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Sensing coverage reflects the quality of surveillance of a field by a wireless sensor network (WSN). This paper investigates the problem of minimum connected k-coverage in WSNs, where each point in a field is covered (or sensed) by at least k active sensors while minimizing the necessary total number of active sensors and ensuring connectivity between them. In this paper, we propose two k-coverage protocols using different scheduling approaches. In the first protocol, called self-scheduling driven k-coverage (SSCk), each sensor turns itself on based on the local information it has about its sensing neighbors in order to k-cover its sensing range. The second protocol, called triggered-scheduling driven k-coverage (TSCk), allows a sensor to trigger a necessary number of its sensing neighbors to become active in order to achieve k-coverage of its sensing range. Then, we relax some commonly used assumptions for coverage configuration protocols in WSNs to promote the use of SSCk and TSCk in realworld sensing applications. Simulation results show that TSCk outperforms SSCk with regard to the number of sensors required for connected k-coverage as well as the network lifetime. We find that SSCk outperforms an existing connected k-coverage protocol for WSNs.