This study presents a mixed-integer second-order conic programming (MISOCP) model for the robust reconfiguration of electrical distribution systems, considering the minimisation of active power losses and reliability constraints. The uncertainty at the reliability data is considered by using a linear and adjustable robust approach. The indices used to evaluate the reliability of the system are the system average interruption frequency index (SAIFI) and the system average interruption duration index (SAIDI), both of which are calculated as functions of the switch status (open or closed). Under radiality constraints, the solution generated by the proposed model is robust in terms of the reliability, i.e. the SADI and SAIFI limits imposed by regulators are not violated even if the uncertain data changes stochastically. The use of an MISOCP model guarantees convergence to optimality by using existing convex optimisation solvers. To evaluate the robustness of each solution generated by the proposed model, a set of Monte Carlo simulations were deployed. Finally, all tests were executed in a 136-node real distribution system.