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In this work, we investigate grid-forming control for power systemscontaining three-phase and single-phase converters connected to unbalanceddistribution and transmission networks, investigate self-balancing betweensingle-phase converters, and propose a novel balancing feedback forgrid-forming control that explicitly allows to trade-off unbalances in voltageand power. We develop a quasi-steady-state power network model that allows toanalyze the interactions between three-phase and single-phase power convertersacross transmission, distribution, and standard transformer interconnections.We first investigate conditions under which this general network admits awell-posed kron-reduced quasi-steady-state network model. Our main contributionleverages this reduced-order model to develop analytical conditions forstability of the overall network with grid-forming three-phase and single-phaseconverters 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-balancedoperating point and (ii) single-phase converters phase-balance throughsynchronization with three-phase converters. Moreover, we establish that theconditions 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 illustratethe results for (i) a power system containing a mix of transmission anddistribution connected converters and, (ii) a power system solely usingdistribution-connected converters at the grid edge.

Grid-forming control of three-phase and single-phase converters across unbalanced transmission and distribution systems