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Mean Field Control of Thermostatically Controlled Loads as Piecewise Deterministic Markov Processes

Thomas Le Corre, Adrien Séguret, Ana Bušić

TL;DR

This work addresses coordinating a large population of Thermostatically Controlled Loads (TCLs) modeled as Piecewise Deterministic Markov Processes (PDMPs) through mean-field control. It develops a PDMP-specific mean-field framework with a dual formulation and a stochastic gradient algorithm to compute optimal jump-intensity controls under a quality-of-service constraint, demonstrated on water heater control. A key extension introduces an additional jump intensity to keep temperatures within bounds, enabling safe, decentralized coordination. Numerical results show the approach can track time-varying signals and manage electricity costs, with potential extensions to stronger constraints, online operation, and sensitivity analyses for different population sizes.

Abstract

This paper presents a mean-field control approach for Piecewise Deterministic Markov Processes (PDMPs), specifically designed for controlling a large number of agents. By modeling the interactions of a large number of agents through an aggregate cost function, the proposed method mitigates the high dimensionality of the problem by focusing on a representative agent. The contribution of this work is the application of a PDMP-based mean-field control framework to the coordination of a large population of Thermostatically Controlled Loads (TCLs). Adapting this framework to TCLs requires incorporating a quality-of-service constraint ensuring that each agent's temperature remains within a specified comfort range. To achieve this, an additional jump intensity is introduced so that agents are very likely to switch between heating and cooling modes when they reach the boundaries of their temperature range. This extension to TCLs is demonstrated through Water Heaters (WHs) control, with a decentralized algorithm based on a dual formulation and stochastic gradient descent. The numerical results obtained illustrate this approach on two examples (signal tracking and taking into account energy price).

Mean Field Control of Thermostatically Controlled Loads as Piecewise Deterministic Markov Processes

TL;DR

This work addresses coordinating a large population of Thermostatically Controlled Loads (TCLs) modeled as Piecewise Deterministic Markov Processes (PDMPs) through mean-field control. It develops a PDMP-specific mean-field framework with a dual formulation and a stochastic gradient algorithm to compute optimal jump-intensity controls under a quality-of-service constraint, demonstrated on water heater control. A key extension introduces an additional jump intensity to keep temperatures within bounds, enabling safe, decentralized coordination. Numerical results show the approach can track time-varying signals and manage electricity costs, with potential extensions to stronger constraints, online operation, and sensitivity analyses for different population sizes.

Abstract

This paper presents a mean-field control approach for Piecewise Deterministic Markov Processes (PDMPs), specifically designed for controlling a large number of agents. By modeling the interactions of a large number of agents through an aggregate cost function, the proposed method mitigates the high dimensionality of the problem by focusing on a representative agent. The contribution of this work is the application of a PDMP-based mean-field control framework to the coordination of a large population of Thermostatically Controlled Loads (TCLs). Adapting this framework to TCLs requires incorporating a quality-of-service constraint ensuring that each agent's temperature remains within a specified comfort range. To achieve this, an additional jump intensity is introduced so that agents are very likely to switch between heating and cooling modes when they reach the boundaries of their temperature range. This extension to TCLs is demonstrated through Water Heaters (WHs) control, with a decentralized algorithm based on a dual formulation and stochastic gradient descent. The numerical results obtained illustrate this approach on two examples (signal tracking and taking into account energy price).

Paper Structure

This paper contains 13 sections, 17 equations, 5 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. PROBLEM FORMULATION
  4. Context
  5. Definition of the problem
  6. Assumptions
  7. Dualization
  8. ALGORITHM
  9. NUMERICAL RESULTS
  10. Presentation of the use case
  11. Signal Tracking
  12. Electricity cost
  13. Future works

Figures (5)

  • Figure 1: Three trajectories of temperatures generated according to the nominal control
  • Figure 2: Aggregated Consumptions for different values of $\kappa$ and $M=10^5$
  • Figure 3: $\alpha+\hat{\alpha}$ for the jump from $0$ to $1$ (On to Off) for $\kappa=100$
  • Figure 4: An electricity cost is added
  • Figure 5: Agents are divided into three classes, each with a different pricing structure, to smooth out the consumption