Moment Constrained Optimal Transport for Control Applications

Thomas Le Corre, Ana Busic, Sean P. Meyn

Abstract: This paper concerns the application of techniques from optimal transport (OT) to mean field control, in which the probability measures of interest in OT correspond to empirical distributions associated with a large collection of controlled agents. The control objective of interest motivates a one-sided relaxation of OT, in which the first marginal is fixed and the second marginal is constrained to a “moment class”: a set of probability measures defined by generalized moment constraints. This relaxation is particularly interesting for control problems as it enables the coordination of agents without the need to know the desired distribution beforehand. The inclusion of an entropic regularizer is motivated by both computational considerations, and also to impose hard constraints on agent behavior. A computational approach inspired by the Sinkhorn algorithm is proposed to solve this problem. This new approach to distributed control is illustrated with an application of charging a fleet of electric vehicles while satisfying grid constraints. An online version is proposed and applied in a case study on the ElaadNL dataset containing 10,000 electric vehicle charging sessions in the Netherlands. This empirical validation demonstrates the applicability of the proposed approach to optimizing flexibility while respecting grid constraints.

Submission Type: Regular submission (no more than 12 pages of main content)

Changes Since Last Submission:

Main changes:
- Clarifying constributions and the comparison with the literature
- Adding a discussion on the applicability of the method in other contexts in the conclusion
- Adding a new appendix "Costs of the solutions shown in Section 3" to address questions raised by Reviewer DLci: *Can the authors also provide the cost of the solutions the algorithms produce in Sections 3 and 4? This would help the readers understand how much the additional constraints penalize the cost of the solution, along with another point of comparison against prior work, e.g. in Figure 4.*
- Implementing the answers to remarks of Reviewer Hpy6. The exception is remark 33 that we are still working on: *Report proofs right below the statements in an appendix..[...]*
- Adding a new appendix with the application of MCOT-C on another control problem (water heaters control).
- Implementing the new answers to remarks of Reviewer Hpy6.

Assigned Action Editor: ~Amir-massoud_Farahmand1

Submission Number: 6261