The vision of the Smart Grid includes the creation of intelligent electricity supply networks to allow efficient use of energy resources, reduce carbon emissions and are robust to failures. One of the key assumptions underlying this vision is that it will be possible to manage the trading of electricity between homes and micro-grids while coping with the inherent real-time dynamism in electricity demand and supply. The management of these trades needs to take into account the fact that most, if not all, of the actors in the system are self-interested and transmission line capacities are constrained. Against this background, we develop and evaluate a novel market-based mechanism and novel trading strategies for the Smart Grid. Our mechanism is based on the Continuous Double Auction (CDA) and automatically manages the congestion within the system by pricing the flow of electricity. We also introduce mechanisms to ensure the system can cope with unforeseen demand or increased supply capacity in real time. Finally, we develop new strategies that we show achieve high market efficiency (typically over 90%).