  Wireless power charging enables portable devices to be permanently unplugged.
Due to its low transmission power and low transmission efficiency, it requires
much longer time slot to charge users compared with that for data transmission
in wireless communication networks. Besides, each user's demand urgency needs
to be taken into consideration for power allocation. Therefore, new algorithms
are essential for wireless power allocation in multi-user wireless charging
networks. In this paper, this problem is formulated as a static noncooperative
game. It is shown that there exists a unique Nash equilibrium, which is the
static state of the wireless power charging network. A distributed power
allocation algorithm is proposed to compute the Nash equilibrium of the game.
The main result of the paper consists of rigorous analysis of the distributed
algorithm for power allocation. The algorithm is shown to converge to Nash
equilibrium of the game with exponentially convergence rate for arbitrary
initial value with synchronous scheduling. Moreover, the distributed algorithm
is also convergence guaranteed with asynchronous scheduling under communication
delay and packet drops. Numerical simulations prove the correctness of the
analysis and demonstrate the fast convergence of the algorithm and the
robustness to synchronous scheduling.
