  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.
