In this paper, we present theory and a design of the online double auction for the trading of energy within a smart grid with microgrids (MGs). The online double auction has the potential to enable the allocation of surplus electricity to the MGs that need electricity with the highest gain in the real-time market. Nonetheless, two critical issues remain challenging when designing an effective online double auction scheme in such a system. First, as the agents are allowed to arrive and depart at any time, the auctioneer needs to make decisions without the information of further bids and asks. Second, the economic properties of strategy-proof, individual rational, and (weak) budget balance should be satisfied. To address these issues and enable multiunit electricity trading among local MGs, in this paper, we propose a strategy-proof online double auction (SODA) scheme, in which the surplus and insufficient MGs in the system are treated as sellers and buyers, respectively, and the MG center controller is capable of maximizing the social welfare of MGs by appropriately matching buyers and sellers. Via theoretical analysis, we prove that SODA can achieve the properties of individual rationality, (weak) budget balance, strategy-proofness, and computational efficiency. Experiments also show that SODA is capable of reducing the energy purchasing cost of the MGs and shifting the peak-load, while achieving great performance with respect to social welfare, seller/buyer satisfaction ratio, social efficiency, and computation overhead.