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An experiment is carried out to measure the power consumption of households. The analysis on the real measurement data shows that the significant change of power consumption arrives in a Poisson manner. Based on this experiment, a novel wireless communication scheme is proposed for the advanced metering infrastructure (AMI) in smart grid that can significantly improve the spectrum efficiency. The main idea is to transmit only when a significant power consumption change occurs. On the other hand, the policy of transmitting only when change occurs may bring a security issue; i.e., an eavesdropper can monitor the daily life of the house owner, particularly the information of whether the owner is at home. Hence, a defense scheme is proposed to combat this vulnerability by adding artificial spoofing packets. It is shown by numerical results that the defense scheme can effectively prevent the security challenge.