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In this paper, we formulate a problem of power profile market clearing and develop a distributed market clearing scheme with explicit consideration of high battery penetration. The power profile market is a multiperiod electricity market in which each aggregator aims at making the highest profit by transacting a power profile, i.e., a time sequence of energy amounts at several time slots, that is generated by dispatchable power generation as well as the charge and discharge of batteries. It is theoretically shown that the clearing price profile during the time period of interest tends to level off in the high penetration of batteries. This finding enables to develop a distributed market clearing scheme that is implemented as a bidding strategy for the total energy amount during the period followed by a distributed iterative algorithm for profile imbalance minimization. Numerical simulations demonstrate the price leveling-off led by high battery penetration and the efficiency of the proposed distributed scheme.