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Vehicle-to-grid (V2G) is emerging as an attractive paradigm in smart grid, and provides power and information services by periodically collecting power status of battery vehicles (BVs). During a BV's interaction with power grid, it may be in one of the following states: charging, fully-charged (FC), and discharging. In this paper, we identify that there are unique security challenges in a BV's different battery states. Accordingly, we propose a battery status-aware authentication scheme (BASA) to address the issue for V2G networks. In BASA, 1) an aggregated-identifier is proposed during the charging-to-FC state transition to ensure that BVs can be authenticated without disclosing their real identities; 2) selective disclosure based challenge-response authentication is presented during the FC-to-discharging phase to realize anonymous data transmission; 3) an aggregated-status is reported during the discharging-to-charging transition in order to hide a BV's power level from an aggregator. In addition, we perform comprehensive security analysis, which shows that BASA achieves both privacy preservation and security protection during battery state transitions. The analysis also indicates that battery status awareness is crucial for BVs' secure operations for V2G networks in smart grid.