currently recommended, whereas the bending moment at 10 m water depth was underestimated by 25–28%. This underestimation was mainly due to ignoring the short-term variability in the response. The bias associated with contour construction, an effect much discussed in recent publications, was of much smaller magnitude.

Comparison of environmental contour and response-based approaches for system reliability analysis of floating structures

Long-term extreme design loads have typically been estimated using an environmental contour (EC) approach, in which the maximum extreme responses of a structural system associated with critical sea states along the contours are applied for mechanical design purposes, to avoid the tedious task of direct integration. The variabilities in short-term extreme response and the contributions of all sea states have been ignored, and this has caused unreliable results. Considering concerns over the sensitivity of the hydrodynamic performance of floating structures to wave data, the objective of this study is to evaluate the effect of environmental conditions on the short-term extreme response parameters and present a full long-term analysis for the failure probability evaluation of floating structures. Numerical simulations are conducted to establish a representative and reliable database which consists of wave variables and the corresponding short-term extreme response parameters for a semi-submersible platform based on a discrete scheme, using the specialised finite element software ANSYS/AQWA. Tension and offset extreme data are derived from time-domain simulations of the coupled system using the peak-over-threshold method to determine the short-term load distribution. Starting with the database, a linear interpolation scheme and intelligent algorithm are utilised to predict extreme response parameters for any other environmental conditions. Surrogate models are employed for three scenarios: (1) design load estimations obtained from various ECs as a basis for long-term load evaluation, (2) full long-term extreme response distribution calculation accounting for all possible sea states, and (3) system reliability analysis of floating systems. A simplified long-term analysis based on EC-based design loads is proposed, and the results are compared with those of the full long-term methodology. The system reliability of a floating system is calculated considering mooring line breaking, fatigue, and floater offset. These expressions provide a means of preliminary design evaluation of floating structures, accounting for all sea state contributions, which can be used to address the design loads and failure probability of an entire system.

Prediction of Long-Term Extreme Response of Fish Cage Using Environmental Contour Method

Prediction of Long-Term Extreme Response of Fish Cage Using Environmental Contour Method

Long-term extreme buffeting response of cable-supported bridges with uncertain turbulence parameters

Although the full long-term method (FLM) is recognized as the appropriate way to identify the design stresses of marine structures subjected to stochastic environmental loading, the FLM has not yet been adopted for the design of wind excited long-span bridges. The results presented in this study show that the current design practice, through short-term extreme response analyses with deterministic turbulence parameters, may significantly underestimate the long-term design stresses of long-span bridges. Both the variability of the turbulence parameters and the uncertainty in the short-term extreme response are found to be important when estimating the design stresses. In addition, the long-term extreme acceleration responses have been compared with the acceleration responses measured in full scale at the Hardanger Bridge, showing considerable improvements to the current design practice.