Novel multivariate design concept for floating wind turbines by Gaidai multivariate reliability method and deconvolution scheme

Abstract

Development of novel risk and reliability assessment methods is intended to support safer construction of offshore structures, subjected to environmental wave loads. Current study investigated 10-MW FWT (i.e., Floating Wind Turbine), operating under realistic environmental conditions. While increasing operating safety, enhanced risk and reliability assessment methods may eventually help reduce manufacturing and maintenance costs. Excessive structural dynamics being usually caused by environmental stressors, acting on structural system. Environmental loads resulting from ambient wind and wave motions are typical for offshore structures. Current work advocates a novel risk and reliability assessment methodology that allows for reliable forecasting of failure/damage risks, arising from excessive FWT structural dynamics. Recently developed Gaidai multivariate reliability methodology along with state-of-the-art deconvolution method had been employed. Unlike existing reliability approaches such as Weibull-type, GP (i.e., Generalized Pareto), POT (i.e., Peaks Over the Threshold), etc., the recommended methodology does not rely on any pre-assumed functional class, when extrapolating failure probability functional tail. Practical advantages of the suggested multivariate reliability methodology combined with deconvolution scheme over, that is, 4-parameter Weibull’s extrapolation method had been demonstrated. Suggested methodology makes effective use of even limited underlying datasets, enabling robust and accurate projections of multidimensional structural system failure/damage risks. Overall methodological performance suggests that numerically stable and accurate extreme dynamics forecasts for FWT structural bending moments might be obtained, utilizing suggested multivariate reliability methodology. Deconvolution extrapolation approach being more numerically stable than parametric extrapolation techniques, due to its non-parametric nature.

Keywords

Floating wind turbine deconvolution wind power green energy reliability

Introduction

Modern FWT systems assist reaching 2050 net-zero emissions targets.¹ The International Electrotechnical Commission (IEC) has released guidelines stating that FWTs have to be robustly constructed to safely function for at least 10 years under in situ ambient wind-wave conditions.² Five distinct types of operational FWTs are shown in Figure 1. State of the art deconvolution extrapolation scheme has been presented in the current work, suitable for nonlinear structural extreme dynamic analysis. For 10 MW FWT, the recommended multivariate reliability methodology, coupled with deconvolution scheme allows for the reliable yet efficient estimates of excessive structural loadings/responses, given in situ environmental conditions. Utilizing measured or MC (i.e., Monte Carlo) numerically simulated datasets, deconvolution enables precise structural dynamics PDF (i.e., Probability Density Function) tail extrapolation, offering a novel non-parametric method to evaluate design/characteristic values.^4–15 Deconvolution approach does not rely on any asymptotic-type PDF class, as it does not assume that the underlying distribution is of the GEV (or Generalized Extreme Value) type. When structural dynamic FWT system is not yet at the asymptotic level, the suggested deconvolution technique might outperform. For examination of offshore wind turbines, fastened to the seabed, see Refs 16–21. For using probabilistic approaches to extrapolate extreme and fatigue loads, see Refs 22–27. For modified Weibull-type extrapolation technique, see Refs 28–33.

Figure 1.

Different FWT types. Only semi-submersible (second from the left) type is considered.³

Due to FWT structural failures/damages risks, caused by excessive in situ environmental loadings, DNV (Det Norske Veritas) safety standards have being introduced.³⁴ Areal/sectional extremes can be modeled using, for example, MSP (Max-Stable Processes) theory.³⁵ If data are gathered within a specific geographic or structural geometrical domain, MSP models can be fitted to estimate areal exceedances, while taking into consideration spatial interdependence of the extremes. MSP being theoretically based on the Extreme Value Theory (EVT) paradigm. The deconvolution technique that has been recommended here can be successfully incorporated into MSP, by replacing the EVT part with the sub-asymptotic distribution functional class.

Figure 2 schematically illustrates suggested long-term multivariate reliability workflow.

Figure 2.

Long-term multivariate reliability diagram.

Structural description and environmental conditions

The main FWT structural characteristics and the environmental modeling methodology are described in this Section.

FWT system’s description

The 10-MW RWT, also known as the Reference Wind Turbine, has been selected for this study, based on its compliance with the NREL 5-MW RWT standards. For conventional three-bladed, clockwise rotating upwind fixed-wing turbines, the IEC Class 1A suggests using variable speed control system and collective pitch. At DTU, or Danish Technical University, a number of research studies investigated 10-MW RWT numerical models.^36–39 Brief technical synopsis of DTU’s 10-MW RWT is shown in Table 1.

Table 1.

DTU’s 10-MW reference FWT key parameters.⁴⁰

Parameters	Values
Rating	10-MW
Type	Upwind with 3 blades
Cut-in, rated, cut-out windspeeds (m/s)	4, 11, 25
Min, max rotor speed (rpm)	6.0, 9.6
FWT rotor-diameter (m)	178
FWT hub-height (m)	119
FWT nacelle-mass (tones)	446
FWT tower-mass (kg)	1.3 $\cdot$ 10⁶

FWT’s floating structure of the studied semi-submersible 10-MW RWT had been built with assistance from the Olav Olsen (OO) AS LIFES 50 + project.^40,41

FWT dynamic parts

FWT structural loadings at the 2 selected measurement sites, shown in Figure 3, to be analyzed. Those are: A) $M_{1}$ -root flatwise bending moment); B) $M_{3}$ -tower bottom fore-aft bending moment).

Figure 3.

Left: 10-MW OO-Star FWT. Right: locations of selected FWT bending moments are marked with 2 stars.⁴⁰

Figure 3 left schematically presents 10-MW FWT analyzed in the current investigation.

Load cases, environmental conditions

The current study made use of ambient winds and waves, measured at in situ offshore sites in the North Sea, throughout the recent decade of 2010–2020. The joint wind-wave PDF,⁴² includes 1-hour mean windspeed U₁₀, measured at 10 m above Mean Sea Level (MSL), wave spectral peak-period T_p, and significant wave-height H_s. Joint PDF of U₁₀, H_s, and T_p is given by

f_{U_{10,} H_{s,} T_{p}} (u, h, t) = f_{U_{10}} (u) \cdot f_{H_{s} {ǀ U}_{10}} (h ǀ u) \cdot f_{T_{p} {ǀ U}_{10,} H_{s}} (t ǀ u, h)

(1)

with

f_{U_{10}} (u), f_{H_{s} {ǀ U}_{10}} (h ǀ u)

f_{T_{p} {ǀ U}_{10,} H_{s}} (t ǀ u, h)

being marginal PDF of U₁₀, conditional on U₁₀ and conditional PDF of T_p conditional on U₁₀, and H_s.

LCs and environmental conditions, such as sea current, waves, and wind loading conditions have been discussed in detail in Ye and Ji (2019).⁴³ In Selected LCs being listed in Table 2. Joint PDF, supplied by equation (1) reflects in situ representative windspeed, spectral peak-period, and significant wave-height. Proper numerical models of turbulent winds and irregular waves had been utilized. The windspeed profile can be roughly represented as follows, by using windspeed’s PL (i.e., Power-Law), formulation

U_{w} (z) = {U_{h u b} (\frac{z}{z_{h u b}})}^{α}

(2)

with U_w(z) being mean directional windspeed at the height z above SWL, and U_hub is the mean windspeed at the FWT hub-height (see Table 1), z_hub being the FWT’s hub-height above SWL at 119 m. PL exponent, represented by α, was set equal to 0.14, in accordance with IEC 61,400-3-2 standard.⁴⁴

Table 2.

Selected LCs (Load Cases) for MC simulation.

LC	$U_{10}$ (m/s)	$H_{s}$ (m)	$T_{p}$ (s)	Samples	Simulation duration
LC1	8	1.9	9.7	24	1 hour
LC2	12	2.5	10.1	24	1 hour
LC3	16	3.2	10.7	24	1 hour

Reliability methodology for series-type multidimensional structural system

Current section to present theoretical details of advocated state-of-the-art reliability methodology.

Gaidai multivariate reliability methodology for series-systems

Let’s take into consideration structural dynamic MDOF (i.e., Multi-Degree Of Freedom) FWT system’s critical/key components (i.e., response/load components), composed into vector $(X (t), Y (t), \dots)$ , being measured/simulated over a sufficiently long (representative) timelapse $(0, T)$ . 1D (1-Dimensional) FWT system’s critical/key component’s global maxima being denoted here as $X_{T}^{\max} = \max_{0 \leq t \leq T} X (t)$ , $Y_{T}^{\max} = \max_{0 \leq t \leq T} Y (t)$ , $\dots$ . By sufficiently long (representative) timelapse $T$ one means large enough duration $T$ wrt dynamic energy system auto-correlation, as well as relaxation times. Let $X_{1}, \dots, X_{N_{X}}$ be temporally consequent local maxima of dynamic system’s component process $X = X (t)$ recorded at discrete, temporally increasing times $t_{1}^{X} < \dots < t_{N_{X}}^{X}$ within $(0, T)$ . Identical types of definitions being valid for other MDOF components: $Y (t), Z (t), \dots$ , namely, $Y_{1}, \dots, Y_{N_{Y}};$ $\dots$ . For simplicity, all dynamic system component’s local maxima had been assumed to be non-negative. Next

P = ∭_{(0, 0, \dots)}^{(η_{X}, η_{Y}, \dots)} p_{X_{T}^{\max}, Y_{T}^{\max}, \dots} (x_{T}^{\max}, y_{T}^{\max}, \dots) d x_{T}^{\max} d y_{T}^{\max} \dots

(3)

Being target probability of dynamic system’s survival, and $η_{X}$ , $η_{Y}$ , ... being system components critical/hazard/limit values; $p_{X_{T}^{\max}, Y_{T}^{\max} \dots}$ being FWT system’s joint PDF of individual system component’s global maxima. Is the dynamic FWT system’s NDOF (i.e., Number of Degrees Of Freedom) being large; it may not be practically feasible to estimate directly joint PDF $p_{X_{T}^{\max}, Y_{T}^{\max} \dots}$ and hence the FWT system’s target survival chances/probability $P$ . Dynamic FWT system’s 1D key/critical components $X, Y, \dots$ being re-scaled, and then non-dimensionalized

X \to \frac{X}{η_{X}}, Y \to \frac{Y}{η_{Y}}, \dots

(4)

Making all dynamic FWT system’s critical/key components non-dimensional, having same target hazard/failure/risk limits $λ = 1$ , having target failure probability $P = P (1)$ . Survival probability $P (λ)$ being a smooth $C^{1}$ function of the non-dimensional level $λ$ . 1D dynamic system’s critical components local maxima, being merged into one temporally increasing vector $R (t) \equiv \vec{R} = (R_{1}, R_{2}, \dots, R_{N})$ according with corresponding merged system temporal vector $t_{1} < \dots < t_{N}$ , $N \leq N_{X} + N_{Y} + \dots$ . Each local maxima of $R_{j}$ have been encountered dynamic FWT system’s critical/key component’s local maxima, corresponding to either $X (t)$ , or $Y (t)$ , … or other dynamic FWT system’s critical components.

Deconvolution scheme

Current section provides a brief overview of the MC-based numerical method, used to evaluate FWT structural risks and reliability. Recently, a novel deconvolution technique has been successfully proven.^68,73,76 Assessment of excessive system dynamics is frequently a difficult design and engineering endeavor, especially when the underlying dataset is small.^{33–41,43,45–48} Therefore, from the standpoint of practical design, it is essential to create state-of-the-art, accurate, and efficient extrapolation methods. $X (t)$ is a quasi-ergodic stochastic/random process that can be represented as the total of two separate quasi-ergodic component sub-processes, $X_{1} (t)$ and $X_{2} (t)$ , that are observed, measured, or MC-simulated over a typical timelapse, $0 \leq t \leq T$

X (t) = X_{1} (t) + X_{2} (t)

(5)

Due to the generic character of advocated methodology, a wide range of naval, marine, subsea, and offshore structures may well benefit from its use, when evaluating extreme values for pertinent structural dynamic reactions and related stresses.^49–56 Two different approaches to treating the stochastic process of interest $X (t)$ may be selected to analyze target marginal PDF $p_{X}$ :

A) Deconvolution: straightforward deconvolution, assessing $p_{X} \approx p_{X}^{A}$ from the unfiltered/raw underlying dataset, that is, from timeseries $X (t)$ ,

B) Convolution: separate sub-process component’s $X_{1} (t)$ , $X_{2} (t)$ PDFs extraction, that is, $p_{X_{1}}$ , $p_{X_{2}}$ , respectively, while holding $p_{X} \approx p_{X}^{B} = conv (p_{X_{1}}, p_{X_{2}})$ .

Target PDF $p_{X}$ may be approximated in two distinct ways, that is, by the PDFs $p_{X}^{A}$ or $p_{X}^{B}$ . Case A) being straightforward—while Case B) offering potentially more precise estimate of the target PDF $p_{X}$ . Application of the convolution case B) offers an advantage of delivering direct empirical PDF $p_{X}^{A}$ assessment, without any presumptions. It should be noted that the majority of extrapolation techniques, currently in engineering/design use either presume particular GEV functional class, such as Gumbel-plot, or they involve so-called tail’s markers, such as Weibull-type, POT, and GP (i.e., Generalized Pareto).^{42,44,53,57–62} Given unknown extrapolation parameters (i.e., using parametric extrapolation), forecasts may suffer accuracy loss and numerical stability.^63–68 For applications of bivariate modified Weibull-type, Gumbel-fit, and Weibull-fit see Refs 69–78. In view of equation (5), considering the rare availability of 2 independent components, one could try to evaluate 2 independent PDF sub-components $p_{X_{1}}$ and $p_{X_{2}}$ , or, in the simplest case, find 2 identically distributed sub-processes $X_{1} (t)$ and $X_{2} (t)$ , having $p_{X_{1}} = p_{X_{2}}$ . Case A) corresponds to the situation, where there exist 2 sub-processes with identical PDFs $X_{1} (t)$ , $X_{2} (t)$ has been preferred here. Thus, the current inquiry would aim to evaluate the sub-component PDF $p_{X_{1}}$ in order to directly assess target PDF $p_{X}$ , that is, as in case A)

p_{X} = conv (p_{X_{1}}, p_{X_{1}})

(6)

Convolution of two vectors, $u$ and $v$ , given support-domain overlap, where $v$ “sliding” over $u$ . Convolution’s algebraic action is the same as multiplying 2 polynomials, with $u$ and $v$ elements serving as discrete coefficients. If $n = length (v)$ , $m = length (u)$ , then the length of the $w$ vector is $m + n - 1$ , and it $k$ -th element is equal to

w (k) = \sum_{j = 1}^{m} u (j) v (k - j + 1)

(7)

Summation performed over all legal subscripts for $u (j)$ and $v (k - j + 1)$ , that is, $\max (1, k + 1 - n) \leq j \leq \min (k, m)$ . As it had been assumed $m = n$ , one has

w (1) = u (1) \cdot v (1)

w (2) = u (1) \cdot v (2) + u (2) \cdot v (1)

w (3) = u (1) \cdot v (3) + u (2) \cdot v (2) + u (3) \cdot v (1)

(8)

\dots

w (n) = u (1) \cdot v (n) + u (2) \cdot v (n - 1) + \dots + u (n) \cdot v (1)

\dots

w (2 n - 1) = u (n) \cdot v (n)

Equation (8) highlights gradually reduced extraction of the $w$ -vector parts: $w (n + 1), . ., w (2 n - 1)$ , when $j$ -index increases from $n + 1$ up to $2 n - 1$ . The latter is done after evaluating $u = v = (u (1), . ., u (n))$ . Deconvolution grows vector $w$ towards broader support-domain, doubling the original PDF $p_{X}$ support-domain: $(2 n - 1) \cdot ∆ x \approx 2 n \cdot ∆ x = 2 X_{L}$ , wrt the initial PDF’s support length of $n \cdot ∆ x = X_{L}$ , where $∆ x$ being constant bin-length. With $n$ being the discontinuous length of the PDF’s support $[0, X_{L}]$ , Vector $w = (w (1), . ., w (n))$ may be viewed as a discrete representation of the empirical target PDF $p_{X}$ . The current study only looked at positive random variables, that is, one-sided, or $\geq 0$ . Only case A) of self-deconvolution alone will be investigated further on, that is, when equation (6) holds $u = v$ . Probability functions $p_{X}$ , $p_{X_{1}}$ can be PDFs, represented in a discrete form by 2 vectors, $w$ , $u$ , respectively, generated by utilizing equation (8). Given that $w = (w (1), . ., w (n))$ is known, equation (6) makes it clear that one may now sequentially deduce the unknown vector’s components $u = v = (u (1), . ., u (n))$ starting with the 1st component, $u (1) = \sqrt{w (1)}$ , and proceeding to the 2^nd, $u (2) = \frac{w (2)}{2 u (1)}$ , etc., until $u (n)$ . Next, linear extrapolation of the self-deconvoluted vector $(u (1), . ., u (n))$ to be performed, towards $(u (n + 1), . ., u (2 n - 1))$ ; that is, linear extrapolation of the PDF $p_{X_{1}}$ tail, within range $(X_{L}, 2 X_{L}) .$ One may refer to PDF $p_{X_{1}}$ as the “deconvoluted PDF,” having vector $u$ as its discrete form. Vector $w$ should be extended (in terms of vector’s length) and then extrapolated towards a wider PDF support-domain $2 \times$ as long, as the original PDF’s support-domain. In contrast to the original PDF’s support-length of $n \cdot ∆ x = X_{L}$ , the vector will double ( $2 \times$ ) PDF’s support-length $(2 n - 1) \cdot ∆ x \approx 2 n \cdot ∆ x = 2 X_{L}$ . The procedure for $C^{1}$ smoothing the PDF tail needs to be introduced as the numerically calculated dataset-based PDF tail being corrupted/discontinuous due to numerical errors. To $C^{1}$ -smoothen the original PDF $p_{X} (x)$ tail, it is now advised to $C^{1}$ -smooth PDF $p_{X}$ tail, using the proper interpolation. The latter is because at high PDF’s tail $x$ -values, the complementary CDF, or $\bar{CDF} \equiv$ (1-CDF), tail is more $C^{1}$ -smooth. A modified 4-parameter Weibull-type extrapolation method, had been utilized for $\geq x_{0}$ , where the PDF tail with fitted 4 parameters, $a, b, c, d$ , shows behavior similar to $\exp {- {(a x + b)}^{c} + d}$ , provided suitable PDF/ $\bar{CDF}$ tail marker-value $x_{0}$ , .^74–80 Linear extrapolation of the PDF $p_{X_{1}}$ tail had been regarded as an adequate, straightforward, yet numerically stable solution. If parametric extrapolation techniques were to be utilized in place of deconvolution scheme, additional biases and numerical instability would be introduced. The proposed deconvolution approach having basic limitations that are essentially the same for any extrapolation method: representativity, quasi-ergodicity, and quality of the underlying dataset.^81–88

Results

Deconvolution scheme to be demonstrated “in action” in the current Section, by means of damage/failure risks assessment for FWT excessive dynamic bending moments, see Figure 3. A total of 20 1-hour MC simulations had been run. The suggested methodology had been effectively utilized for available limited underlying dataset, to forecast extreme/design values with predetermined return periods. Results have indicated that the deconvolution scheme well handles nonlinear, non-stationary process dynamics, producing reliable yet accurate predictions.^89–91 For a thorough explanation of the modified 4-parameter Weibull-type extrapolation method, see⁵⁸. Figure 4 illustrates $\vec{R}$ vector, introduced in Section 3.1, composed of FWT’s 2 bending moments M₁, and M₃, see also Figure 3. Note that advocated methodology can be applied to arbitrary number of synchronous structural responses.

Figure 4.

$\vec{R}$ vector illustration.

Note that $\vec{R}$ vector, plotted in Figure 4, only combines different system’s components with potentially different dimensions, hence it does not have physical meaning, but it represents system’s dynamics.

Figure 5 (b)) displays the final, unscaled results, which are based on the deconvolution technique (blue line), along with extrapolation by modified (the 4-parameter) Weibull-type extrapolation (cyan line).^92–98 The modified 4-parameter Weibull-type extrapolation is compared with novel deconvolution scheme, as shown in Figure 5 (b))—it is seen that deconvolution scheme delivered more accurate and conservative FWT failure probability forecast (indicated by star), while 4-parameter Weibull predicted non-conservative value, lying outside 95% CI. Presented results serve as a benchmark and cross-validation for the novel deconvolution approach.^99–106 Primary benefit of deconvolution methodology over parametric extrapolation techniques is its non-parametric nature, which allows for more numerically stable extrapolation¹⁰⁷.

Figure 5.

Forecasts for FWT combined bending moments: (a) Linear extrapolation of $p_{X_{1}}$ (b) Deconvolution-based extrapolation versus 4-parameter Weibull extrapolation. Dashed lines indicate 95% CI. Failure probability $P_{F} = 1 - P$ on the decimal log scale on the vertical axis.

Discussion

Table 3 outlines design/engineering merits of the suggested Gaidai multivariate reliability and risks evaluation methodology. Existing reliability/design methods being restricted to 2D (i.e., bivariate) structural systems, while the advocated multivariate risks evaluation methodology having virtually no NDOF restrictions.

Table 3.

Gaidai multivariate risk evaluation prognostics/design benefits.

	Gaidai multivariate risk evaluation methodology	Existing 1D/2D reliability methods
Potentials for structural multivariate system prognostics	NDOF = $\infty$ D	NDOF $\leq 2 D$
Potentials to fully incorporate WFT system’s nonlinearities	Full	Partial/Full
Potentials to assess reliability of damaged, nonstationary structural systems	Full	Partial

The following are the suggested methodology’s benefits and drawbacks:

▪ The suggested structural risk assessment methodology has an advantage over existing reliability methods, as it can handle high-dimensional dynamic systems;

▪ Challenges, associated with predicting an underlying system’s trend may be seen as certain limitations of the advocated approach—however, this is common challenge for any reliability method.

The primary aim of this research has been to provide a reliable, user-friendly, multi-dimensional approach for assessing the risk of wide range of dynamic systems.

Conclusions

A novel deconvolution technique has been introduced and employed to assess target design/characteristic values of the FWT. The blade-root along with tower-bottom bending moments of FWT had been examined, under realistic in situ environmental circumstances. Various load situations had been accounted for to describe ambient wind-wave loads. A realistic illustration of the long-term multivariate reliability strategy had been provided. The advantages of the applied multivariate reliability coupled with deconvolution scheme are:

• State-of-the-art Gaidai multivariate reliability methodology has no limit on the number of structural system dimensions/components, while existing reliability methods are limited to 2D.

• Novel deconvolution extrapolation scheme being stable and independent of the distribution tail marker choice, that is, being non-parametric.

• Unlike IFORM/SORM, the novel deconvolution scheme does not discard model nonlinearities and it does not rely on EVT-assumption.

• In contrast to existing extrapolation methods, that is, Weibull-type, GP, Gumbel-type, POT, the suggested methodology does not depend on any presumptive asymptotic functional (parametric) class, to carry out PDF tail’s extrapolation.

The proposed methodology being not restricted only to the examined FWT example, hence having broad variety of potential design/engineering applications.

Footnotes

Acknowledgments

The authors declare no financial interests. This study advocates a similar methodology as already had been presented by authors in,⁸² however, floating wind turbine dynamic components, analyzed in this study are different.

Author’s contributions

All authors had contributed equally.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethical Statement

ORCID iD

Oleg Gaidai

Data availability statement

Data will be made available on request from the corresponding author.*

Appendix

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