In this paper, a secure and efficient access authentication scheme is designed between edge devices in the smart grid, to achieve secure communication between devices in the smart grid. Considering the characteristics of power grid equipment, identity based cryptography is introduced to embed the equipment identity, timestamp, etc., into the process of identity recognition and private key update. Combined with the characteristics of private key update in the cryptographic system, bilinear pairing is used to negotiate the communication key in each time interval, and only hash, XOR operation and concatenation operation are used to achieve lightweight authentication scheme. The scheme can not only meet the security requirements, but also minimize the burden on edge device resources. Experimental results show that the communication cost and computation cost of our scheme are less than the existing schemes.