1 An Exact SOCP Formulation for AC Optimal Power Flow P. Naga Yasasvi, Snehil Chandra, A. Mohapatra and S. C. Srivastava Department of Electrical Engineering, Indian Institute of Technology Kanpur, India 208016 Email: yasasvi@iitk.ac.in; snehilc@iitk.ac.in; abheem@iitk.ac.in; scs@iitk.ac.in Abstract—AC Optimal Power Flow (ACOPF) formulation, in general, is nonlinear and non-convex in nature and hence its application to problems like robust system operation, optimal transmission switching, etc. is a herculean task. Convexification of OPF is desired in order to reduce the complexity of solving such problems. Existing convex formulations such as Semi- Definite Programming (SDP), conventional Second Order Conic Programming (SOCP), and so on and their variants result either in significant optimality gap (OG) or reduced OG, albeit with the presence of non-convexity or approximations which bridge the gap between convexity and non-convexity. Due to these challenges, the current practice in the electricity sector is still to use the DCOPF (approximation of ACOPF). Hence in this paper, an exact and novel SOCP formulation of actual ACOPF has been proposed which aims for getting an equivalent convex formulation similar to the actual nonlinear, non-convex ACOPF. This also ensures attaining zero OG for mesh as well as radial networks and hence, the possibility of easily integrating this formulation in OPFs with data uncertainty. The proposed methodology has been tested on a 3 bus, IEEE 14 bus and IEEE 118 bus test systems on GAMS platform. The results thus obtained, have been compared with those obtained from DCOPF, nonlinear, non-convex ACOPF, conventional SOCP formulation for radial networks (SOCP-I) and conventional SOCP formulation for mesh networks (SOCP-II). The results ensure zero OG of the proposed SOCP formulation. Index Terms—Optimal power flow, convexification, semi- definite programming, second order conic programming, opti- mality gap. I. I NTRODUCTION O PTIMAL power flow (OPF) is a powerful and funda- mental tool for decision making problems in electrical power system analysis. It was first introduced by Carpentier for the economic dispatch problem [1]. ACOPF is generally a nonlinear, non-convex, large scale problem which aims at determining optimal values for a set of decision variables, based on a certain objective while subject to power balance constraints (constructed based on Kirchhoff’s voltage and current laws) and operational constraints which include voltage magnitudes, branch flows, real and reactive power generation limits, etc. It finds application in power system planning, operational planning, system operation, control, electricity markets etc. [2]. Since decades, one of the major challenges faced in ACOPF solution process is to attain the global optimal solution and this issue originates from the very inherent characteristic of ACOPF i.e. the nonlinear and the non-convex feature. Hence to partially rectify this issue, linearized approximation of ACOPF i.e. DCOPF, is solved initially and further power flow equations are solved to evaluate feasible reactive powers and voltages. Also, in literature network losses are compensated within the DCOPF framework [3]. However, this approach does not ensure optimal solution for ACOPF [4]. In the aim of improving the quality of ACOPF solution, convex relaxations of OPF have gained tremendous attention in the past few years. For a non-convex optimization problem, global optimality of the solution cannot be ensured and the convergence of solving algorithms cannot be guaranteed [5]. Hence, literature reports convex relaxation techniques for OPF which include SDP [6], conventional SOCP [7], Quadratic Convex (QC) [8] and moment based [9] relaxations. Convexi- fication of OPF is desired as it further facilitates critical studies like robust optimization in the presence of uncertainties with respect to power system operation and planning. Also, convex relaxation of OPF eases in modeling mixed integer nonlinear programming (MINLP) optimization problems such as trans- mission expansion planning, transmission switching, etc. [10]. Convex OPF with non-zero OG is practically meaningful than a non-convex OPF with zero OG. SDP relaxation as in [6] deals with appropriate choice of a positive semidefinite matrix in order to optimize a linear function with respect to linear constraints. SDP relaxation convexifies OPF at the expense of high OG of around 30% for few case studies as reported in [8]. Convergence failure issues, numerical accuracy warnings and memory run out problems for large node system have also been encountered by OPF with SDP relaxation [8]. Also, global optimal solution is attained only if the SDP relaxation is exact and is restricted for few class of problems like radial networks under load over- satisfaction [11], [12] , non existence of real power generation lower bounds [13] and lossless networks with cyclic graphs [14]. In recent times, research focus has shifted to SOCP [7], [15] in which the non-convexities in power flow equations are explicitly dealt with by means of rotated second order cone constraints and arctangent function constraints with respect to bus voltage angles. In [7], the arctangent function constraints have been completely ignored in SOCP which makes the ACOPF formulation a convex problem but does not guar- antee global optimal solution, especially in mesh networks. A strong SOCP relaxation of ACOPF is also presented in [7] by including arctangent functions but at the expense of retaining non-convexity in the ACOPF formulation. A study 978-1-5386-6159-8/18/$31.00 c  2018 IEEE Proceedings of the National Power Systems Conference (NPSC) - 2018, December 14-16, NIT Tiruchirappalli, India