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We consider a network of rechargeable sensors, deployed redundantly in a random sensing environment, and address the problem of how sensor nodes should be activated dynamically so as to maximize a generalized system performance objective. The optimal sensor activation problem is a very difficult decision question, and under Markovian assumptions on the sensor discharge/recharge periods, it represents a complex semi-Markov decision problem. With the goal of developing a practical, distributed but efficient solution to this complex, global optimization problem, we first consider the activation question for a set of sensor nodes whose coverage areas overlap completely. For this scenario, we show analytically that there exists a simple threshold activation policy that achieves a performance of at least 3/4 of the optimum over all possible policies. We extend this threshold policy to a general network setting where the coverage areas of different sensors could have partial or no overlap with each other, and show by simulations that the performance of our policy is very close to that of the globally optimal policy. Our policy is fully distributed, and requires the sensor nodes to only keep track of the node activation states in its immediate neighborhood. We also consider the effects of spatial correlation on the performance of the threshold activation policy, and the choice of the optimal threshold.