We propose an extended bidding structure to allow more realistic demand characteristics and behaviors to be expressed via flexible bids. In today's ISO-run energy markets, demand bid formats are all separable over time. However, a significant and growing segment of demand can be shifted across time and therefore has no way to bid its true valuation of consumption. We propose additional bid types that allow deferrable, adjustable and storage-type loads to better express their value, and thus elicit demand response in the most natural way - via direct participation in the market. We show that the additional bid types are easily incorporated into the existing market with no technological barrier and that they preserve the market's efficiency and incentive-compatibility properties. Using real market data, we give a numerical demonstration that the extended bid format could substantially increase social welfare, and also present additional insight on storage expansion scenarios. (c) 2015 Elsevier Ltd. All rights reserved.