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This paper proposes a random network model for a device-to-device (D2D) underlaid cellular system using stochastic geometry and develops centralized and distributed power control algorithms. The goal of centralized power control is twofold: ensure that the cellular users have sufficient coverage probability by limiting the interference created by underlaid D2D users, while scheduling as many D2D links as possible. For the distributed power control method, the optimal on-off power control strategy is proposed, which maximizes the sum rate of the D2D links. Expressions are derived for the coverage probabilities of cellular, D2D links, and the sum rate of the D2D links in terms of the density of D2D links and the path-loss exponent. The analysis reveals the impact of key system parameters on the network performance. For example, the bottleneck of D2D underlaid cellular networks is the cross-tier interference …