Optimal Placement of Energy Relays in a Three-Hop Wireless RF Power Transfer System

Abstract

The rapid growth of large-scale wireless sensor networks (WSNs) for green Internet of Things (IoT) applications requires reliable and scalable energy delivery, positioning wireless power transfer (WPT) as a key enabler. While two-hop WPT systems have been extensively studied, extending them to a threehop architecture introduces additional non-linear complexities arising from the non-convex coupling between harvest-thentransfer (HTT) duty cycles and multi-path coherent spatial superposition. This paper develops a theoretical three-hop WPT model that integrates free-space path loss (FSPL), linear RFto-DC conversion, and the HTT protocol, thereby providing an upper-bound benchmark for relay placement analysis. To maximize the total power harvested at the target node, we formulate a non-convex relay placement problem and solve it using the $\alpha$-Branch-and-Bound $(\alpha \mathbf{B B})$ global optimization framework. Numerical evaluation of the optimized geometry reveals two key insights. First, actively constraining the relay transmit power extends the energy harvesting duty cycle, which enlarges the coherent integration window and yields a 16.8% energy gain over the single-hop direct link. Second, the optimized topology demonstrates robustness, reliable receiver activation, and maintains the performance advantage over fully efficient twohop baselines, even when rectifying efficiency degrades by 50%.