**1. Introduction**

The offshore wind industry has experienced significant growth over the last decade [1]. As a result, the number of offshore wind turbines operating in Europe has reached 5402 in 2020 [2], with much more planned to be installed worldwide in the close future [3]. The typical lifetime of an offshore wind turbine ranges between 20 and 25 years, which means that over the coming years a large number of these structures reach their intended lifetime, and operators will have to take actions regarding their assets. Potential actions, denoted as decision models, can be to decommission, re-power, perform inspections, or extend lifetime. An optimal decision depends on what specific business model the operator pursues, but, regardless of the business aspect, an accurate and precise estimation of the structural reliability is key in making such a decision [4].

A digital twin-defined as a digital replica of a physical asset [5,6]-can help us to assess the structural integrity of existing structures more accurately and precisely compared to predictions from generic design practices because consistent and updated information of the structure is available. This has been successfully demonstrated in the oil and gas industry [7,8], in aerospace engineering [9], and in the offshore wind industry as well [10]. In fact, a number of wind standardization committees, including Det Norske Veritas (DNV) [11,12], International Electrotechnical Commission (IEC) [13], and Federal Maritime and Hydrographic Agency (BSH) [14], are working on design recommendations on how to use measurement data and inspection information to optimize decision models for existing

**Citation:** Augustyn, D.; Ulriksen, M.D.; Sørensen, J.D. Reliability Updating of Offshore Wind Substructures by Use of Digital Twin Information. *Energies* **2021**, *14*, 5859. https://doi.org/10.3390/en14185859

Academic Editor: Eugen Rusu

Received: 5 August 2021 Accepted: 9 September 2021 Published: 16 September 2021

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wind turbines. Currently, a key missing aspect is how to use the improved structural models contained in digital twins to subsequently improve the decision models.

Although fully physics-based digital twins have not yet been applied to improve decision models for wind turbines, some publications already indicate how measurement data can be used to achieve such an improvement. Nielsen and Sørensen [15] applied dynamic Bayesian networks to calibrate a Markov deterioration model based on past inspection data of wind turbine blades. Ziegler and Muskulus [16] investigated the feasibility of lifetime extension for offshore wind monopile substructures, with particular focus on identifying important parameters to monitor during the operational phase of the turbines. Leser et al. [17] presented a general framework for fatigue damage estimation based on in situ measurements. Mai et al. [18] focused on prediction of the remaining useful lifetime of wind turbine support structure joints using met-ocean in situ data. Augustyn et al. [19] extended a conceptual framework for updating decision models based on information from a digital twin, initially proposed by Tygesen et al. [7], to be applied to offshore wind substructures. In the framework, a digital twin is established with an updated structural and load model, and subsequently the digital twin is used to quantify uncertainty and update the structural reliability.

In the present paper, we outline the framework by Augustyn et al. [19] beyond its conceptual level and propose a probabilistic method for updating the structural reliability of offshore wind turbine substructures based on new information obtained from digital twins. Depending on the information type available, various methods for updating reliability can be used [20]. If information on the structural integrity becomes available, for example, by an inspection of joints to identify potential cracks, risk-based inspection methods can be applied [21–24]. Even though the inspection planning methodology is matured and well-proven in industrial applications [25], its feasibility for the majority of offshore wind applications is questionable due to the profound inspection costs [26]. A more economically feasible alternative, in the form of condition-based monitoring, is typically investigated for offshore wind applications [27,28]. In this context, condition monitoring data can be applied to identify structural damage, and then the resulting integrity information can be employed for updating reliability [29]. Application studies have been presented for mechanical components in turbine [30] and wind turbine blades [4]. However, in these studies, the condition monitoring data merely provide structural integrity information at a global level-that is, if damage is present or not. In the present study, we aim at enhancing the spatial resolution of the integrity assessment and hereby provide information at a local (joint) level. Consequently, this paper proposes a framework where condition monitoring data are used to update structural models; these updated models are subsequently used to update structural reliability, including uncertainty stemming from the updating procedure.

The contribution of this paper consists of: (1) proposing a method on how the uncertainties related to the structural dynamics and load modeling in fatigue damage accumulation can be quantified and updated based on updated distribution functions of model parameters, which can be acquired with the aid of a digital twin. Subsequently, (2) we present a framework where the updated uncertainty is used to update the structural reliability based on a well-established probabilistic model [31,32]. Generally, the framework can be used for optimization of operation and maintenance of existing turbines and design of new structures. The framework is exemplified based on two numerical case studies, in which digital twins established in previous studies by the authors [33,34] are included.

The remainder of this paper is organized as follows. In Section 2, we outline the concept of structural reliability estimation and convey the motivation for the proposed structural reliability updating framework, which is presented in Section 3. The two following sections address the numerical case studies used to exemplify the framework for existing and new substructures; Section 4 describes the setup of the case studies and Section 5 presents the appertaining results. Finally, this paper closes with concluding remarks in Section 6.

#### **2. Background and Problem Statement**

A wind turbine consists of structural components, for which reliability analysis is performed using structural reliability theory [35], and electrical/mechanical components, for which classical reliability models can be used, with the main descriptor being the failure rate or the mean time between failure (MTBF). Regardless of the component type being addressed in the reliability analysis, a probabilistic model describing the component's integrity is required. The reliability of electrical/mechanical components is typically modeled by a Weibull model for the time to failure and the components are assumed to be statistically independent. Using, for example, failure tree analysis (FTA) and failure mode and effect analysis (FMEA), system reliability models can be established and the reliability update can be performed when new information becomes available [36–38]. In the present paper, jacket-type steel wind turbine substructures are considered, so structural reliability techniques are required to model loads, resistances, and model uncertainties and to account for the correlation between the components. The fatigue damage is often design driving for the structural components of offshore wind substructures, such as joints. In this instance, fatigue damage accumulation can be expressed in terms of probability of failure or, equivalently, by the reliability index [39].

Let *<sup>g</sup>*(*t*) be the fatigue limit state at year *<sup>t</sup>* <sup>∈</sup> <sup>N</sup> for an offshore wind substructure; then [32,40],

$$\log(t) = \Delta - \sum\_{i=1}^{l} \sum\_{j=1}^{z} \frac{N\_{l,j} p\_i t}{K \Delta s\_{i,j}^{-m}} \left( X\_d X\_l X\_s \right)^m \tag{1}$$

where Δ is the fatigue resistance and the double summation expresses the accumulated fatigue damage. In particular, Δ is a stochastic variable representing the limit value of the accumulated fatigue damage estimated using, for example, SN curves, including the uncertainty related to application of Miner's rule for linear fatigue damage accumulation. In the expression for the fatigue damage, *pi* is the yearly probability of occurrence for sea state *i* (including wind and wave parameters), *Ni*,*<sup>j</sup>* is the number of cycles for the *i*th sea state and *j*th stress range Δ*si*,*j*, and *K* and *m* are the parameters related to the SN curve, with *m* being the Wöhler exponent [41]. The uncertainties related to the SN curve approach are included by modeling *K* as a stochastic variable. *Xd*, *Xl*, and *Xs* are stochastic variables that model the uncertainties associated with the structural dynamics, load modeling and stress concentration.

If *g*(*t*) ≤ 0, the limit state is exceeded and the structure fails, while *g*(*t*) > 0 implies that the structure is safe. The probability of fatigue failure in the time interval *t* ∈ [0, *T*], *Pf*(*t*) = *P*(*g*(*t*) ≤ 0) can be estimated by first-order and second-order reliability methods [39] or, as is the case in this paper, by Monte Carlo methods [42]. The corresponding reliability index, *<sup>β</sup>*, can be computed as *<sup>β</sup>*(*t*) = −Φ−<sup>1</sup> - *Pf*(*t*) , where Φ is the standard normal distribution function. The annual reliability index, Δ*β*, can be calculated analogically assuming a reference period of one year.

We note that (1) *Xd* and *Xl* may be correlated, and, in this instance, they should be modeled by a joint probability density function with correlation coefficient *ρ* and (2) a linear formulation of the limit state equation can be readily generalized for a bi-linear formulation of the SN curve. The parameters in model (1) are elaborated in Section 2.1.

#### *2.1. Uncertain Parameters and Their Modeling*

The uncertainty modeling related to structural reliability due to fatigue damage is summarized in Figure 1. In the framework proposed in Section 3, we focus on updating stochastic variables related to structural dynamics and loading uncertainty, as schematically indicated by the dark blue boxes in Figure 1. The remaining part of the uncertainty (the light blue boxes in Figure 1) can be quantified based on experiments and data. This is not considered in the proposed framework, but a brief discussion is provided in the present subsection for the sake of completeness.

**Figure 1.** Stochastic variables modeling uncertainty in fatigue damage accumulation. The stochastic variables from the probabilistic model (1) are represented by separate boxes. The light blue boxes indicate stochastic variables estimated based on generic, design-based recommendations. The dark blue boxes indicate stochastic variables that can be quantified and updated based on new information from a digital twin.

#### 2.1.1. Met-Ocean Model

The joint probability distributions of the wind-wave climate is discretized by a finite number of short-term sea state simulations including random wind and wave seeds to model a stochastic process [40]. Met-ocean uncertainty is included in (1) by the yearly probability of each sea state, denoted *pi*. The met-ocean uncertainty can be quantified if long-term climate parameters are monitored [18,43].
