The optimal reactive power dispatch (ORPD) problem is an alternating current optimal power flow (ACOPF) problem where discrete control devices for regulating the reactive power, such as shunt elements and tap changers, are considered. The ORPD problem is modelled as a mixed-integer nonlinear optimization problem and its complexity is increased compared to the ACOPF problem, which is highly nonconvex and generally hard to solve. Recently, convex relaxations of the ACOPF problem have attracted a significant interest since they can lead to global optimality. We propose a tight conic relaxation of the ORPD problem and computational results show that a round-off technique applied with this relaxation instead of the continuous relaxation of the ORPD problem leads to near-global optimal solutions with very small guaranteed optimality gaps on selected MATPOWER test cases with up to 3375 buses.