# Moment Constrained Optimal Transport for Thermostatically Controlled Loads

## TL;DR

This paper introduces a novel Moment Constrained Optimal Transport framework for distributed control of thermostatically controlled loads, effectively balancing grid stability with individual device constraints while reducing computational complexity.

## Contribution

The work presents a new optimal transport-based control method incorporating moment constraints, enabling efficient, scalable, and distributed management of TCLs with global and physical constraints.

## Key findings

- Effective coordination of TCLs demonstrated in water heater case study.
- Significant reduction in computational complexity compared to existing methods.
- Method extends to online control scenarios with practical applicability.

## Abstract

Controlling large populations of thermostatically controlled loads (TCLs), such as water heaters, poses significant challenges due to the need to balance global constraints (e.g., grid stability) with individual requirements (e.g., physical limits and quality of service). In this work, we introduce a novel framework based on Moment Constrained Optimal Transport (MCOT) for distributed control of TCLs. By formulating the control problem as an optimal transport problem with moment constraints, our approach integrates global consumption constraints and physical feasibility conditions into the control design. This problem with high (or infinite) dimensionality can be reduced to a much lower finite-dimensional problem. The structure of this problem allows for computing the gradient with Monte Carlo methods by generating trajectories of TCLs. Contrary to all previous work, in our MCOT framework, it is possible to choose the sampling law, which considerably speeds up the calculations. This algorithm mitigates the need for extensive state-space discretization and significantly reduces computational complexity compared to existing methods. Numerical experiments in a water heater case study demonstrate that our MCOT-based method effectively coordinates TCLs under various constraints. We further extend our approach to an online setting, illustrating its practical applicability on simulated data.

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Source: https://tomesphere.com/paper/2508.20059