  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.
