A Unified Stochastic Hybrid System Approach to Aggregate Modeling of Responsive Loads

TL;DR

This paper introduces a comprehensive stochastic hybrid system framework for aggregate modeling of diverse load types, enabling more accurate analysis of demand response strategies beyond traditional models.

Contribution

It develops a unified hybrid system approach and PDE-based density evolution model applicable to general thermostatic and deferrable loads, extending existing methods.

Findings

Derivation of PDE governing load population density.

Unified model encompassing existing TCL and deferrable load models.

Capability to handle complex switching behaviors in load dynamics.

Abstract

Aggregate load modeling is of fundamental importance for systematic analysis and design of various demand response strategies. Instead of keeping track of the trajectories of individual loads, the aggregate modeling problem focuses on characterizing the density evolution of the load population. Most existing models are only applicable to Thermostatically Controlled Loads (TCL) with first-order linear dynamics. This paper develops a unified aggregate modeling approach that can be used for general TCLs as well as deferrable loads. We propose a deterministic hybrid system model to describe individual load dynamics under demand response rules, and develop a general stochastic hybrid system (SHS) model to capture the population dynamics. We also derive a set of partial differential equations (PDE) that governs the probability density evolution of the SHS. Our results cannot be obtained using the exiting SHS tools in the literature as the proposed SHS model involves both random and deterministic switchings with general switching surfaces in multi-dimensional domains. The derived PDE model includes many existing aggregate load modeling results as special cases and can be used in many other realistic modeling scenarios that have not been studied in the literature.