Complex Couplings -- A universal, adaptive and bilinear formulation of power grid dynamics

The paper is now published in PRX Enegry. Please refer to the PRX version from now on. Anna B\"uttner and Frank Hellmann. "Complex Couplings-A Universal, Adaptive, and Bilinear Formulation of Power Grid Dynamics." The energy transition introduces new classes of dynamical actors into the power grid. There is a growing need for so-called grid-forming inverters (GFIs) that can contribute to dynamic grid stability as the share of synchronous generators decreases. Understanding the collective behavior and stability of future grids, featuring a heterogeneous mix of dynamics, remains an urgent and challenging task. Two recent advances in describing such modern power grid dynamics have made this problem more tractable: First, the normal form for grid-forming actors provides a uniform, technology-neutral description of plausible grid dynamics, including grid-forming inverters and synchronous machines. Secondly, the notion of the complex frequency has been introduced to effortlessly describe how the nodal dynamics influence the power flows in the grid. The major contribution of this paper is to show how the normal form approach and the complex frequency dynamics of power grids combine, and how they relate naturally to adaptive dynamical networks and control affine systems. Using normal form and complex frequency, we derive a remarkably elementary and universal equation for the collective grid dynamics. Notably, we obtain an elegant equation entirely in terms of a matrix of complex couplings, in which the network topology does not explicitly appear. These complex couplings give rise to new adaptive network formulations of future power grid dynamics. We give a new formulation of the Kuramoto model with inertia as a special case.