Convexification of power flow problem over arbitrary networks

Consider an arbitrary power network with PV and PQ buses, where active powers and voltage magnitudes are known at PV buses, and active and reactive powers are known at PQ buses. The classical power flow (PF) problem aims to find the unknown complex voltages at all buses. This problem is usually solved approximately through linearization or in an asymptotic sense using Newton's method, given that the solution belongs to a good regime containing voltage vectors with small angles. The question arises as to whether the PF problem can be cast as the solution of a convex optimization problem over that regime. The objective of this paper is to show that the answer to the above question is affirmative. More precisely, we propose a class of convex optimization problems with the property that they all solve the PF problem as long as angles are small. Each convex problem proposed in this work is in the form of a semidefinite program (SDP). Associated with each SDP, we explicitly characterize the set of complex voltages that can be recovered via that convex problem. Since there are infinitely many SDP problems, each capable of recovering a potentially different set of voltages, designing a good SDP problem is cast as a convex problem.