# Ensemble of Thermostatically Controlled Loads: Statistical Physics Approach

## TL;DR

This paper applies statistical physics methods to analyze the relaxation dynamics of large ensembles of thermostatically controlled loads, revealing how switching policies influence their response to perturbations and informing demand response strategies.

## Contribution

It introduces a theoretical framework for understanding TCL ensemble relaxation using statistical physics, linking switching policies to oscillatory behavior and relaxation speed.

## Key findings

- Derived the spectrum of non-equilibrium statistical systems for TCL ensembles.
- Showed how switching policies affect relaxation oscillations.
- Provided insights into demand response applications.

## Abstract

Thermostatically Controlled Loads (TCL), e.g. air-conditioners and heaters, are by far the most wide-spread consumers of electricity. Normally the devices are calibrated to provide the so-called bang-bang control of temperature -- changing from on to off, and vice versa, depending on temperature. Aggregation of a large group of similar devices into a statistical ensemble is considered, where the devices operate following the same dynamics subject to stochastic perturbations and randomized, Poisson on/off switching policy. We analyze, using theoretical and computational tools of statistical physics, how the ensemble relaxes to a stationary distribution and establish relation between the 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 the non-equilibrium (detailed balance broken) statistical system and uncover how switching policy affects oscillatory trend and speed of the relaxation. Relaxation of the ensemble is of a practical interest because it describes how the ensemble recovers from significant perturbations, e.g. forceful temporary switching off aimed at utilizing flexibility of the ensemble in providing "demand response" services relieving consumption temporarily to balance larger power grid. We discuss how the statistical analysis can guide further development of the emerging demand response technology.