Blockchain gives a new method for distributed data ledgering. The smart grid obtains efficient two-way data transmission and information control. It effectively monitors and regulates the grid by collecting real-time electricity consumption data of users. However, online data collection brings privacy leakage. To solve the problem of privacy leakage in the electricity data collection in the smart grid, a privacy-aware electricity data collection model is proposed. Firstly, we propose a new group blind signature scheme by introducing the blind feature into the identity-based encryption method. Secondly, by applying the proposed group blind signature scheme to the electricity data collection process, we propose a privacy-aware electricity data collection model. The proposed model ensures the conditional anonymity and traceability of user identity and the privacy protection and unforgeability of electricity consumption data.