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The probabilistic design of offshore wind turbines aims to ensure structural safety in a cost-effective way. This involves conducting structural reliability assessments for different design options and considering different structural responses. There are several structural reliability methods, and this paper will apply and compare different approaches in some simplified case studies. In particular, the well known environmental contour method will be compared to a more novel approach based on sequential sampling and Gaussian processes regression for an ultimate limit state case study. For one of the case studies, results will also be compared to results from a brute force simulation approach. Interestingly, the comparison is very different from the two case studies. In one of the cases the environmental contours method agrees well with the sequential sampling method but in the other, results vary considerably. Probably, this can be explained by the violation of some of the assumptions associated with the environmental contour approach, i.e. that the short-term variability of the response is large compared to the long-term variability of the environmental conditions. Results from this simple comparison study suggests that the sequential sampling method can be a robust and computationally effective approach for structural reliability assessment.

Longevity and safety of lithium-ion batteries are facilitated by efficient monitoring and adjustment of the battery operating conditions. Hence, it is crucial to implement fast and accurate algorithms for State of Health (SoH) monitoring on the Battery Management System. The task is challenging due to the complexity and multitude of the factors contributing to the battery degradation, especially because the different degradation processes occur at various timescales and their interactions play an important role. Data-driven methods bypass this issue by approximating the complex processes with statistical or machine learning models. This paper proposes a data-driven approach which is understudied in the context of battery degradation, despite its simplicity and ease of computation: the Multivariable Fractional Polynomial (MFP) regression. Models are trained from historical data of one exhausted cell and used to predict the SoH of other cells. The data are characterised by varying loads simulating dynamic operating conditions. Two hypothetical scenarios are considered: one assumes that a recent capacity measurement is known, the other is based only on the nominal capacity. It was shown that the degradation behaviour of the batteries under examination is influenced by their historical data, as supported by the low prediction errors achieved (root mean squared errors from 1.2% to 7.22% when considering data up to the battery End of Life). Moreover, we offer a multi-factor perspective where the degree of impact of each different factor is analysed. Finally, we compare with a Long Short-Term Memory Neural Network and other works from the literature on the same dataset. We conclude that the MFP regression is effective and competitive with contemporary works, and provides several additional advantages e.g. in terms of interpretability, generalisability, and implementability., Comment: 45 pages, 7 figures, 3 tables

Environmental contours are widely used as basis for design of structures exposed to environmental loads. The basic idea of the method is to decouple the environmental description from the structural response. This is done by establishing an envelope of environmental conditions, such that any structure tolerating loads on this envelope will have a failure probability smaller than a prescribed value. Specifically, given an $n$-dimensional random variable $\mathbf{X}$ and a target probability of failure $p_{e}$, an environmental contour is the boundary of a set $\mathcal{B} \subset \mathbb{R}^{n}$ with the following property: For any failure set $\mathcal{F} \subset \mathbb{R}^{n}$, if $\mathcal{F}$ does not intersect the interior of $\mathcal{B}$, then the probability of failure, $P(\mathbf{X} \in \mathcal{F})$, is bounded above by $p_{e}$. As is common for many real-world applications, we work under the assumption that failure sets are convex. In this paper, we show that such environmental contours may be regarded as boundaries of Voronoi cells. This geometric interpretation leads to new theoretical insights and suggests a simple novel construction algorithm that guarantees the desired probabilistic properties. The method is illustrated with examples in two and three dimensions, but the results extend to environmental contours in arbitrary dimensions. Inspired by the Voronoi-Delaunay duality in the numerical discrete scenario, we are also able to derive an analytical representation where the environmental contour is considered as a differentiable manifold, and a criterion for its existence is established., Comment: 23 pages, 9 figures

Environmental contours are used in structural reliability analysis of marine and coastal structures as an approximate means to locate the boundary of the distribution of environmental variables, and hence sets of environmental conditions giving rise to extreme structural loads and responses. Outline guidance concerning the application of environmental contour methods is given in recent design guidelines from many organisations. However there is lack of clarity concerning the differences between approaches to environmental contour estimation reported in the literature, and regarding the relationship between the environmental contour, corresponding to some return period, and the extreme structural response for the same period. Hence there is uncertainty about precisely when environmental contours should be used, and how they should be used well. This article seeks to provide some assistance in understanding the fundamental issues regarding environmental contours and their use in structural reliability analysis. Approaches to estimating the joint distribution of environmental variables, and to estimating environmental contours based on that distribution, are described. Simple software for estimation of the joint distribution, and hence environmental contours, is illustrated (and is freely available from the authors). Extra assumptions required to relate the characteristics of environmental contour to structural failure are outlined. Alternative response-based methods not requiring environmental contours are summarised. The results of an informal survey of the metocean user community regarding environmental contours are presented. Finally, recommendations about when and how environmental contour methods should be used are made.