In this paper we consider a two-hop relay network consisting of a source, a destination and multiple relay nodes. For a given quality of service (QoS) requirement such as transmission rate between the source and the destination, relay nodes have an adaptive power control scheme to meet the QoS requirement. The adaptive power control scheme is assumed to be based on the channel condition between relay nodes and the destination. It is then an interesting problem to design an optimal relay network that minimizes total average energy consumption. By considering the details on the operations of the relay selection protocol considered in this paper, we first develop a mathematical model for the relay network and analyze total average energy consumption. Based on our analytic results, we formulate an optimization problem with which we can design an optimal relay network that minimizes total average energy consumption. We provide some numerical examples to investigate the behavior of total average energy consumption of the relay network.