The dynamic clustering-based hierarchical routing protocols are one of the methods used to save energy and increase the lifetime of wireless sensor networks, however, that most of the researches are neglecting the energy expended in election of the heads and formation of clusters in the network. In this paper, we examine the overhead energy caused by hierarchical routing protocols based on dynamic clustering and study its impact on the stability period of the wireless sensor networks. Also, we proposed a solution to limit this energy by reducing the consumed energy in election of heads and clusters formation operations. It is shown through the simulation results that the energy consumed in LEACH setup phase decreases the stability period of these networks and increases the number of dead nodes. And the use of the proposed solution reduced the energy consumption during the election of the heads and the formation of clusters clearly compared to the normal way followed in LEACH, which has increased stability period and the number of live nodes in the network.