In modern, large scale interconnected power grids, low-frequency oscillation is a key roadblock to improved power transmission capacity. Supplementary generator control, flexible AC transmission system (FACTS), and high voltage direct currents (HVDC) are engineered devices designed to damp such low frequency swings. In this paper a neural network-based approximate dynamic programming method, namely direct heuristic dynamic programming (direct HDP), is applied to power system stability enhancement. Direct HDP is a learning and approximation based approach to addressing nonlinear system control problems under uncertainty, and it is also a model-free design strategy. The action and critic networks of the direct HDP are implemented using multi-layer perceptrons; learning is carried out based on the interactions between the controller and the power system. For this design approach, real time system responses are provided through wide-area measurement system (WAMS). The controller learning objective is formulated as a reward function that reflects global characteristics of the power system under low frequency oscillation, as well as tight coupling effects among system components. Direct HDP control design is illustrated by case studies, which are also used to demonstrate the learning control performance. The proposed direct HDP learning control is also developed as a new solution to a large scale system coordination problem by using the China Southern Power Grid as a major test bed.