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Register a new userUsing extreme value theory for the estimation of risk metrics for capacity adequacy assessment
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Added by
Stan Zachary
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
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This paper investigates the use of extreme value theory for modelling the distribution of demand-net-of-wind for capacity adequacy assessment. Extreme value theory approaches are well-established and mathematically justified methods for estimating the tails of a distribution and so are ideally suited for problems in capacity adequacy, where normally only the tails of the relevant distributions are significant. The extreme value theory peaks over threshold approach is applied directly to observations of demand-net-of-wind, meaning that no assumption is needed about the nature of any dependence between demand and wind. The methodology is tested on data from Great Britain and compared to two alternative approaches: use of the empirical distribution of demand-net-of-wind and use of a model which assumes independence between demand and wind. Extreme value theory is shown to produce broadly similar estimates of risk metrics to the use of the above empirical distribution but with smaller sampling uncertainty. Estimates of risk metrics differ when the approach assuming independence is used, especially when data across different historical years are pooled, suggesting that assuming independence might result in the over- or under-estimation of risk metrics.
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In recent years there has been a resurgence of interest in generation adequacy risk assessment, due to the need to include variable generation renewables within such calculations. This paper will describe new statistical approaches to estimating the joint distribution of demand and available VG capacity; this is required for the LOLE calculations used in many statutory adequacy studies, for example those of GB and PJM. The most popular estimation technique in the VG-integration literature is `hindcast, in which the historic joint distribution of demand and available VG is used as a predictive distribution. Through the use of bootstrap statistical analysis, this paper will show that due to extreme sparsity of data on times of high demand and low VG, hindcast results can suffer from sampling uncertainty to the extent that they have little practical meaning. An alternative estimation approach, in which a marginal distribution of available VG is rescaled according to demand level, is thus proposed. This reduces sampling uncertainty at the expense of the additional model structure assumption, and further provides a means of assessing the sensitivity of model outputs to the VG-demand relationship by varying the function of demand by which the marginal VG distribution is rescaled.
We propose a new method for modeling the distribution function of high dimensional extreme value distributions. The Pickands dependence function models the relationship between the covariates in the tails, and we learn this function using a neural network that is designed to satisfy its required properties. Moreover, we present new methods for recovering the spectral representation of extreme distributions and propose a generative model for sampling from extreme copulas. Numerical examples are provided demonstrating the efficacy and promise of our proposed methods.
We derive bounds on the distribution function, therefore also on the Value-at-Risk, of $varphi(mathbf X)$ where $varphi$ is an aggregation function and $mathbf X = (X_1,dots,X_d)$ is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of $mathbf X$, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Frechet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of $mathbf X$ is known on a subset of its domain, and finally we consider the case where the copula of $mathbf X$ lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Frechet--Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Frechet--Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.
Observed accidents have been the main resource for road safety analysis over the past decades. Although such reliance seems quite straightforward, the rare nature of these events has made safety difficult to assess, especially for new and innovative traffic treatments. Surrogate measures of safety have allowed to step away from traditional safety performance functions and analyze safety performance without relying on accident records. In recent years, the use of extreme value theory (EV) models in combination with surrogate safety measures to estimate accident probabilities has gained popularity within the safety community. In this paper we extend existing efforts on EV for accident probability estimation for two dependent surrogate measures. Using detailed trajectory data from a driving simulator, we model the joint probability of head-on and rear-end collisions in passing maneuvers. In our estimation we account for driver specific characteristics and road infrastructure variables. We show that accounting for these factors improve the head-on collision probability estimation. This work highlights the importance of considering driver and road heterogeneity in evaluating related safety events, of relevance to interventions both for in-vehicle and infrastructure-based solutions. Such features are essential to keep up with the expectations from surrogate safety measures for the integrated analysis of accident phenomena.
In multivariate extreme value theory (MEVT), the focus is on analysis outside of the observable sampling zone, which implies that the region of interest is associated to high risk levels. This work provides tools to include directional notions into the MEVT, giving the opportunity to characterize the recently introduced directional multivariate quantiles (DMQ) at high levels. Then, an out-sample estimation method for these quantiles is given. A bootstrap procedure carries out the estimation of the tuning parameter in this multivariate framework and helps with the estimation of the DMQ. Asymptotic normality for the proposed estimator is provided and the methodology is illustrated with simulated data-sets. Finally, a real-life application to a financial case is also performed.
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