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In this paper, we explore a practical system setting where a rack-aware storage system consists of racks, each containing a few parity checks, referred to as a rack-aware system with locality. To minimize cross-rack bandwidth in this system, we organize the repair sets of locally repairable codes into racks and investigate the problem of repairing erasures in locally repairable codes beyond the code locality. We devise two repair schemes to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. We then establish a cut-set bound for locally repairable codes under the rack-aware model with locality. Using this bound we show that our second repair scheme is optimal. Furthermore, we consider the partial-repair problem for locally repairable codes under the rack-aware model with locality, and introduce both repair schemes and bounds for this scenario.