Smart grid (SG) is a new technology which allows the electrical power grid to be smart. Cyber security plays an essential role in the smart grid in order to secure the bidirectional communication between the utility and the consumer side. To pursue confidentiality and integrity, a key management scheme is considered a critical challenge in the SG. As a result, several key management protocols have been proposed in literature. However, many of those proposed protocols have a centralized feature that depends on a single server called Meter Data Management System (MDMS) to distribute keys and update them. In this paper, we propose a distributed key management scheme with authentication capability to avoid the single-point failure problem by removing the need for using an MDMS. Furthermore, we apply our scheme to the microgrid which can be considered a small-scale smart grid and can be isolated from the main grid. The security and performance analysis are conducted to show the effectiveness and scalability of our scheme by considering the communication and computation cost.