© 2022 Elsevier B.V.Because of the lack of a proper real-time monitoring system at distribution power networks, topology identification can not be performed properly. While in traditional radial structures the topology is unique, in the presence of an open-ring structure, this issue constitutes a risk for the operation of the system, and the safety of the field staff in contact with the system components. Many studies in literature considers this problem in the scope of observability, however having a completely observable power system does not necessarily indicate having a uniquely identifiable topology. Therefore, the operator should know the branch statuses. One may utilize probabilistic approaches to determine the status of each branch, however those methods are vulnerable to the measurement errors and quality of the pseudo-measurements. This paper develops a numerical method that aims to find the identifiable branches, such that the branches whose status can be identified uniquely, based on the available measurement set, but not measurement readings. The proposed method employs a strategical ordering and Cholesky factorization to improve the computational performance. Moreover, the paper presents the optimum location of the measurement devices that is required to satisfy complete topology identifiability. The proposed method is assessed by means of a real-life system and the well-known IEEE test systems.