  The concept of device-to-Device (D2D) communication as an underlay
coexistence with cellular networks gains many advantages of improving system
performance. In this paper, we model such a two-layer heterogenous network
based on stochastic geometry approach. We aim at minimizing the expected power
consumption of the D2D layer while satisfying the outage performance of both
D2D layer and cellular layer. We consider two kinds of power control schemes.
The first one is referred as to independent power control where the transmit
powers are statistically independent of the networks and all channel
conditions. The second is named as dependent power control where the transmit
power of each user is dependent on its own channel condition. A closed-form
expression of optimal independent power control is derived, and we point out
that the optimal power control for this case is fixed and not relevant to the
randomness of the network. For the dependent power control case, we propose an
efficient way to find the close-to-optimal solution for the power-efficiency
optimization problem. Numerical results show that dependent power control
scheme saves about half of power that the independent power control scheme
demands.
