Deconvolution approach for floating wind turbines

Green renewable energy is produced by floating offshore wind turbines (FOWT), a crucial component of the modern offshore wind energy industry. It is a safety concern to accurately evaluate excessive weights while the FOWT operates in adverse weather conditions. Under certain water conditions, dangerous structural bending moments may result in operational concerns. Using commercial FAST software, the study's hydrodynamic ambient wave loads were calculated and converted into FOWT structural loads. This article suggests a Monte Carlo‐based engineering technique that, depending on simulations or observations, is computationally effective for predicting extreme statistics of either the load or the response process. The innovative deconvolution technique has been thoroughly explained. The suggested approach effectively uses the entire set of data to produce a clear but accurate estimate for severe response values and fatigue life. In this study, estimated extreme values obtained using a novel deconvolution approach were compared to identical values produced using the modified Weibull technique. It is expected that the enhanced new de‐convolution methodology may offer a dependable and correct forecast of severe structural loads based on the overall performance of the advised de‐convolution approach due to environmental wave loading.


| INTRODUCTION
A significant amount of the world's energy demands might be met by wind energy, a substantial ecologically benign renewable energy source. Offshore wind farms are commonly constructed to harness plentiful wind energy and produce power. Due to the fact that offshore wind speeds are frequently higher than onshore wind speeds, the floating offshore wind turbine (FOWT)'s contribution to energy generation is essential for the sector.
Predicting FOWT design loads may be done largely in one of the two methods. The recommended design procedure might assist in choosing the ideal wind turbine characteristic values, thereby reducing the risk of structural damage to FOWTs: (a) prediction of severe events and corresponding system responses with a low probability of occurrence, resulting in excessive structural loads and reactions; and (b) extrapolation of system responses and load levels toward cricilevels, while system being simulated/measured under typical operating environmental circumstances. [1][2][3][4] Both strategies are advised by the IEC 61400-1 standard. [5][6][7] The second approach (b), which is elaborated in this research study, promotes techniques that are already shown to be effective for a range of maritime constructions, including various offshore platforms and vessels. [8][9][10][11][12][13] There are numerous recently published research studies related to FOWT reliability aspects. 14 In Zhang et al. 15 authors studied dynamic system fault tree analysis to evaluate quantitatively FOWT failure rates. In Li et al. 16 authors utilized Bayesian Network to analyze FOWT reliability function and reported that the results were in good agreement with the available verified data. In Xu et al. 17 authors have proposed a multidimensional reliability approach to assess site-specific FOWT environmental loads. In Song et al. 18 authors conducted dynamic reliability structural analysis, given FOWT subjected to wind-wave joint excitations, using the so-called probability density evolution method. In Liu et al. 19 authors have used the intelligent teaching-learning-based optimization technique to assess FOWT mooring system reliability functions. In Sultania and Manuel 20 authors have utilized three-dimensional (3D) inverse first-order reliability method to estimate 50-year return period longterm loads acting on FOWT. In Zhang et al. 21 authors used the fuzzy set theory to handle FOWT failure statistics, reported results showed advantages; for example, failure rate reduced errors.
This study uses unique de-convolution to boost the effectiveness of using measured or simulated data. To characterize the tail behavior of the extreme value distribution, the structural load data that are now available are paired with an appropriate class of parametric functions. Then, a comprehensive methodology to estimate extreme values is devised that not only just relies on an asymptotic distribution (peak over threshold) but also is independent of conventional methods like Gumbel, Pareto, Weibull, and POT. With much fewer simulations and observations than the direct MC technique, the recommended MC-based approach for engineering design has the benefit of producing predictions of severe load and response that are comparable to accuracy. Figure 1 left shows the National Data Buoy Center's (NDBC) National Oceanic and Atmospheric Administration (NOAA) database that was used in this study. The Cape Elizabeth location's accessible in situ measured hourly historical metocean data, gathered between 2010 and 2017, have been used to calculate joint wind-wave statistics. 22 The flow chart for the described long-term MC-based statistical/reliability study is shown in Figure 1 right.
The authors use the term "environmental sea state" to refer to the whole set of locally accounted environmental factors, such as wave height and wind speed, that are included in the in situ environmental sea state, see International Electrotechnical Commission and colleagues [23][24][25][26][27][28][29][30][31][32][33] for current studies on FOWTs and their engineering dependability.

| METHOD
Extreme value prediction problems in engineering are frequent and challenging, especially when the available data set is limited. [34][35][36][37][38] Let's take a look at a stationary (ideally ergodic) stochastic of either the load or the response process X t ( ), which may be described as the sum of two distinct stationary component processes X t ( ) 1 and X t ( ) 2 X t X t X t ( ) = ( ) + ( ). 1 2 (1) It should be noted that this work promotes a general method that may be used to predict extreme values for a F I G U R E 1 Left: Typical data measurement buoy. 11 Right: Flow chart for long-term environmental statistical/reliability analysis. LIU ET AL.
| 2743 variety of loads and reactions for different ships and offshore structures. One can derive PDF p X for the process of interest load/response X t ( ) in two distinct ways: A) With the time series X t ( ) as the available data set, direct de-convolution is used to estimate p X A , B) Separate PDF extraction is performed from the corresponding process components X t ( ) 1 and X t ( ) 2 , namely p X 1 and p X 2 before convolution is applied Target PDF p X estimates both p X A and p X B . Approach (A) is more straightforward to use; however, (B) would yield a more accurate target PDF p X . Convolution has the distinct advantage of enabling direct extrapolation of the empirical PDF p X A toward target design probability levels without assuming any extrapolation parametric or functional class, such as from the family of generalized extreme value distributions (GEV), which is necessary to extrapolate PDF tail. It should be emphasized that the majority of extrapolation techniques that are often used in offshore engineering practice do, in fact, rely on assuming certain extrapolation parametric/ functional classes. [39][40][41] Among the widely used techniques, the techniques currently in use are peak over the threshold (POT), 31 Pareto, modified Weibull method, [42][43][44][45][46][47] bivariate modified Weibull, 48,49 traditional Weibull fit, and Gumbel fit; these are just a few examples of fitting techniques. In the simplest instance, PDFs pX1 and p X 2 may represent two identically distributed processes, X t ( ) 1 and X t ( ) 2 , with p p = X X 1 2 . The alternative (A) scenario, in which processes X t ( ) 1 and X t ( ) 2 have similar distributions, is the subject of this study. Therefore, the objective of the current study would be to locate component PDF p X 1 such that it can produce a directly calculated PDF p X as in instance (A) restricting our investigation to a single deconvolution illustration to illustrate the later idea of improving a given empirical PDF, p X , by robustly estimating the unknown PDF, p X 1 . Convolution of the two vectors, u and v, occurs at the area where the vectors' constituent parts (supports) overlap, with vector v gliding over vector u. Convolution is analogous to multiplying two polynomials, whose coefficients are parts of u and v, in algebra. Let w be a vector of length m n + − 1, with the k-th element being m n + − 1, and n be a vector of All j values that produce acceptable subscripts for u j ( ) and v k j ( − + 1), particularly j k n = max(1, + 1 − ): k m 1 : min ( , ), are summed. The main scenario in this section is when m n = , and one may obtain Having established u = v = u u n ( (1), …, ( )), one may now infer from Equation (4) that one obtains increasingly lower amounts of the w --components w n w n ( + 1), …, (2 − 1) as the index increases from n + 1 to n 2 − 1. The latter, which is twice as long as the original u -distribution support domain and doubles the length of the p X distribution support, x X · = L , with ∆x being a constant discrete distribution bin width, clearly extends vector w into a wider support domain. The empirical target PDF p X is discretely represented by the vector w = w w n ( (1), …, ( )), where n is the length of the distribution support, X [0, ] L , and this research study is limited to the situation of non-negative valued one-sided random variables, that is,  X 0, for simplicity. In this study, only the same distribution scenario-that is, the case when Equation (4) holds equality and u = vwill be investigated. Vectors w and u are represented by the PDFs p X and p X 1 in Equation (2), respectively. The unknown components u = v = u u n ( (1), …, ( )) may be estimated progressively, starting with the first component u w 2 (1) , and so on, until the last one u n ( ), according to Equation (4), given the values of w = w w n ( (1), …, ( )). In this article, the authors promote a straightforward linear extrapolation of the self-deconvoluted vector u u n ( (1), …, ( )) toward u n u n ( ( + 1), …, (2 − 1)); in other words, the PDF tail of p X 1 will be linearly extrapolated within the support range X X ( , 2 ) L L . Given that PDF p X 1 is a discrete representation of the associated estimated vector u, it is possible to refer to it as a deconvoluted PDF. Using Equation (3), the initial vector w will be doubled in length and projected into the PDF support domain, resulting in a p X support length as the original PDF support length ∆ n x X · = L . Since the original (raw) PDF tail, calculated by MC simulations or observations, p X is typically not smooth, authors suggest smoothening the original PDF p x ( ) X tail by interpolation, as cumulative density function (CDF) tail being more regular for higher tail values x. The modified Weibull approach has been used; for  x x 0 , the PDF tail behaves very similarly to , , , are appropriate constants fitted for the appropriate x 0 , see Equations (6) and (7). Authors have used linear tail extrapolation of p X 1 since it is an objective, numerically more stable choice. Biases and assumptions are commonly used in nonlinear extrapolation approaches.
The assumptions of the underlying data's stationarity, ergodicity, quality, and sufficiency are identical to the typical restrictions of any extrapolation approach of this type and apply to the recommended strategy as well. As previously mentioned, the PDF/ CDF distribution tail may be extrapolated using the de-convolution extrapolation technique without the need for a specific extrapolation functional or parametric class. Since projecting exceedance probability is crucial in the majority of reliability analysis engineering applications, 1 − CDF extrapolation is required rather than marginal PDF. The complementary cumulative density function 1 − CDF will thus be denoted in this research using the same notation as the marginal PDF, f X . A portion of the original data set (called here "shorter" data set) has been extrapolated to validate the extrapolation procedure indicated above, and estimations based on the entire (called here "longer") data set are compared with the extrapolated data. Therefore, the purpose of this work is to demonstrate the effectiveness of the recommended extrapolation technique over at least a few orders of magnitude.
In contrast to the marginal PDF, where one may utilize 1-CDF and then use integration to create a new, smoother CDF, an iterative approach may be used, as the previous manner of explanation demonstrates. To estimate the deconvoluted 1-CDF distribution f X 1 given an empirical distribution f X , described in the preceding section, the discrete convolution approach, or rather deconvolution, has been built on sequentially solving Equation (4). As was previously predicted for the empirical parent PDF/CDF distribution f X , the resultant deconvoluted vector u = u u n ( (1), …, ( )) components u j ( ) are often monotonously decreased with increasing index j. Some of the final values of the resulting vector u, such as u n L u n ( ( − ), …, ( )), may go negative for some L n < . The pivot value is the lowest positive value f L of a particular distribution tail of f X . The scaling is then a linear transformation on a decimal-log scale along the PDF's vertical y-axis with the reference level f L being constant and g x ( ) X being a scaled log 10 version of the empirical base distribution f X . To prevent the occurrence of negative components in the resultant f X 1 , the scaling coefficient μ may be selected. When f f f = conv( , ) X X X 1 1 as in Equation (2) was used to get f X 1 , the original scale was restored by performing an inverse scaling μ −1 with f̃X being the target extrapolated version of f X .

| Modified Weibull extrapolation
We now include a comment on the "shorter" data record PDF/CDF distribution tail f X interpolation problem. The latter interpolation was required since the empirical PDF f X is frequently an inappropriate input for Equation (4) due to its inherently extremely irregular tail section. Because of this, a straightforward modified Weibull (Naesss-Gaidai) extrapolation form has been used with the probability distribution tail (x x > 0 ) becoming preasymptotically regular at the beginning of the extrapolation tail area, where x 0 serves as an appropriate tail marker. There are several methods to construct the weight function h, such as h is the confidence interval (CI), which is experimentally calculated using simulated or observed data.
When performing modified Weibull multiparameter fit, such as parametric nonlinear extrapolation, room for error and prediction instability can be significant due to variability in estimated model parameters and extrapolation's nonlinearity. This is one advantage of the proposed methodology over other extrapolation methods.
Without the use of any presumptions, linearizations, or other oversimplifications, the environmental data from the monitored buoys were post-processed into the empirical multidimensional joint probability distribution function (PDF). This study used the power law formula to extrapolate wind speed, where U z ( ) and U z ( ) r represent the wind speed at height z and the reference wind speed at height z r , respectively. Surface roughness length is denoted by z 0 , and the power law constant is equal to α = 0.14. The in situ metocean data were then used to quickly estimate the joint PDF p U H T ( , ) s p , which produced a three-dimensional (3D) dispersed diagram with H s and T p standing for significant wave height and peak-spectral period, respectively. This strategy promotes the direct long-term MC simulation method, 40,[42][43][44][45][46]48,49 which has the benefit of not relying on any assumptions or simplifications. A semisubmersible FOWT model with one main column and three outer offset columns is shown in Figure 2.
The primary structural dimensions of the semisubmersible FOWT are shown in Table 1. The term "center of mass" (CM).
On top of the DeepCwind platform that was partially submerged, the 5-MW NREL baseline wind turbine was built. To correlate the relevant aerodynamic and gravitational FOWT stresses with its in situ structural dynamics, FAST and AeroDyn were used. 50 The validation of floating offshore wind turbine modeling methods using experimental data had been the subject of substantial experimental work. 2,3,33 For this work, the aero-hydro-servo-elastic simulation code OpenFAST 25 was used. TurbSim, 27 with generated random wind fields on a 31 × 31 square grid having 145 m width, using the Kaimal spectral and exponential coherence models. When using the blade element momenta approach and taking into account rotor-wake effects, dynamic stall, and baseline responses, the Open-FAST code module AeroDyn is adequate to describe FOWT aerodynamics. The motion equations of the coupled rigid-flexible system have been solved to determine the structural dynamic responses in the time domain. The Kane technique was used to create these equations of motion, using HydroDyn, 28 incorporating potential flow theory and Morison's equation for largediameter constructions, to estimate hydrodynamic stresses. Potential flow theory has been used to forecast hydrodynamic coefficients in the frequency domain, such as additional mass and potential damping coefficients. To account for viscous drag forces occurring on FOWT, Morison's formulation incorporated a drag force component. The MoorDyn mooring module, which is based on the lumped mass theory, is used to represent the three catenary mooring lines of the NERL 5 MW semisubmersible FOWT. For extrapolation to predict ultimate loads with a desired return time of 50 years, at least 15 F I G U R E 2 Left: DeepCwind semi-submersible floating offshore wind turbine (FOWT) platform 1/50 scale model. 30,32 Right: An example of operating FOWT. The fore-aft bending moment average value of the tower base is larger than zero as a result of the aerodynamic and hydrodynamic loadings. When subjected to large in situ wind forces, excessive structural loads are more likely to reach a particular, extreme value. Additionally, an apparent fluctuation component has been noted; hence, strong structural stresses are produced by aerodynamic and hydrodynamic forces. Extensive experimental work has been done within the context of the OC3 projects, [26][27][28][29][30][31][32][33] to provide experimental data and test FOWT numerical modeling methodologies. The modeling skills of OpenFAST, the numerical simulation tool chosen for this investigation, have been confirmed by the numerical and experimental results.

| RESULTS
This work presents the methodology for computing the FOWT severe bending moment response. The groundbreaking de-convolution method has been described in previous sections. The recommended method efficiently uses all the available information to precisely predict extreme values. Based on the overall effectiveness of the suggested strategy, it was found that the novel deconvolution methodology could include environmental information and provide more accurate forecasts. [36][37][38][51][52][53][54][55][56] Figure 3 shows the final un-scaled results of the proposed de-convolution technique in this paper, namely the "shorter" decimal log scale f X PDF tail, extrapolated by de-convolution, along with "longer" data distribution tail and modified Weibull Naess-Gaidai (NG) extrapolation. Equidistant sampling was utilized to reduce the size of the "shorter" data collection by 50 times compared to the entire "longer" simulated data set. Figure 4, similar to Figure 3, presents the final unscaled results of the proposed de-convolution technique in this paper, namely the "shorter" decimal log scale f X PDF tail, extrapolated by de-convolution, along with "longer" data distribution tail and modified Weibull (NG) extrapolation. Different wind speeds U have been combined according in situ wind speed probabilities scatter diagram, following the long-term analysis flowchart Figure 1 right, to obtain realistic extreme response predictions. In Figures 3 and 4, red arrows indicate the directional difference between "shorter" and "longer" data sets, indicating that proposed deconvolution technique yields more accurate results than the modified Weibull fit.
Equidistant sampling was utilized to reduce the size of the "shorter" data collection by 50 times compared to the entire "longer" simulated data set. It should be emphasized that it is hard to come to a firm judgment on the precision of the suggested de-convolution approach using the FOWT response data set; yet, it can be seen from Figures 3 and 4 that the proposed method agrees well with the modified Weibull method, being based on the "shorter" data set, and delivering distribution quite close to the one based on the "longer" data set. It is also seen from Figures 3 and 4 that the proposed de-convolution technique performs slightly better than modified Weibull fit. It has been suggested to estimate the FOWT characteristic design values using a unique de-convolution technique. This study analyzed FOWT anchor tensions, as well as its structural bending moments, occurring as a result of in situ environmental loads.
The described technique has the following advantages: • Various datasets, such as those that are numerically simulated or quantitatively measured, may be explored. • Unlike other techniques like three-parameter Weibull, modified Weibull, Gumbel, and POT, the proposed strategy does not depend on any pre-assumed functional family to perform reliable distribution tail extrapolation.
Additionally, it should be kept in mind that the offered technique may have technical benefits beyond simply anticipating severe FOWT responses. The reason for better performance of the suggested deconvolution technique is that, opposite to the modified Weibull method, the proposed deconvolution method is less reliant on preassumed functional class, highly nonlinear, and potentially unstable in extrapolation.
A major limitation of the advocated approach lies within the system stationarity assumption, which is reasonable within short-term 3 h stationary sea state conditions, often used in offshore engineering, but may not be the case for long-term analysis. Dynamic FOWT systems may become nonstationary in the presence of an underlying trend; for example, system degradation due to corrosion or fatigue damage. If the latter is the case, one should first identify and subtract the underlying trend and only then use the methods advocated in this study. This study advocated a general-purpose reliability approach, being well suitable not only for wave-and wind-induced loads but for any combination of in situ environmental loads.

DATA AVAILABILITY STATEMENT
Data will be available on request. For environmental data, see [23].