Stability of a Stochastic Model for Demand-Response

We study the stability of a Markovian model of electricity production and consumption that incorporates production volatility due to renewables and uncertainty about actual demand versus planned production. We assume that the energy producer targets a fixed energy reserve, subject to ramp-up and ramp-down constraints, and that appliances are subject to demand-response signals and adjust their consumption to the available production by delaying their demand. When a constant fraction of the delayed demand vanishes over time, we show that the general state Markov chain characterizing the system is positive Harris and ergodic (i.e., delayed demand is bounded with high probability). However, when delayed demand increases by a constant fraction over time, we show that the Markov chain is non-positive (i.e., there exists a non-zero probability that delayed demand becomes unbounded). We exhibit Lyapunov functions to prove our claims. In addition, we provide examples of heating appliances that, when delayed, have energy requirements corresponding to the two considered cases.