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The energy transition introduces new classes of dynamical actors into thepower grid. There is a growing need for so-called grid-forming inverters (GFIs)that can contribute to dynamic grid stability as the share of synchronousgenerators decreases. Understanding the collective behavior and stability offuture grids, featuring a heterogeneous mix of dynamics, remains an urgent andchallenging task. Two recent advances in describing such modern power griddynamics have made this problem more tractable: First, the normal form forgrid-forming actors provides a uniform, technology-neutral description ofplausible grid dynamics, including grid-forming inverters and synchronousmachines. Secondly, the notion of the complex frequency has been introduced toeffortlessly describe how the nodal dynamics influence the power flows in thegrid. The major contribution of this paper is to show how the normal formapproach and the complex frequency dynamics of power grids combine, and howthey relate naturally to adaptive dynamical networks and control affinesystems. Using normal form and complex frequency, we derive a remarkablyelementary and universal equation for the collective grid dynamics. Notably, weobtain an elegant equation entirely in terms of a matrix of complex couplings,in which the network topology does not explicitly appear. These complexcouplings give rise to new adaptive network formulations of future power griddynamics. We give a new formulation of the Kuramoto model with inertia as aspecial case. Starting from this formulation of the grid dynamics, the questionof the optimal design of future grid forming actors becomes treatable bymethods from affine and bilinear control theory. We demonstrate the power ofthis perspective by deriving a quasi-local control dynamics that can stabilizearbitrary power flows, even if the effective network Laplacian is not positivedefinite.

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