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The widespread adoption of distributed generation (DG) is changing the operation of distribution networks. To increase the control of power flows and avoid the detriments of DG intermittency, active distribution networks (ADN) emerge as a feasible solution. In this paper, we propose a convex model for ADN expansion planning. Four active network management (ANM) schemes are considered: DG active power curtailment, DG reactive power control, on-load tap changer (OLTC) tap adjustment, and shunt compensation device control. The problem is formulated as a mixed integer second-order cone programming problem (MISOCP) using the distflow equations and second-order cone relaxations (SOCR). By scenario analysis, we address the uncertainty associated with power demand, renewable generation, and electricity prices. The proposed model is used to study the benefits of different combinations of ANM schemes to a modified IEEE 33-bus system and compare them with the individual benefits of each scheme. The results indicate that ANM schemes can reduce total network costs, reduce losses, delay network reinforcement, and improve the voltage profile. In addition, some combinations of schemes develop synergies in which the combined economic benefits are greater than the sum of the individual benefits. [Display omitted] [ABSTRACT FROM AUTHOR]