Convexity/nonconvexity certificates for power flow analysis

Abstract

Optimal Power Flow problem is considered as minimization of quadratic performance function subject to linear and quadratic equality/inequality constraints, AC power flow equations specify the feasibility domain. Similar quadratic problems arise in discrete optimization, uncertainty analysis, physical applications. In general they are nonconvex, nevertheless, demonstrate hidden convexity structure. We investigate the “image convexity” property. That is, we consider the image of the space of variables under quadratic map defined by power flow equations (the feasibility domain). If the image is convex, then original optimization problem has nice properties, for instance, it admits zero duality gap and convex optimization tools can be applied. There are several classes of quadratic maps representing the image convexity. We aim to discover similar structure and to obtain convexity or nonconvexity certificates for the individual quadratic transformation. We also provide the numerical algorithms exploiting convex relaxation of quadratic mappings for checking convexity. We address such problems as membership oracle and boundary oracle for the quadratic image. Finally the results are illustrated through some examples of 3-bus systems, namely, we detect nonconvexity of them.