In a smart grid, distributed energy management (DEM) is a promising approach to realize reliable and efficient operation. Since cyberattack is one of the crucial threats faced by smart grid, the investigation of the effect of cyberattacks on DEM is both a theoretical merit and a practical value. This paper considers the typical DEM problem, i.e., distributed economic dispatch (ED) problem, under attacks. Under a well-developed consensus-based ED protocol, we first define the stealthy attack through false data injection for offline and online cases, respectively. The necessary and sufficient conditions are provided to guarantee convergence of the algorithm when the attacker only injects false data into the broadcast information. And the offline stealthy attack can be realized under this kind of attack. Then, we prove that there exists no stealthy attack injecting false data into the broadcast information for the online case. Further, we provide the stealthy attack by merely injecting constant false data into generation cost parameters for the online case. We also prove that for every node, there exists false data injection into generation cost parameters, which reduces the generation efficiency. Simulation studies validate the theoretical results.