# Optimal Routing for Time-Driven EH-WSN under Regular Energy Sources
## 1 Introduction
Finally, [11] and [12] propose modifications to the Low-Energy Adaptive Clustering Hierarchy (LEACH) protocol [13], which is the most well-known hierarchical protocol for battery-powered sensor networks. Particularly, an Energy Potential (EP) function is introduced in [11] to quantify the node capability for harvesting energy from the environment. The resulting protocol, called EP-LEACH, uses the EP function in the cluster head selection strategy. Compared to LEACH, it is shown to exhibit better performance in terms of lifetime and throughput. A similar approach is followed in [12], but in this case a solar energy prediction model based on a neural network is used to estimate the energy that is to be harvested by a node in a short term horizon. This estimate and the node current residual energy are then combined to determine the probability of the node to become a cluster head. Again, simulation results demonstrate that the proposed clustering method outperforms that of traditional LEACH with respect to average residual energy of nodes and network throughput. A common characteristic of the above routing protocols (except La-CTP) is that they assume an oversimplified MAC layer, where the only sources of energy consumption are the pure transmission and reception of packets. Whereas this is true for MAC protocols based on Time Division Multiple Access (TDMA), it is far from reality for the rest of them, since they typically involve other sources of energy expenditure, namely idle listening, packet retransmissions, etc. Another drawback of current routing protocols for EH-WSN is their reliance on heuristic algorithms, which do not necessarily lead to optimal performance and/or incur significant overhead in terms of computational time or control traffic. Whereas these approaches may constitute the only viable methodology for EH-WSN subject to unpredictable traffic loads and/or energy sources, the regular patterns exhibited by some energy-harvesting processes, like those based on solar power, and some traffic loads, like those resulting from periodic monitoring applications, suggest the possibility of analytical treatment. This paper focuses on EH-WSN devoted to periodic monitoring, also known as time-driven EH-WSN. Under the time-driven paradigm, nodes periodically sense the environment and report the corresponding data to the base station, usually via multi-hop communication. This is the case of numerous sensor-based applications, which span a great diversity of monitored variables. Typically, the design of the MAC layer is based on duty-cycling the communication activity of nodes, in order to avoid fast depletion of energy resources, due to idle listening. Since the reporting frequencies imposed by monitoring applications are commonly very low, every reporting period may contain thousands of unused duty cycles, due to the relatively small number of packets being transmitted or forwarded. Therefore, in spite of having duty-cycled the communication activity, idle listening is still a dominant component in the energy wasted by time-driven sensor networks. Note that this is true for both, battery-powered and energy-harvesting sensor networks, though it is expected that the latter allows for larger duty cycles, or for larger network sizes (in terms of the number of nodes) under the same duty cycle. Particularly, the present work follows an analytical approach to address the routing problem in time-driven (duty-cycled) EH-WSN operating under regular or quasi-periodic energy sources, among which solar radiation is the most representative example. Such approaches rely on a comprehensive energy consumption model for the MAC layer, which takes into account all sources of energy consumption, and on the application of the so-called energy neutral condition for EH-WSN. The following assumptions are adopted:  
* Both planar and hierarchical topologies are taken into account. In the latter case, the routing problem focuses on interconnecting the cluster heads to the base station.