Fast High-Order Consensus Time Synchronization Protocol in Industrial Wireless Sensor Networks

Slow convergence remains a critical limitation hindering the practical deployment of consensus-based time synchronization protocols (CTSPs). Increasing algebraic connectivity is a key mechanism for improving the convergence speed of distributed algorithms. However, existing strategies inevitably introduce redundant packet communication, while forwarding stale timing information may degrade synchronization accuracy. To address this challenge, this paper proposes a high-order consensus time synchronization protocol (HCTSP). Unlike traditional CTSPs, HCTSP incorporates previous clock states from two-hop neighboring nodes to establish a virtual topology and further employs this information to enhance the estimation of logical clock parameters, thereby achieving fast estimation of clock parameters while effectively suppressing fluctuations in parameter estimation caused by a large initial synchronization error. Although the proposed method utilizes single-hop communication to relay information from non-adjacent nodes—an indirect transmission mechanism that inherently introduces additional communication overhead—we further develop an accumulator-based redundant information optimization scheme. Furthermore, the consensus algorithm integrates both an accumulator-based update mechanism and multi-hop historical memory, partially alleviating the impact of Gaussian communication delay jitter on clock correction. Theoretical proofs have verified the fast convergence of the proposed protocol. Extensive simulation experiments also demonstrate the superior efficiency of HCTSP in terms of convergence speed, communication overhead, and synchronization accuracy. Specifically, in random networks with 25 nodes, there is an approximately 50% reduction in single-round synchronization message length and a 66.67% decrease in total packet exchange volume compared to the virtual topology-based time synchronization protocol (VTSP). In the typical ring topology, where the convergence speed of the consensus algorithm is slow, HCTSP has a 59.33% increase in convergence speed compared to VTSP and a 75.29% increase compared to the gradient time synchronization protocol (GTSP).