This paper presents a mixed-integer second-order conic programming (MISOCP) model to solve the reconfiguration problem of electrical distribution systems, considering the simultaneous minimization of total active power losses and improvement of customer-oriented reliability indices. The reliability indices considered in this paper are the system average interruption frequency index (SAIFI), the system average interruption duration index (SAIDI), and the energy not supplied (ENS). Under radiality, the proposed model satisfies the operational constraints of the reconfiguration problem, i.e., the voltage magnitude limits of the nodes and the current capacities of the conductors are not violated. The use of an MISOCP model guarantees convergence to optimality via convex optimization software tools. 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