The technology of differential privacy, adding a noise drawn from the Laplace distribution, successfully overcomes a difficulty of keeping both the privacy of individual data and the utility of the statistical result simultaneously. Therefore, it is prevalent to use a rechargeable battery as the noise for achieving differential privacy in the application of smart grid. Unfortunately, to the best of our knowledge, we observe that the existing privacy protection mechanisms cannot satisfy differential privacy, when considering physical restrictions of the battery in practice. In this paper, we first classify two types of challenges caused by two physical restrictions, the maximum charging/discharging rate and the capacity of battery. We then propose a stateless privacy protection scheme by exploring a boundary-changeable distribution for noise, and prove this scheme satisfies differential privacy, with regard to the first type of challenge. We further explain the difficulty to achieve differential privacy under the second type of challenge, and formalize the definition of a relaxed differential privacy. Finally, we present a stateful privacy protection scheme that satisfies the relaxed differential privacy. Experimental analysis shows that the maximum privacy leakage of our privacy protection schemes at each time point stably outperforms that of the existing work.