Modeling Unbalanced Power Flow with Δ-connected Devices

In this tutorial we present a simple approach to modeling unbalanced three-phase power flows. We allow general non-ideal models of voltage sources, ZIP loads as well as distribution lines and transformers. The basic idea is to explicitly separate a device/transformer model into an internal model, that depends on the characteristics of the single-phase devices or transformers, and a conversion rule, that depends on their configuration. This approach provides two benefits. First it facilitates the modeling of secondary distribution circuits where only the end devices are directly controllable, not the currents or powers at the secondary transformers. Second it allows us to exploit common structures across different device/transformer variants and derive their external models that are general and unified. We illustrate these benefits by extending a three-phase backward forward sweep method in the literature to allow secondary circuits and formulating a three-phase optimal power flow problem as a quadratically constrained quadratic program.