Consider a distributed storage system where parts of the source file fragments in storage nodes are lost. We denote a storage node that lost a part of its fragments as a faulty storage node and a storage node that lost non of its fragment as a complete storage node. In a process, termed as partial repair, a set of storage nodes (among faulty and complete storage nodes) transmit repairing fragments to other faulty storage nodes to recover the lost fragments. We first investigate the optimal partial repair in which the required bandwidth for recovering the lost fragments is minimal. Next, we assume that an eavesdropper wiretaps a subset of links connecting storage nodes, and overhears a number of repairing fragments. We then study optimal secure partial-repair in which the partial-repair bandwidth is minimal and the eavesdropper obtains no information about the source file by overhearing the repairing fragments. We propose optimal secure codes for exact partial-repair in a special scenario.