Recently an online electric vehicle (OLEV) concept has been introduced, where vehicles are propelled through the wirelessly transmitted electrical power from the infrastructure installed under the road while moving. The absence of secure-and-fair billing is one main hurdle to widely adopt this promising technology. This paper introduces a secure and privacy-aware fair billing framework for OLEV on the move through the charging plates installed under the road. We first propose two extreme lightweight mutual authentication mechanisms, a direct authentication and a hash chain-based authentication between vehicles and the charging plates that can be used for different vehicular speeds on the road. Second we propose a secure and privacy-aware wireless power transfer on move for the vehicles with bidirectional auditability guarantee by leveraging game-theoretic approach. Each charging plate transfers a fixed amount of energy to the vehicle and bills the vehicle in a privacy-aware way accordingly. Our protocol guarantees secure, privacy-aware, and fair billing mechanism for the OLEVs while receiving electric power from the road. Moreover our proposed framework can play a vital role in eliminating the security and privacy challenges in the deployment of power transfer technology to the OLEVs.