In this paper a neural network-based approximate dynamic programming method, namely direct heuristic dynamic programming (direct HDP), is applied to power system stability control. Direct HDP makes use of learning and approximation to address nonlinear system control problems under uncertainty. The contribution of the paper includes a convergence proof of the direct HDP algorithm using an LQR framework. Under this setting, the paper proposes a direct HDP learning control algorithm for a static var compensator (SVC) supplementary damping control in a standard benchmark power system. The results are used to evaluate the online learning ability of the proposed direct HDP controller, and also to demonstrate that the learning controller does converge to the theoretical limit as derived.