Statistical Mechanics of Thermostatically Controlled Multi-Zone Buildings

We study the collective phenomena and constraints associated with the aggregation of individual cooling units from a statistical mechanics perspective. These units are modelled as Thermostatically Controlled Loads (TCLs) and represent zones in a large commercial or residential building. Their energy input is centralized and controlled by a collective unit -- the Air Handling Unit (AHU) -- delivering cool air to all TCLs, thereby coupling them together. Aiming to identify representative qualitative features of the AHU-to-TCL coupling, we build a realistic but also sufficiently simple model and analyze it in two distinct regimes: the Constant Supply Temperature (CST) and the Constant Power Input (CPI) regimes. In both cases, we center our analysis on the relaxation dynamics of individual TCL temperatures to a statistically steady state. We observe that while the dynamics are relatively fast in the CST regime, resulting in all TCLs evolving around the control setpoint, the CPI regime reveals emergence of a \emph{bi-modal probability distribution and two, possibly strongly separated, time scales}. We observe that the two modes in the CPI regime are associated with all TCLs being in the same low and high-temperature states, respectively, with occasional (and therefore possibly rare) collective transition between the modes akin in the Kramer's phenomenon of statistical physics. To the best of our knowledge, this phenomenon was overlooked in the context of the multi-zone energy building engineering, even thought it has direct implications on the operations of centralized cooling systems in buildings. It teaches us that a balance needs to be struck between occupational comfort -- related to variations in the individual temperatures -- and power output predictability -- the main focus of the DR schemes.