Florian Dorfler

Florian Dorfler, Francesco Bullo

Consider a weighted and undirected graph, possibly with self-loops, and its corresponding Laplacian matrix, possibly augmented with additional diagonal elements corresponding to the self-loops. The Kron reduction of this graph is again a graph whose Laplacian matrix is obtained by the Schur complement of the original Laplacian matrix with respect to a subset of nodes. The Kron reduction process is ubiquitous in classic circuit theory and in related disciplines such as electrical impedance tomography, smart grid monitoring, transient stability assessment in power networks, or analysis and simulation of induction motors and power electronics. More general applications of Kron reduction occur in sparse matrix algorithms, multi-grid solvers, finite--element analysis, and Markov chains. The Schur complement of a Laplacian matrix and related concepts have also been studied under different names and as purely theoretic problems in the literature on linear algebra. In this paper we propose a general graph-theoretic framework for Kron reduction that leads to novel and deep insights both on the mathematical and the physical side. We show the applicability of our framework to various practical problem setups arising in engineering applications and computation. Furthermore, we provide a comprehensive and detailed graph-theoretic analysis of the Kron reduction process encompassing topological, algebraic, spectral, resistive, and sensitivity analyses. Throughout our theoretic elaborations we especially emphasize the practical applicability of our results.

Florian Dorfler, Francesco Bullo

The Kuramoto model captures various synchronization phenomena in biological and man-made systems of coupled oscillators. It is well-known that there exists a critical coupling strength among the oscillators at which a phase transition from incoherency to synchronization occurs. This paper features four contributions. First, we characterize and distinguish the different notions of synchronization used throughout the literature and formally introduce the concept of phase cohesiveness as an analysis tool and performance index for synchronization. Second, we review the vast literature providing necessary, sufficient, implicit, and explicit estimates of the critical coupling strength for finite and infinite-dimensional, and for first and second-order Kuramoto models. Third, we present the first explicit necessary and sufficient condition on the critical coupling to achieve synchronization in the finite-dimensional Kuramoto model for an arbitrary distribution of the natural frequencies. The multiplicative gap in the synchronization condition yields a practical stability result determining the admissible initial and the guaranteed ultimate phase cohesiveness as well as the guaranteed asymptotic magnitude of the order parameter. Fourth and finally, we extend our analysis to multi-rate Kuramoto models consisting of second-order Kuramoto oscillators with inertia and viscous damping together with first-order Kuramoto oscillators with multiple time constants. We prove that the multi-rate Kuramoto model is locally topologically conjugate to a first-order Kuramoto model with scaled natural frequencies, and we present necessary and sufficient conditions for almost global phase synchronization and local frequency synchronization. Interestingly, these conditions do not depend on the inertiae which contradicts prior observations on the role of inertiae in synchronization of second-order Kuramoto models.

Linbin Huang, Jeremy Coulson, John Lygeros et al.

We apply a novel data-enabled predictive control (DeePC) algorithm in grid-connected power converters to perform safe and optimal control. Rather than a model, the DeePC algorithm solely needs input/output data measured from the unknown system to predict future trajectories. We show that the DeePC can eliminate undesired oscillations in a grid-connected power converter and stabilize an unstable system. However, the DeePC algorithm may suffer from poor scalability when applied in high-order systems. To this end, we present a finite-horizon output-based model predictive control (MPC) for grid-connected power converters, which uses an N-step auto-regressive-moving-average (ARMA) model for system representation. The ARMA model is identified via an N-step prediction error method (PEM) in a recursive way. We investigate the connection between the DeePC and the concatenated PEM-MPC method, and then analytically and numerically compare their closed-loop performance. Moreover, the PEM-MPC is applied in a voltage source converter based HVDC station which is connected to a two-area power system so as to eliminate low-frequency oscillations. All of our results are illustrated with high-fidelity, nonlinear, and noisy simulations.

Claudio De Persis, Erieke Weitenberg, Florian Dorfler

A novel power consensus algorithm for DC microgrids is proposed and analyzed. DC microgrids are networks composed of DC sources, loads, and interconnecting lines. They are represented by differential-algebraic equations connected over an undirected weighted graph that models the electrical circuit. A second graph represents the communication network over which the source nodes exchange information about the instantaneous powers, which is used to adjust the injected current accordingly. This give rise to a nonlinear consensus-like system of differential-algebraic equations that is analyzed via Lyapunov functions inspired by the physics of the system. We establish convergence to the set of equilibria consisting of weighted consensus power vectors as well as preservation of the weighted geometric mean of the source voltages. The results apply to networks with constant impedance, constant current and constant power loads.

Mohit Sinha, Florian Dorfler, Brian B. Johnson et al.

This paper examines the dynamics of power-electronic inverters in islanded microgrids that are controlled to emulate the dynamics of Van der Pol oscillators. The general strategy of controlling inverters to emulate the behavior of nonlinear oscillators presents a compelling time-domain alternative to ubiquitous droop control methods which presume the existence of a quasi-stationary sinusoidal steady state and operate on phasor quantities. We present two main results in this work. First, by leveraging the method of periodic averaging, we demonstrate that droop laws are intrinsically embedded within a slower time scale in the nonlinear dynamics of Van der Pol oscillators. Second, we establish the global convergence of amplitude and phase dynamics in a resistive network interconnecting inverters controlled as Van der Pol oscillators. Furthermore, under a set of non-restrictive decoupling approximations, we derive sufficient conditions for local exponential stability of desirable equilibria of the linearized amplitude and phase dynamics.

Jules Authier, Rabab Haider, Anuradha Annaswamy et al.

To maintain a reliable grid we need fast decision-making algorithms for complex problems like Dynamic Reconfiguration (DyR). DyR optimizes distribution grid switch settings in real-time to minimize grid losses and dispatches resources to supply loads with available generation. DyR is a mixed-integer problem and can be computationally intractable to solve for large grids and at fast timescales. We propose GraPhyR, a Physics-Informed Graph Neural Network (GNNs) framework tailored for DyR. We incorporate essential operational and connectivity constraints directly within the GNN framework and train it end-to-end. Our results show that GraPhyR is able to learn to optimize the DyR task.

Stefania Ionescu, Robin Forsberg, Elsa Lichtenegger et al.

Throughout application domains, we now rely extensively on algorithmic systems to engage with ever-expanding datasets of information. Despite their benefits, these systems are often complex (comprising of many intricate tools, e.g., moderation, recommender systems, prediction models), of unknown structure (due to the lack of accompanying documentation), and having hard-to-predict yet potentially severe downstream consequences (due to the extensive use, systematic enactment of existing errors, and many comprising feedback loops). As such, understanding and evaluating these systems as a whole remains a challenge for both researchers and legislators. To aid ongoing efforts, we introduce a formal framework for such visibility allocation systems (VASs) which we define as (semi-)automated systems deciding which (processed) data to present a human user with. We review typical tools comprising VASs and define the associated computational problems they solve. By doing so, VASs can be decomposed into sub-processes and illustrated via data flow diagrams. Moreover, we survey metrics for evaluating VASs throughout the pipeline, thus aiding system diagnostics. Using forecasting-based recommendations in school choice as a case study, we demonstrate how our framework can support VAS evaluation. We also discuss how our framework can support ongoing AI-legislative efforts to locate obligations, quantify systemic risks, and enable adaptive compliance.

Florian Dorfler, Francesco Bullo

Motivated by recent interest for multi-agent systems and smart power grid architectures, we discuss the synchronization problem for the network-reduced model of a power system with non-trivial transfer conductances. Our key insight is to exploit the relationship between the power network model and a first-order model of coupled oscillators. Assuming overdamped generators (possibly due to local excitation controllers), a singular perturbation analysis shows the equivalence between the classic swing equations and a non-uniform Kuramoto model. Here, non-uniform Kuramoto oscillators are characterized by multiple time constants, non-homogeneous coupling, and non-uniform phase shifts. Extending methods from transient stability, synchronization theory, and consensus protocols, we establish sufficient conditions for synchronization of non-uniform Kuramoto oscillators. These conditions reduce to and improve upon previously-available tests for the standard Kuramoto model. Combining our singular perturbation and Kuramoto analyses, we derive concise and purely algebraic conditions that relate synchronization and transient stability of a power network to the underlying system parameters and initial conditions.