An analytical approach to calculating stationary PDFs for reflected random walks with an application to BESS-based ramp-rate control

TL;DR

This paper derives a rigorous analytical method using Wiener-Hopf integral equations to compute stationary PDFs of reflected random walks, applied to BESS inverter sizing and ramp-rate control for renewable energy.

Contribution

It introduces a novel analytical approach based on Fredholm integral equations and Neumann series for modeling BESS power distributions, aiding system design.

Findings

Analytical solutions match numerical and simulation results.

Truncated solutions enable simplified design rules.

Provides insights into inverter sizing for renewable ramp-rate control.

Abstract

A Wiener-Hopf-type integral equation for the stationary PDF of a reflected random walk is derived rigorously based on modern probability theory, and an application to battery energy storage systems (BESS), specifically the sizing of the inverter, is discussed in depth. The methodological steps include the construction of a Markov kernel, the derivation of a Fredholm integral equation of the second kind for the PDF of the BESS power, and an analytical solution of the equation based on a Neumann series. The analytical results were compared against numerical solutions obtained with the Nystrom method, as well as against the results of an algorithmic simulation using simulated input time series. The use of truncated versions of the analytic solution allows for the construction of simplified design rules for the power systems practitioner. General insights into inverter sizing criteria of storage systems for ramp-rate control of variable renewable energy (VRE) sources such as wind and solar are provided.