  In distribution systems, power injection variability due to growing
penetrations of distributed energy resources (DERs) and dispatchable loads can
lead to power quality issues such as severe voltage unbalance. To ensure the
safe operation of phase-balance-sensitive components such as transformers and
induction motor loads, the amount of voltage unbalance must be maintained
within specified limits for a range of uncertain loading conditions. This paper
builds on existing "solvability conditions" that characterize operating regions
for which the power flow equations are guaranteed to have a unique high-voltage
solution. We extend these existing solvability conditions to be applicable to
distribution systems and augment them with a "balancibility" condition which
quantifies an operating region within which a unique, adequately balanced power
flow solution exists. To build this condition, we consider different unbalance
definitions and derive closed-form representations through reformulations or
safe approximations. Using case studies, we evaluate these closed-form
representations and compare the balancibility conditions associated with
different unbalance definitions.
