  In this work, we investigate grid-forming control for power systems
containing three-phase and single-phase converters connected to unbalanced
distribution and transmission networks, investigate self-balancing between
single-phase converters, and propose a novel balancing feedback for
grid-forming control that explicitly allows to trade-off unbalances in voltage
and power. We develop a quasi-steady-state power network model that allows to
analyze the interactions between three-phase and single-phase power converters
across transmission, distribution, and standard transformer interconnections.
We first investigate conditions under which this general network admits a
well-posed kron-reduced quasi-steady-state network model. Our main contribution
leverages this reduced-order model to develop analytical conditions for
stability of the overall network with grid-forming three-phase and single-phase
converters connected through standard transformer interconnections.
Specifically, we provide conditions on the network topology under which (i)
single-phase converters autonomously self-synchronize to a phase-balanced
operating point and (ii) single-phase converters phase-balance through
synchronization with three-phase converters. Moreover, we establish that the
conditions can be relaxed if a phase-balancing feedback control is used.
Finally, case studies combining detailed models of transmission systems (i.e.,
IEEE 9-bus) and distribution systems (i.e., IEEE 13-bus) are used to illustrate
the results for (i) a power system containing a mix of transmission and
distribution connected converters and, (ii) a power system solely using
distribution-connected converters at the grid edge.
