Power flow models are fundamental to power systems analyses ranging from short-term market clearing and voltage stability studies to long-term planning. Due to the nonlinear nature of the AC power flow equations and the associated computational challenges, linearized approximations (like the DC power flow) have been widely used to solve these problems in a computationally tractable manner. The linearized approximations have been justified using traditional engineering assumptions that under “normal” operating conditions, voltage magnitudes do not significantly deviate from nominal values and phase differences are “small”. However, there is only limited work on rigorously quantifying when it is safe to use these linearized approximations. In this paper, we propose an algorithm capable of computing rigorous bounds on the approximation error in the DC power flow (and, in future extensions, more general linearized approximations) using convex relaxation techniques. Given a set of operational constraints (limits on the voltage magnitudes, phase angle differences, and power injections), the algorithm determines an upper bound on the difference in injections at each bus computed by the AC and DC power flow power flow models within this domain. We test our approach on several IEEE benchmark networks. Our experimental results show that the bounds are reasonably tight (i.e., there are points within the domain of interest that are close to achieving the bound) over a range of operating conditions.