Multi-level optimization with the koopman operator for data-driven, domain-aware, and dynamic system security

Cyber–Physical Systems (CPSs) like the power grid are critically important but also increasingly vulnerable; ensuring reliable system operation in the face of disruptions is becoming more and more challenging. Multi-Level Optimization (MLO) is a powerful way to model adversarial interactions, which naturally makes it applicable to studying CPS security. However, MLO typically does not address underlying system dynamics, and incorporating nonlinear dynamics is generally infeasible. In this paper, we show how to combine MLO with the Koopman Operator (KO) to remedy this. The KO maps nonlinear dynamics to a lifted space in which those dynamics are linear, thus making it ideal for use with MLO. Moreover, the structure of the KO also provides convenient ways to incorporate domain knowledge into the data-driven process of learning the KO representation of a given system. Our contribution is a proposed, fairly general method for incorporating nonlinear dynamics into a MLO using a learned linear representation of the KO. We also demonstrate the use and tractability of this approach through experiments on small instances of a reliability-focused power grid problem. We conclude by discussing the scalability and computational cost of this physics-informed MLO-KO approach, and identify future research directions for this work.