Super-relaxation of space-time-quantized ensemble of energy loads to curtail their synchronization after demand response perturbation

TL;DR

This paper demonstrates that nonlinear feedback control can induce super-relaxation in space-time-quantized ensembles of thermostatically controlled loads, effectively reducing synchronization after demand response events while considering realistic device operation constraints.

Contribution

It introduces a discrete probabilistic model incorporating space-time quantization and shows super-relaxation is achievable and stable in this realistic setting, with criteria to prevent oscillations.

Findings

Super-relaxation is stable against randomness in the stochastic matrix.

Super-relaxation remains sensitive to the discretization scheme.

Analytical criteria can prevent undesirable oscillations.

Abstract

Ensembles of thermostatically controlled loads (TCL) provide a significant demand response reserve for the system operator to balance power grids. However, this also results in the parasitic synchronization of individual devices within the ensemble leading to long post-demand-response oscillations in the integrated energy consumption of the ensemble. The synchronization is eventually destructed by fluctuations, thus leading to the (pre-demand response) steady state; however, this natural desynchronization, or relaxation to a statistically steady state, is too long. A resolution of this problem consists in measuring the ensemble's instantaneous consumption and using it as a feedback to stochastic switching of the ensemble's devices between on- and off- states. A simplified continuous-time model showed that carefully tuned nonlinear feedback results in a fast (super-) relaxation of the ensemble energy consumption. Since both state information and control signals are discrete, the actual TCL devices operation is space-time quantized, and this must be considered for realistic TCL ensemble modelling. Here, assuming that states are characterized by indoor temperature (quantifying comfort) and air conditioner regime (on, off), we construct a discrete model based on the probabilistic description of state transitions. We demonstrate that super-relaxation holds in such a more realistic setting, and that while it is stable against randomness in the stochastic matrix of the quantized model, it remains sensitive to the time discretization scheme. Aiming to achieve a balance between super-relaxation and customer's comfort, we analyze the dependence of super-relaxation on details of the space-time quantization, and provide a simple analytical criterion to avoid undesirable oscillations in consumption.