Data-driven online optimization of dynamical systems: algorithms and applications to power grids

The talk focuses on the synthesis of online algorithms to optimize the behavior of networked and dynamical systems operating in uncertain, dynamically changing environments. Motivating examples are drawn from power systems, with specific applications that include management of distributed energy resources and electric vehicles. The main strategy revolves around the synthesis of online optimization algorithms that effectively act as feedback controllers to drive dynamical systems towards well-defined equilibria. In particular, the desired equilibrium points coincide with optimal solution trajectories of optimization problems formalizing performance metrics and operational constraints that may evolve over time. The first part of the talk considers the case where the dynamics of the plant are fast and algorithms are synthetized based on the algebraic representation of the system; the design of the algorithms capitalizes on an online implementation of data-driven first-order methods, suitably modified to accommodate actionable feedback in the form of measurements from the systems and functional evaluations of the cost. Leveraging contraction arguments, the performance of the closed-loop online algorithm is analyzed in terms of tracking of an optimal solution trajectory; in particular, results in terms of convergence in expectation and in high-probability are presented, with the latter leveraging a sub-Weibull model for the gradient error. The talk then considers the case where the time scales of online algorithms and dynamical systems are comparable; in this case, sufficient conditions on the tunable parameters of the online algorithm are presented to guarantee exponential and input-to-state stability of the interconnection between the online algorithm and the dynamical system. Algorithms are applied to solve problems in power systems such as demand response, optimal power flows, and real-time economic dispatch.