Dispersed storage systems (DSSs) can represent an important near-term solution for supporting the operation and control of active distribution networks (ADNs). Indeed, they have the capability to support ADNs by providing ancillary services in addition to energy balance capabilities. Within this context, this paper focuses on the optimal allocation of DSSs in ADNs by defining a multi-objective optimization problem aiming at finding the optimal trade-off between technical and economical goals. In particular, the proposed procedure accounts for: 1) network voltage deviations; 2) feeders/lines congestions; 3) network losses; 4) cost of supplying loads (from external grid or local producers) together with the cost of DSS investment/maintenance; 5) load curtailment; and 6) stochasticity of loads and renewables productions. The DSSs are suitably modeled to consider their ability to support the network by both active and reactive powers. A convex formulation of ac optimal power flow problem is used to define a mixed integer second-order cone programming problem to optimally site and size the DSSs in the network. A test case referring to IEEE 34 bus distribution test feeder is used to demonstrate and discuss the effectiveness of the proposed methodology.