On the comparison of different convexified power flow models in radial network

In this paper, we review two convex models for power flow in radial distribution network, namely the bus injection and branch flow models. We start with the fundamental equations of voltage drop, power losses, and line power flows between two buses in a distribution line represented by an impedance. These AC circuit analysis equations contain trigonometric functions such as sine and cosine. We show that we can obtain a new set of equivalent AC circuit analysis without trigonometric functions by defining auxiliary variables and/or using linear combination of original equations. Additionally, by treating squared values of voltages and currents we obtain linear form of AC circuit equation whereas the relation of sine and cosine functions can be equivalently embedded in rotated cones. Consequently, we obtain the bus injection and branch flow models, which are theoretically equivalent. Their numerical performance, however, could be different, due to the numerical ill-conditions that may arise when constructing rotated cones.