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As multi-microgrids become readily available, some limited models have been proposed that study operational and power quality constraints with local energy markets independently. This paper proposes a convex optimization model of an energy management system with operational and power quality constraints and interactions in a Local Energy Market (LEM) for unbalanced microgrids (MGs). The LEM consists of a pre-dispatch step and an energy transactions step (ETS). The ETS combines the MGs' objectives while considering two strategies: minimize the cost of buyers or maximize the revenue of sellers. Our proposed model considers harmonic distortion and voltage limit power quality constraints in both steps. Moreover, we model operational constraints such as power flow, power balance, and distributed energy resources behaviors and capacities. We numerically evaluate the proposed model using three unbalanced MGs with residential, industrial, and commercial load profiles, where each microgrid manages its resources locally. Furthermore, we create two groups of cases to analyze the interactions in the local energy market. In the first group, the price of the DSO energy and the surplus from MGs to DSO are the same. The numerical results show that using the increasing revenue strategy promotes MGs to interact more while encouraging them to have high energy prices. When the reducing cost strategy is used, fewer energy interactions occur, and the price of MGs energy is encouraged to be lower.