A distributed approach to the Optimal Power Flow problem for unbalanced and mesh networks ⋆ Giulio Ferro ∗ Michela Robba ∗ David D’Achiardi ∗∗ Rabab Haider ∗∗ Anuradha M. Annaswamy ∗∗ ∗ Department of Informatics, Bioengineering, Robotics and Systems Engineering, University of Genoa, Genoa, Italy (email: giulio.ferro@edu.unige.it) ∗∗ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02141 USA (email: aanna@mit.edu). Abstract: In the present paper we introduce a new distributed optimization approach to the solution of general Optimal Power Flow problem for unbalanced and mesh networks. A new convex formulation, based on McCormick Envelopes, is proposed with a decomposition profile and a distributed approach based on proximal coordination. The resulting algorithm is shown to converge with a rate of o(1/τ ), where τ is the number of iterations. The approach is validated on a modified IEEE 13 bus network, with added distributed energy resources including distributed generation and demand response. Keywords: Smart grids, modeling of power systems, optimal operation and control of power systems, optimization and control of large-scale network systems, decentralized and distributed control. 1. INTRODUCTION AND STATE OF THE ART Over the last decade, the power grid has undergone transformation with the increased penetration of small- scale distributed energy resources (DERs), primarily in the distribution network. These include renewable energy resources (RERs), distributed generation (DG), demand response (DR), and storage devices. The large number of DERs pose a challenge to grid operators, in which the state of art is to employ centralized decision making. With such a large number of DERs, these centralized decision mak- ing tools, which typically include optimization problems for resource dispatch, become intractable. To efficiently integrate these DERs, it is necessary to develop models and methods for optimizing power injections into the grid which are able to deal with large scale systems. As the DERs are situated throughout the distribution network, detailed and accurate models of the electric networks are needed, while still ensuring tractability within an op- timization framework. A detailed representation of the power grid is provided by load flow equations (Kersting, 2006). Optimal Power Flow (OPF) denotes the method utilized for optimizing power injections or flows within the net- work, subject to constraints that correspond to the power physics of the grid (Dommel and Tinney, 1968). Generally, the OPF is challenging to solve since it is a non-convex nonlinear problem. To address this, convex relaxations are often employed to render the problem easier to solve. Of these, the most popular power system models are the bus ⋆ This work was supported in part by the Department of Energy under Award Number DE-IA0000025 for UI-ASSIST Project. injection model (BIM) based on semidefinite programming (SDP), and the branch flow model (BF) based on second order cone programming (SOCP) (Farivar et al., 2011; Lavaei and Low, 2012). It should be noted that the BF model guarantees optimality for a limited set of networks, namely networks with radial topology and balanced struc- ture, with limits on how active the network can be (Farivar and Low, 2014; Christakou et al., 2017). This is highly problematic for distribution systems where lines are un- balanced and consist of many single-phase loads. To address these challenges and limitations, we first pro- pose a new convex formulation of the OPF problem which can be used for any general distribution grid (including meshed and unbalanced topologies). The formulation is based on the current injection (CI) method, and uses McCormick Envelopes (MCE) (Mccormick, 1976) to con- vexify bilinear constraints. We then reformulate the OPF problem posed using the CI model, in the form of a distributed algorithm based on the Proximal Atomic Co- ordination method (PAC) Romvary (2018) so as to make it computationally tractable. In the current literature there are many works that apply distributed optimization to power distribution networks. In (Li et al., 2012) DR is studied in a radial distribution network, by formulating it as an OPF minimizing energy costs and power line losses, subject to the power flow constraints, with the BF model, and operating constraints. The results therein are derived based on assumptions that include a balanced structure, radial topology and network’s passivity (i.e. inverse power flow). In contrast, our proposed approach removes all of these assumptions. Preprints of the 21st IFAC World Congress (Virtual) Berlin, Germany, July 12-17, 2020 Copyright lies with the authors 13470