The Ubiquitous Power Internet of Things (UPIoT) is a concrete manifestation of the Internet of things (IoT) in the power industry, which is a deep integration of the interconnected power network and communication network, realizing full perception of the system status and full business penetration in all links of power production, transmission, and consumption. The introduction of edge computing in UPIoT fully meets the requirements of rapid response, real-time perception, and to some extent, privacy protection. However, there is currently no comprehensive investigation on the application of edge computing technology in UPIoT. First, this paper introduces the development background and construction of UPIoT and its technical architecture. Then the challenges faced by UPIoT in the process of construction are analyzed. Furthermore, the paper elaborates on the functions and features of edge computing, proposes that the support of edge computing technology can solve the challenges of efficient, fast, and secure processing of massive edge data faced by the traditional cloud-based centralized big data processing technology of UPIoT, and analyzes the architecture of the edge computing-assisted UPIoT. For the three typical scenarios of UPIoT, namely power monitoring system, smart energy system and power metering system, the edge computing architecture of the three scenarios are analyzed, and the specific application methods and roles played by edge computing in the three scenarios are also elaborated. Finally, we discuss the challenges of edge computing in UPIoT, in terms of policy challenges, market challenges, and technical challenges, as well as outline the outlooks of the technical challenges.