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We are witnessing pervasive use of wireless sensor networks (WSN)s in a wide variety of applications such as monitoring of road infrastructure. As they are expected to be deployed in harsh environments for long durations, the research community have turned their attention to tapping on ambient energy to power such networks. However, since energy harvesting rates are still significantly lower than the power consumption in each wireless sensor node, the energy availability is sporadic, making the design of runtime policies in WSN powered by ambient energy harvesting (WSN-HEAP) to maximize performance an important but challenging task. In this paper, using extensive simulations, we evaluate the efficacy of transmission power control for 2-D WSN-HEAP deployed in a grid topology in terms of throughput, data delivery ratio and fairness. When a fixed power is assigned to all nodes, we observe a trade-off between throughput and fairness: throughput is maximized at lower powers at the expense of fairness and vice versa. When nodes are assigned powers according to their proximity from the sinks, we observe that assigning the minimum transmission power required for each node to communicate with its nearest sink maximizes all performance metrics. This indicates that minimizing interference dominates over multi-sink redundancy in a 2-D WSN-HEAP.