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Moment Constrained Optimal Transport for Thermostatically Controlled Loads

Thomas Le Corre, Julien Cardinal, Ana Bušić

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.

Moment Constrained Optimal Transport for Thermostatically Controlled Loads

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.

Paper Structure

This paper contains 20 sections, 3 theorems, 34 equations, 8 figures, 1 table, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. TCL CONTROL PROBLEM
  3. Water Heaters
  4. Problem for N Water Heaters
  5. MOMENT CONSTRAINED OPTIMAL TRANSPORT
  6. Mean Field Formulation
  7. Problem Formulation
  8. Dual problem
  9. ALGORITHM
  10. Monte Carlo Methods
  11. Gradient Descent Algorithm
  12. NUMERICAL RESULTS
  13. Presentation of the use case
  14. Results
  15. Comparison with Kullback Leibler Quadratic Optimal Control for heterogeneous Water Heaters
  16. ...and 5 more sections

Key Result

Proposition 1

(i) The minimum e:DualMCOTC gives: (ii) The maximizer is $\pi^\lambda(x,y)=P^\lambda(x,y)\mu_1(x)$ with $\forall x=(x_0,y_{1:T})\in\mathcal{X}, \forall y=(x_0,y_{1:T})\in \mathcal{X}$ (iii) There is no duality gap: there is a unique $\lambda^*\in\mathbb{R}_+^A$ satisfying

Figures (8)

  • Figure 1: Example of a nominal trajectory of a single water heater, starting at $\theta_0=54$°c and $m_0=0$
  • Figure 2: Average heat loss generated by the water drains during the day
  • Figure 3: Example of a controlled trajectory of a single water heater, starting at $\theta_0=54$°c and $m_0=0$ with two switches, one at $t_1=$1:00 and the other at $t_2=$14:20
  • Figure 4: Comparison of the KLQ aggregated consumption, in evaluation, on average drains. We chose $N_v$ points of discretization of the volume range $[0.1,0.3]$ m$^3$.
  • Figure 5: Aggregated consumption of a population of 2000 WHs on the tracking problem; (a) The solution in the ideal case; (b) Its evaluation on a finite population of WHs; (c) The MPC version by generating (a) and (b) every $\delta t = 10$ minutes.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof