Energy harvesting brings a key solution to the increasing energy bill and environmental concerns but, at the same time, the network availability may be deteriorated due to potential energy shortage. In this paper, we analyze the performance of off-grid small-cell base stations (scBS) with finite battery capacity and design a new power-availability-aware cell association based on periodical broadcast of the scBS battery level. Each mobile terminal (MT) targets its own set of available scBSs before association, i.e. the set of scBSs that can guarantee service provided (i) the scBS battery level, (ii) the power required to satisfy a received power constraint at each MT, given the scBS-MT distance and the shadowing attenuation, and (iii) the estimated power consumed to serve other MTs potentially associated to the same scBS, which is computed using stochastic geometry tools. Next, we develop for it a tractable performance analysis and derive closed-form expressions for the probability of power outage and the coverage probability. By dynamically adapting to the fluctuations of the base station battery and user power requirement, the proposed cell association allows a more even distribution of the available energy in the network, brings robustness against harvesting impairment and thereby, significantly outperforms conventional strategies.