We consider an energy-limited network, where a source node intends to communicate with two energy harvesting (EH) near-users and one far-user. We design a nonlinear EH-based alternate cooperative nonorthogonal multiple access (CNOMA) scheme, where the source node alternately superimposes the data of each near-user and the far-user and broadcasts the composite signal. When one near-user decodes the data of both the far-user and itself by using the adaptive interference cancellation technique, the other near-user can harvest energy from the received signal. The two near-users alternately and adaptively forward data to the far-user. We properly model the energy statuses of both the two near-users by defining a group of unequally-spaced discrete energy levels. We analyze the average success probability and throughput of each user, and use the differential evolution (DE) algorithm to efficiently achieve the maximum system throughput by jointly optimizing both the time and power allocation factors. We also present two benchmark EH-based transmission schemes for comparison. Extensive simulations are conducted to validate our theoretical analysis and reveal the impacts of various parameters.