Elliptic Approximation Theory for the Joint Distribution of Photovoltaic Power and Irradiance

This paper rigorously proves, from a probabilistic perspective, that in the vicinity of the normal operating point of a photovoltaic (PV) station, the level sets of the joint probability density function of normalized irradiance \\(G\\) and normalized power \\(P\\) are approximately elliptical. Based on the facts that irradiance follows a Beta distribution and that, given irradiance, power follows a heteroscedastic normal distribution, we derive the gradient and Hessian matrix of the negative log-likelihood function, prove that the Hessian matrix is positive definite, and then, using a Taylor expansion with Peano remainder, show that the local shape of the level set is exactly elliptical, with a higher-order error that is second-order infinitesimal. This proof provides a solid mathematical foundation for unsupervised anomaly detection methods based on ellipse fitting. This repository contains the supplementary material for the paper submitted to the IEEE International Symposium on Circuits and Systems (ISCAS 2027).