  Energy reduction for wireless systems becomes more and more important due to
its impact on the operation cost and global carbon footprint. In this paper, we
investigate three kinds of energy-efficient association schemes under open loop
power control for uplink heterogeneous cellular networks, which are formulated
as a whole energy efficiency maximization problem, a sum energy efficiency
maximization problem and a utility maximization problem respectively. The third
case takes account of load balancing level and user's fairness in the
energy-efficient association. Considering that the first problem is in a
fractional mixed-integer form, we introduce an energy efficiency parameter to
convert it into a parametric subtractive form, and then design an effective
iterative algorithm to achieve the optimal solutions. As for the third problem,
we first introduce a dual variable to decouple the constraint and then develop
a distributed algorithm using dual decomposition. In addition, we also give the
convergence proof for the proposed algorithms. In order to confirm the
effectiveness of energy-efficient user association algorithms, we introduce
other association rules for comparison, and investigate the influences of
different parameters on the association performance of these association rules.
