AC OPF in radial distribution networks - Part I: On the limits of the branch flow convexification and the alternating direction method of multipliers

- The inexact formulation of the Farivar and Low optimal power flow formulation is discussed.
- Extending the Farivar-Low OPF formulation to exact line models can provide infeasible solutions.
- Cases are shown for which the ADMM method, applied to the non-relaxed OPF, fails to converge.

The optimal power-flow problem (OPF) has always played a key role in the planning and operation of power systems. Due to the non-linear nature of the AC power-flow equations, the OPF problem is known to be non-convex, therefore hard to solve. During the last few years several methods for solving the OPF have been proposed. The majority of them rely on approximations, often applied to the network model, aiming at making OPF convex and yielding inexact solutions. Others, kept the non-convex nature of the OPF with consequent increase of the computational complexity, inadequateness for real time control applications and sub-optimality of the identified solution. Recently, Farivar and Low proposed a method that is claimed to be exact for the case of radial distribution systems under specific assumptions, despite no apparent approximations. In our work, we show that it is, in fact, not exact. On one hand, there is a misinterpretation of the physical network model related to the ampacity constraint of the lines' current flows. On the other hand, the proof of the exactness of the proposed relaxation requires unrealistic assumptions and, in particular, (i) full controllability of loads and generation in the network and (ii) no upper-bound on the controllable loads. We also show that the extension of this approach to account for exact line models might provide physically infeasible solutions. In addition to the aforementioned convexification method, recently several contributions have proposed OPF algorithms that rely on the use of the alternating direction method of multipliers (ADMM). However, as we show in this work, there are cases for which the ADMM-based solution of the non-relaxed OPF problem fails to converge. To overcome the aforementioned limitations, we propose a specific algorithm for the solution of a non-approximated, non-convex OPF problem in radial distribution systems. In view of the complexity of the contribution, this work is divided in two parts. In this first part, we specifically discuss the limitations of both BFM and ADMM to solve the OPF problem.