**Abstract :**This paper investigates a method to distributively solve a
Network Utility Maximization (NUM) problem with coupled variables and applies it to study power control in wireless sensor networks (WSNs). We present a dual decomposition-based consistency price algorithm to solve the coupled problem. However, the
consistency price algorithm suffers from slow convergence. We
then propose a two-step method to address the given issue. The
first step is to build up a global consensus problem by introducing slack variables to transform the NUM problem with globally
coupled variables into a NUM problem with coupled constraints.
The second step is to design a distributed algorithm that combines
the first-order gradient/subgradient method and a local consensus
algorithm to solve the global consensus problem. The proposed algorithm is a primary algorithm which has faster convergence speed
than the consistency price algorithm which is a primary-dual algorithm. Experimental results have demonstrated the effectiveness of
our proposed approach.

**Index terms :**Consensus algorithm, network utility maximization, power control, primary-dual algorithm, wireless sensor networks.