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|a Wireless sensor network (WSN) nodes are usually battery-driven devices with limited power supply and the network lifetime has undoubtedly become a crucial issue. Normally, in WSNs, the data are sent to a base station or cluster heads through multi-hop routing. Under the condition of multiple hops, the sensor nodes closer to the base station or cluster heads tend to die faster because of the imbalance traffic among sensor nodes that consequently lead to an energy hole problem. For mitigating the problem, a mathematical formula to calculate the optimal number of coronas in both random uniform and non-uniform deployment nodes is proposed in circular networks also known as corona-based networks. To formulate it, the energy consumption was calculated for each corona. Besides that, a lemma and two propositions have been proven in this research. Based on this lemma, the critical corona can be identified. In addition, the two proposed propositions can determine the situation in which the interior corona is the one with the greatest energy hole. In addition, an energy-balanced method, Intermittent Varied Range (IVR) data transmission method is proposed to prolong the overall network lifetime and simulations showed that the second inner-most corona has the most positive effect. For IVR method, only two transmission ranges could increase the network lifetime. There would be an increase more than doubled for environment with path loss exponent of two. In this research, a new relay nodes deployment technique is also proposed to balance the energy consumption in the coronas. Using the proposed mathematical equations, the number of relay nodes for maximizing the lifetime of corona-based WSNs could be calculated.
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