Novel Reliability Method Validation for Floating Wind Turbines

Wind turbines and associated parts are susceptible to cyclic stresses, including torque, bending, and longitudinal stress, and twisting moments. Therefore, research on the resilience of dynamic systems under such high loads is crucial for design and future risk‐free operations. The method described in this study is beneficial for multidimensional structural responses that have undergone sufficient numerical simulation or measurement. In contrast to established dependability methodologies, the unique technique does not need to restart the numerical simulation each time the system fails. Herein, it is demonstrated that it is also possible to accurately predict the probability of a system failure in the event of a measurable structural reaction. In contrast to well‐established bivariate statistical methods, which are known to predict extreme response levels for 2D systems accurately, this study validates a novel structural reliability method that is particularly suitable for multidimensional structural responses. In contrast to conventional methods, the novel reliability approach does not invoke a multidimensional reliability function in the Monte Carlo numerical simulation case. As demonstrated in this study, it is also possible to accurately anticipate the likelihood of a system failure in the case of a measurable structural reaction.

DOI: 10.1002/aesr.202200177 Wind turbines and associated parts are susceptible to cyclic stresses, including torque, bending, and longitudinal stress, and twisting moments. Therefore, research on the resilience of dynamic systems under such high loads is crucial for design and future risk-free operations. The method described in this study is beneficial for multidimensional structural responses that have undergone sufficient numerical simulation or measurement. In contrast to established dependability methodologies, the unique technique does not need to restart the numerical simulation each time the system fails. Herein, it is demonstrated that it is also possible to accurately predict the probability of a system failure in the event of a measurable structural reaction. In contrast to well-established bivariate statistical methods, which are known to predict extreme response levels for 2D systems accurately, this study validates a novel structural reliability method that is particularly suitable for multidimensional structural responses. In contrast to conventional methods, the novel reliability approach does not invoke a multidimensional reliability function in the Monte Carlo numerical simulation case. As demonstrated in this study, it is also possible to accurately anticipate the likelihood of a system failure in the case of a measurable structural reaction. failure probability, a better design or control mechanism for the FWT might be created and implemented. The best wind turbine parameters could prevent mechanical damage. Effective component sizing will also be made feasible by precisely predicted extreme loads.
Application of the multivariate extreme value theory is important from a practical standpoint since there is a link between bending moments at various structural support points. The capacity to effectively use a small nonstationary data set and subsequently estimate the likelihood of extreme events is a big requirement in new statistical approaches. This study promotes a unique Monte Carlo (MC) statistical method that may naturally address underlying nonlinear effects. By comparing it to methods that have been previously benchmarked in various applications, the unique methodology presented in this study has been verified.  Note that the novel Gaidai-Fu-Xing (GFX) method is MDOF (multi-degree of freedom), while average conditional exceedance rate 2 dimensional (ACER2D) is only limited to 2 degrees of freedom (2DOF) (two dimensions); moreover, ACER2D does not provide confidence intervals (CIs), while the new GFX method can.

10-MW RWT
The DTU 10 MW Reference Wind Turbine is used in this investigation (RWT). The 10 MW RWT was created by DTU Wind Energy and Vestas Wind System as a part of the Light Rotor project. DTU started the Light Rotor project to provide a design framework for upcoming 10 þ MW wind turbine improvements. The DTU 10 MW RWT, which is utilized as the powertrain design in this study, was created by scaling the national renewable energy laboratory (NREL) 5 MW RWT, which includes a medium-speed motor and an efficient, lightweight rotor. Since we required a well-known and approved turbine for our reference, we started our inquiry with the DTU-10 MW RWT. The main design element of the 10 MW DTU wind turbine is displayed in Table 1.

OO-Star Semisubmersible Wind Floater and Mooring System
In the LIFES 50þ project, Dr. Techn. Olav Olsen AS built the semisubmersible floating substructure that supports the 10 MW RWT. [44] A center column and three lateral columns with cylindrical top portions and tapering bottom sections each make up the floating substructure. The slab is linked at the pontoon's base, where the four columns are positioned on a three-legged star-shaped platform ( Figure 1).

Load Cases and Environmental Conditions
The Northern North Sea site's hindcast data from 2001 to 2010 is where the wind and wave information used in this paper's analysis came from. The wind and wave data from the aforementioned site are used to estimate the environmental conditions, and a combined wind and wave distribution was produced using a 1 h mean wind speed at 10 m above sea level (U 10 ), significant wave height (H S ), and peak period (T p ) f U 10 , H S , T p ðu, h, tÞ ¼ f U 10 ðuÞ ⋅ f H S jU 10 ðhjuÞ ⋅ f T p jU 10 , H S ðtju, hÞ. Table 2 outlines the three real load scenarios that were considered for this research and have a high possibility of occurring in typical operating circumstances. The operating ranges of the turbines, which fluctuate between cut-in and cut-out zones with a bin size of 4 m s À1 increments, are the basis for the mean wind speed utilized in this article.
The vertical wind shear is generally described using a simple wind power law UðzÞ ¼ U ref being wind speed at level z; U ref being wind speed at a given reference height; Z ref being reference height; α being empirical wind shear exponent (power law exponent is given as α = 0.14 m for all wind speeds). For operational circumstances, 10 random samples of wind and wave are typically used for each sea state, but 20 random seeds of wave and wind are utilized to estimate severe wind and wave situations accurately. All simulations are performed for 4000 s, removing the 400 s beginning transient period, to produce a simulation that lasts 1 h.
For numerical simulation, only three particular load case scenarios were adopted, and twenty 1 h samples were adopted to represent these operational load cases. The latter may be regarded as a short-term statistical analysis. For long-term statistical analysis, one would require a full in situ load cases scattered diagram, covering a large number of environmental states and thus requiring substantial computational effort. The reason for limiting this study to three load cases, as in Table 2, was that the authors aimed to demonstrate novel reliability techniques, not to deliver long-term technical analysis. Regarding the time length of each simulation, namely twenty 1 h samples, authors have halved this simulation length and found statistical prediction still contained within a 95% CI, predicted based on the complete non-reduced time series data set.

Experimental Section
Consider an MDOF structure subjected to random ergodic environmental loadings (stationary in time), for example, from the surrounding waves and wind. The other alternative is to view the process as being dependent on specific environmental parameters whose variation in time may be modeled as an ergodic process on its own. The MDOF dynamic response vector process RðtÞ ¼ ðX ðtÞ, YðtÞ, ZðtÞ, : : : Þ is measured and/or simulated over a sufficiently long time interval ð0, TÞ. Unidimensional global maxima over the entire time span ð0, TÞ are denoted as X max ZðtÞ, : : : . By sufficiently long time T, one primarily means a large value of T with respect to the dynamic system autocorrelation time.
Let X 1 , : : : , X N X be consequent in time local maxima of the process XðtÞ at monotonously increasing discrete time instants t X 1 < : : : < t X N X in ð0, TÞ. The analogous definition follows for other MDOF response components YðtÞ, ZðtÞ, : : : with Y 1 , : : : , Y N Y ; Z 1 , : : : , Z N Z , and so on. For simplicity, all RðtÞ components, and therefore its maxima are assumed to be non-negative. The aim is to estimate the system failure probability being the probability of non-exceedance for response components η X , η Y , η Z ,… critical values; ∪ denotes logical unity operation «or»; and p X max T , Y max T , Z max T , : : : being joint probability density of the global maxima over the entire time span ð0, TÞ.
In practice, however, it is not feasible to estimate the latter joint probability distribution directly p X max T , Y max T , Z max T , : : : due to its high dimensionality and available data set limitations. In other words, the time instant when either X ðtÞ exceeds η X , or YðtÞ exceeds η Y , or ZðtÞ exceeds η Z , and so on, the system being regarded as immediately failed. Fixed failure levels η X , η Y , η Z , … are of course individual for each unidimensional response component of RðtÞ. X max N X ¼ max fX j ; j ¼ 1, : : : , T , and so on, see (Naess and Gaidai, 2009; Naess and Moan, 2013).
Next, the local maxima time instants ½t X 1 < : : : < t X N X ; t Y 1 < : : : < t Y N Y ; t Z 1 < : : : < t Z N Z in monotonously non-decreasing order are sorted into one single merged time vector t 1 ≤ : : : . In this case, t j represents local maxima of one of MDOF system response components either X ðtÞ or YðtÞ, or ZðtÞ, and so on. That means that having RðtÞ time record, one just needs continuously and simultaneously screen for unidimensional response component local maxima and record its exceedance of MDOF limit vector ðη X , η Y , η Z , : : : Þ in any of its components X , Y, Z, : : : . The local unidimensional response component maxima are merged into one temporal non-decreasing vectorR ¼ ðR 1 , R 2 , : : : , R N Þ in accordance with the merged time vector t 1 ≤ : : : ≤ t N . That is to say, each local maxima R j is the actual encountered local maxima corresponding to either X ðtÞ or YðtÞ, or ZðtÞ, and so on. Finally, the unified limit vector  Figure 1. OO-Star Wind Floater Semi-10-MW concept. [44] www.advancedsciencenews.com www.advenergysustres.com Adv. Energy Sustainability Res. 2023, 4, 2200177 ðη 1 , : : : , η N Þ is introduced with each component η j is either η X , η Y , or η Z , and so on, depending on which of X ðtÞ or YðtÞ, or ZðtÞ, etc., corresponding to the current local maxima with the running index j. Next, a scaling parameter 0 < λ ≤ 1 is introduced to artificially simultaneously decrease limit values for all response components, namely the new MDOF limit vector ð η λ X , η λ Y , η λ z , : : : Þ with η λ X ≡ λ ⋅ η X , ≡λ ⋅ η Y , η λ z ≡ λ ⋅ η Z , … is introduced. The unified limit vector ðη λ 1 , : : : , η λ N Þ is introduced with each component η λ j is either η λ X , η λ Y , or η λ z , and so on. The latter automatically defines probability PðλÞ as a function of λ, note that P ≡ Pð1Þ from Equation (1). Non-exceedance probability PðλÞ can be now estimated as follows.
In practice, dependency between neighboring R j is not always negligible; thus, the following one-step (called here conditioning level k ¼ 1) memory approximation is introduced for 2 ≤ j ≤ N (called here conditioning level k ¼ 2). The approximation introduced by Equation (4) can be further expressed as where 3 ≤ j ≤ N (will be called conditioning level k ¼ 3), and so on. The motivation is to monitor each independent failure that happened locally first in time, thus avoiding cascading local intercorrelated exceedances. Equation (5) presents subsequent refinements of the statistical independence assumption. The latter type of approximation captures the statistical dependence effect between neighboring maxima with increased accuracy. Since the original MDOF response process RðtÞ was assumed ergodic and therefore stationary, the probability p k ðλÞ≔ProbfR j > η λ j j R jÀ1 ≤ η λ jÀ1 , R jÀkþ1 ≤ η λ jÀkþ1 g for j ≥ k will be independent of j but only dependent on conditioning level k. Thus non-exceedance probability can be approximated as in the Naess-Gaidai method, 0-0, where P k ðλÞ % exp ðÀN ⋅ p k ðλÞÞ , k ≥ 1 (6) Note that Equation (6) follows from Equation (1) by neglecting ProbðR 1 ≤ η λ 1 Þ % 1, as the design failure probability is usually very small. Further, it is assumed N "k.
Equation (5) is similar to the well-known mean up-crossing rate equation for the probability of exceedance. There is obvious convergence with respect to the conditioning parameter k.
Note that Equation (6) for k ¼ 1 turns into the quite well-known non-exceedance probability relationship with the mean up-crossing rate function where ν þ ðλÞ is the mean up-crossing rate of the response level λ for the aforementioned assembled nondimensional vector RðtÞ assembled from scaled MDOF dynamic system response . In the aforementioned analysis, the stationarity assumption has been used. The proposed methodology can also treat the nonstationary case. An illustration of how the methodology can be used to treat nonstationary cases is provided. Consider a scattered diagram of m ¼ 1, .., M environmental states, each short-term environmental state having a probability q m , so that P M m¼1 q m ¼ 1. The corresponding long-term equation is then where p k ðλ, mÞ being the same function as in Equation (7) but corresponding to a specific short-term environmental state with the number m ( Figure 2).  The previously presented p k ðλÞ functions are often regular in their tail, that is, extreme values of λ approaching extreme level 1. More precisely, for λ ≥ λ 0 , the distribution tail behaves like expfÀðaλ þ bÞ c þ dg with a, b, c, d being fitted constants for appropriate tail cut-on λ 0 value. Optimal values of the parameters a k , b k , c k , p k , q k may be determined using a sequential quadratic programming (SQP) technique implemented in NAG Numerical Library. [45]

Results
The method for calculating the extreme loads under operating conditions for the 10 MW DTU WT-OO-Star is presented in this study. A FAST model-based precise numerical simulation served as the foundation for the empirical data.
The loads at the two places, as in Figure 3, are considered. These are the blade 1 root flapwise bending moment (RootMyb1) and tower bottom fore-aft bending moment (TwrBsMyt).
The proposed methodology provides an accurate bivariate extreme value prediction, utilizing all available data efficiently.
The described approach may be used at the design stage of a large FWT to define FWT parameters that would minimize extreme loads and potential damages. Based on the overall performance of the proposed method, it was concluded that the GFX method could incorporate environmental input and provide a more robust and accurate bivariate prediction based on proper numerical simulations.
This section presents statistical analysis results for M 1 and M 3 bending moments. The focus is on accurately predicting extreme responses, which is vital for safety and reliability at the design stage. The conditioning level k is set to be 5, as it was observed that ACER functions have converged at that level in the distribution tail. Figure 4 left presents responses M 1 and M 3 . Figure 4 right presents the phase space for responses M 1 versus M 3 .
This section illustrates the GFX method's efficiency by application to WFT bending moments data set. Two different WFT bending moments, M 1 and M 3 , were chosen as components X, Y, thus constituting an example of 2D dynamic system. To unify both measured time series X, the following scaling was performed according to Equation (11), making both responses nondimensional and having the same failure limit equal to 1. Next, all local maxima from two measured time series were merged into one single time series by keeping them in time non-decreasing orderR ¼ ðmaxfX 1 , Y 1 g, : : : , maxfX N , Y N gÞ.
To unify both two measured time series X , Y, the following scaling was performed Figure 5 presents ACER2D bivariate contours for WFT bending moments. It is seen from Figure 5 left that ACER2D fits different Gumbel copula to the measured data, and there is an inherent mismatch with original data (indicated by red dots) due to a particular copula choice; for more details on the ACER2D method, see Refs. . The bivariate nondimensional failure point indicated by the star in Figure 5 was chosen purely as an example. The probability level p ¼ 10 À3 corresponding to this contour line was then compared with GFX method estimate. It was found that ACER2D probability level estimate fell outside 95% CI, predicted by the GFX method, meaning that GFX is more accurate than ACER2D.   It is noted that for the prediction of failure probability of the FWT, it is important that all load cases are taken into considerations. This includes the survival under extreme weather load cases which are not illustrated in this paper. Since this paper aimed only at illustration and validation of the novel reliability technique; therefore, the authors purposely kept it short and clear to highlight the novelty of the advocated methodology and not so much focus on numerous technical details. The suggested methodology has a general purpose and is not limited to the dynamic system studied here.

Conclusions
Traditional dependability techniques that deal with time series do not have the benefit of effectively dealing with highly dimensional systems and cross-correlation between various system responses. The methodology's main benefit is the capacity to analyze the dependability of high-dimensional dynamical systems.
In this research, simulated WFT dynamic response and synthetic wind speed data set were both evaluated. The suggested method's theoretical justification is explained in depth. It should be noted that while using direct measurement or MC simulation to analyze the reliability of dynamic systems is appealing, the complexity and high dimensionality of dynamic systems necessitate the development of novel, accurate, and robust techniques that can handle the available data while utilizing it as effectively as possible.
The approach described in this study has already been shown effective when applied to various simulation models, but only for 1D system responses. In general, exact predictions were made. This work focused on a general-purpose, reliable, and user-friendly multidimensional dependability approach.
To sum up, the recommended technique may be used in various engineering fields. By no means does the given naval architecture example restrict the potential applications of the new methodology.