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In this paper, we introduce a new cost function for SIR-Based uplink power control algorithms that takes into account transmission power level as well as target Signal-to-Interference Ratio (SIR) deviation. Existence of Nash equilibrium point, its uniqueness and the algorithm convergence are proved in the framework of so-called standard power control schemes. As shown through simulations, the proposed method results in significant transmit power reduction while achieving almost the same average SIR level. In addition, it is shown that the new scheme results in improved overall network fairness in comparison with another algorithm.