Computing the CDF of the function of a
real-valued random vector with elliptical
distribution: an original level-set based
method
CAYET Pierre
This article proposes a method for calculating the probability of a (potentially non-linear) combination of random events, and finds numerous applications in the study of energy systems under uncertainty. For example, it allows the calculation of the probability distribution of metrics associated with the stability of the electrical system (LOLP, amount of electricity not supplied to end users), the aggregate production of a set of solar and wind power generation units taking into account correlations between sites, or the probability distribution of curtailment for a renewable electricity producer.
This paper proposes an original approach for the computation of the CDF of a continuous function h applied to a real-valued random vector X, when this vector follows an elliptical distribution. Instead of sampling directly from the density of X and computing the proportion of draws for which h(X) exceeds a given threshold α ∈ R, we propose an original level-set approach for calculating the CDF of h(X), by expressing it as a weighted integral computed over the level-sets of its probability density function.
Using Lebesgue integration and the fact that the level-sets of the density function of an elliptical distribution form a family of homothetic hyperellipsoids, we reformulate the probability P(h (X) ≥ α) as a weighted integral taken over this family of hyperellipsoids, which Lebesgue measure can be computed under closed-form. We further generalize our expression to mixtures of elliptical distributions, and establish potentially fruitful connexions with differential forms by reformulating our integral as a weighted integral over the surface our family of hyperellipsoids using the generalized Stokes theorem.
Finally, we show that the probability P(h (X) ≥ α) can be reformulated in terms of the weighted expectation of a specific random function η(U), where U follows a uniform distribution on the unit sphere. We provide a consistent estimator and assess the performances of our methods in terms of empirical bias and variance of the estimator.
Here is a (non-exhaustive) list of potential applications for this method, which aims to use the geometric properties of elliptical distributions (such as the normal distribution) to avoid Monte Carlo-type methods:
- Calculation of metrics related to power system adequacy: calculation of the Loss of Load Probability (LOLP), calculation of the probability distribution of the amount of electricity not supplied to end users
- Calculation of the aggregate output of a set of renewable generation units (e.g., probability of low output over several successive periods), taking into account correlations between generation sites.
- Calculation of the probability distribution of the Levelized Cost of Hydrogen (LCOH), based on correlations between spot electricity prices and the usage/maintenance schedule of electrolysers, for example.
- Calculation of the probability distribution of potential revenues related to storage and flexibility, again using, for example, the spot price of electricity and a battery charge/discharge schedule, which will depend on the difference between electricity production and consumption on the grid in question.
- Calculation of the probability of curtailment for a renewable electricity producer.
- Read the working paper