State estimation of DC microgrids using manifold optimization and semidefinite programming

DC microgrids are becoming more common in modern systems, so computation methodologies such as the power flow, the optimal power flow, and the state estimation require being adapted to this new reality. This paper deals with the latter problem, which consists of reconstructing the state variables given voltage and power measurements. Although the model of DC grids is undoubtedly less complicated than its counterpart AC, it is still a nonlinear/non-convex optimization problem. Our approach is based on the idea of solving the problem in a matrix space. Although it may be counter-intuitive to transform from R-n to R-nxn, a matrix space exhibits better geometric properties that allow an elegant formulation and, in some cases, an efficient form to solve the optimization problem. We compare two methodologies: semidefinite programming and manifold optimization. The former relaxes the problem to a convex set, whereas the latter maintains the geometry of the original problem. A specialized gradient method is proposed to solve the problem in the matrix manifold. Extensive numerical experiments are conducted to showcase the key characteristics of both methodologies. Our study aims to shed light on the potential benefits of employing matrix space techniques in addressing operation problems in DC microgrids and power system computations in general.