Network update enables Software-Defined Networks (SDNs) to optimize the data plane performance. The single update focuses on processing one update event at a time, i.e., updating a set of flows from their initial routes to target routes, but it fails to handle continuously arriving update events in time incurred by high-frequency network changes. On the contrary, the continuous update proposed in ``Update Algebra'' can handle multiple update events concurrently and respond to the network condition changes at all times. However, ``Update Algebra'' only guarantees the blackhole-free and loop-free update. The congestion-free property cannot be respected. In this paper, we propose Coeus to achieve the continuous update while maintaining consistency, i.e., ensuring the blackhole-free, loop-free, and congestion-free properties simultaneously. Firstly, we establish the continuous update model based on the update operations in update events. With the update model, we dynamically reconstruct the operation dependency graph (ODG) to capture the relationship between update operations and link utilization variations. Then, we develop a composition algorithm to eliminate redundant operations in update events. To further speed up the update procedure, we present a partition algorithm to split the operation nodes of the ODG into a series of suboperation nodes that can be executed independently. The partition algorithm is proven to be optimal. Finally, extensive evaluations show that Coeus can improve the update speed by at least 179% and reduce redundant operations by at least 52% compared with state-of-the-art approaches when the arrival rate of update events equals three times per second. IEEE