Power utilities globally are increasingly upgrading to Smart Grids that use bi-directional communication with the consumer to enable an information-driven approach to distributed energy management. Clouds offer features well suited for Smart Grid software platforms and applications, such as elastic resources and shared services. However, the security and privacy concerns inherent in an information-rich Smart Grid environment are further exacerbated by their deployment on Clouds. Here, we present an analysis of security and privacy issues in a Smart Grids software architecture operating on different Cloud environments, in the form of a taxonomy. We use the Los Angeles Smart Grid Project that is underway in the largest U.S. municipal utility to drive this analysis that will benefit both Cloud practitioners targeting Smart Grid applications, and Cloud researchers investigating security and privacy.