Local energy markets, platforms in which pro-sumers in the same Low Voltage network can trade energy among themselves, offer a great opportunity to incentivize the consumption of locally generated energy. Unfortunately, traditionally proposed implementations of local energy markets such as simple double auctions and peer to peer exchanges do not fully exploit the available flexibility in these systems. We design a market mechanism that exploits the characteristics of the players, providing them with expressive bids to represent their flexibility, which we assume is due to energy storage. The proposed market is not obviously manipulable and can be cleared by solving a linear programming problem that grows linearly in the number of participants. Using realistic data, we benchmark the proposed mechanism against sequential auctions and peer to peer exchanges often used in the literature. Our numerical results show that the proposed mechanism outperforms traditional implementations.