Virtual power lines (VPLs) use utility-scale energy storage systems (ESS), acting as an alternative to reinforcing or building new infrastructure in congested transmission or distribution systems. This paper proposes a mathematical programming model for the congestion management of distribution systems by VPLs' optimal operation considering voltage and current magnitude operational limits. The model considers ESS and the optimal operation of lines' switches to minimize the energy curtailment among system nodes. The proposed formulation considers demand response and curtailment in photovoltaic (PV) generation, while facilitating the nodes' collective participation by adopting a Rawlsian social choice function. The model is cast as a mixed-integer second-order cone programming problem and is tested on a radial 34-node test system. Results showed a reduction of the curtailed energy from loads of more than 40% and from PVs of more than 75% using VPLs, while the fairness of the decisions was evaluated using Jain's index.