Keeping the pace of transfer capacity expansion in the power system is becoming increasingly difficult due to the rapid growth of distributed variable renewable energy. A flexible alternative to network reinforcement is virtual power lines. A virtual power line (VPL) consists of large-scale energy storage systems (ESS) located to alleviate transmission constraints. The ESS are operated in coordination to provide extra capacity while simultaneously relieving congestion on constrained transmission elements. We propose a decentralized controller for the real-time operation of virtual power lines. Through proportional-integral control, model-free VPL operation is obtained, and congestion is eliminated with minimal requirements for online measurements. We develop an analytical nonlinear model for controller tuning in the presence of ESS power and energy limitations. A flow sensitivity-based approach is used to estimate how the VPL effectiveness is influenced by the placement and sizing of the ESS. Finally, the performance of the controller under various conditions is demonstrated in the Kundur four-machine two-area and the Nordic 44 test systems.