Emerging 5G communication paradigms, such as machine-type communication, have triggered an explosion in ad-hoc applications that require connectivity among the nodes of wireless networks. Ensuring a reliable network operation under fading conditions is not straightforward, as the transmission schemes and the network topology, i.e., uniform or clustered deployments, affect the performance and should be taken into account. Moreover, as the number of nodes increases, exploiting natural energy sources and wireless energy harvesting (WEH) could be the key to the elimination of maintenance costs while also boosting immensely the network lifetime. In this way, zero-energy wireless-powered sensor networks (WPSNs) could be achieved, if all components are powered by green sources. Hence, designing accurate mathematical models that capture the network behavior under these circumstances is necessary to provide a deeper comprehension of such networks. In this paper, we provide an analytical model for the connectivity in a large-scale zero-energy clustered WPSN under two common transmission schemes, namely, unicast and broadcast. The sensors are WEH-enabled, while the network components are solar-powered and employ a novel energy allocation algorithm. In our results, we evaluate the tradeoffs among the various scenarios via extensive simulations and identify the conditions that yield a fully connected zero-energy WPSN.