Stability of a stochastic model for demand-response

Abstract

We study the stability of a Markovian model of electricity productionand consumption that incorporates production volatility due torenewables and uncertainty about actual demand versus plannedproduction. We assume that the energy producer targets a fixed energyreserve, subject to ramp-up and ramp-down constraints, and thatappliances are subject to demand-response signals and adjust theirconsumption to the available production by delaying their demand. Whena constant fraction of the delayed demand vanishes over time, we showthat the general state Markov chain characterizing the system ispositive Harris and ergodic (i.e., delayed demand is bounded with highprobability). However, when delayed demand increases by a constantfraction over time, we show that the Markov chain is non-positive (i.e., there exists a non-zero probability that delayed demand becomesunbounded). We exhibit Lyapunov functions to prove our claims. Inaddition, we provide examples of heating appliances that, whendelayed, have energy requirements corresponding to the two consideredcases.

Keywords

• General state Markov chain stability
• smart grids
• demand-response
• macroscopic model