Smart meters (SMs) measure and report users' energy consumption to the utility provider (UP) in almost real-time, providing a much more detailed depiction of the consumer's energy consumption compared to their analog counterparts. This increased rate of information flow to the UP, together with its many potential benefits, raise important concerns regarding user privacy. This work investigates, from an information theoretic perspective, the privacy that can be achieved in a multi-user SM system in the presence of an alternative energy source (AES). To measure privacy, we use the mutual information rate between the users' real energy consumption profile and the SM readings that are available to the UP. The objective is to characterize the \textit{privacy-power function}, defined as the minimal information leakage rate that can be obtained with an average power limited AES. We characterize the privacy-power function in a single-letter form when the users' energy demands are assumed to be independent and identically distributed over time. Moreover, for binary and exponentially distributed energy demands, we provide an explicit characterization of the privacy-power function. For any discrete energy demands, we demonstrate that the privacy-power function can always be efficiently evaluated numerically. Finally, for continuous energy demands, we derive an explicit lower-bound on the privacy-power function, which is tight for exponentially distributed loads.