New challenges have emerged from the integration of renewable energy sources within the conventional electrical grid which powers base stations (BS). Energy-aware traffic offloading brings a promising solution to maintain the user performance while reducing the carbon footprint. Focusing on downlink cellular networks consisting of on-grid, off-grid and hybrid BSs, we propose a novel power-aware biased cell association where each user independently partitions BSs into two sets and applies different association biases for each, depending on the type of power, renewable or not, that can be requested for service. The gain provided by such strategy regarding the probability of power outage and the average grid power consumption is investigated. To capture their dual nature, the bias applied for association with a hybrid BS is not constant among users nor over time, and is dynamically tailored to the fluctuations of the BS battery level, the user power requirement and the estimated power consumed to serve other users potentially associated with the same BS. Such approach allows to efficiently share the available energy among BSs and turns high heterogeneity in the BS powering into advantage.