Thermostatically Controlled Loads (TCL), e.g. air-conditioners and heaters,are by far the most wide-spread consumers of electricity. Normally the devicesare calibrated to provide the so-called bang-bang control of temperature --changing from on to off, and vice versa, depending on temperature. Aggregationof a large group of similar devices into a statistical ensemble is considered,where the devices operate following the same dynamics subject to stochasticperturbations and randomized, Poisson on/off switching policy. We analyze,using theoretical and computational tools of statistical physics, how theensemble relaxes to a stationary distribution and establish relation betweenthe relaxation and statistics of the probability flux, associated with devices'cycling in the mixed (discrete, switch on/off, and continuous, temperature)phase space. This allowed us to derive and analyze spectrum of thenon-equilibrium (detailed balance broken) statistical system and uncover howswitching policy affects oscillatory trend and speed of the relaxation.Relaxation of the ensemble is of a practical interest because it describes howthe ensemble recovers from significant perturbations, e.g. forceful temporaryswitching off aimed at utilizing flexibility of the ensemble in providing"demand response" services relieving consumption temporarily to balance largerpower grid. We discuss how the statistical analysis can guide furtherdevelopment of the emerging demand response technology.
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