# Optimal Routing for Time-Driven EH-WSN under Regular Energy Sources
## 4 Energy Neutral Operation  
In contrast to conventional battery-powered WSN, which are designed with the objective of maximizing network lifetime, in the case of EH-WSN, the objective is to maximize performance under self-sustained operation. More formally, this condition is known as Energy Neutral Operation (ENO), which essentially means that, in a given period of time, the energy balance at a node is non-negative. Also, many energy-harvesting mechanisms in sensor networks obey the so-called harvest-store-consume supply alternative, which consists of combining the energy-harvesting subsystem with a buffer for energy storage (rechargeable battery or supercapacitor) [3]. According to this model, and assuming that the energy buffer does not have any inefficiency in charging and does not leak any energy over time, ENO can be mathematically formulated as follows [45] (the notation has been adapted):  
\[E(t)=E(0)+\int_{0}^{t}P_{out}(u)du-\int_{0}^{t}P_{c}(u)du\geq 0,\ \forall t\geq 0. \tag{11}\]  
In this expression, \(E(t)\) denotes the energy balance at time \(t\), \(P_{out}(t)\) is the output power delivered by the harvesting subsystem at time \(t\), \(P_{c}(t)\) is the power consumed by the device at time \(t\) and, obviously, \(E(0)\) is the energy initially stored in the buffer. Let us assume that the energy source exhibits a regular pattern, with periodicity \(T_{S}\) (energy-harvesting period). Accordingly, ENO can be formulated for one period \(T_{S}\), since this is typically a very large multiple integers of the reporting period given by \(T_{rnd}\). For instance, the most representative periodic source is the sunlight (really, it is quasi-periodic, but this will be considered further), as photovoltaic circuits constitute the most efficient form of energy conversion, at least for current sensor networks. In this case, \(T_{S}\) corresponds to a one-day interval, which is much larger than usual reporting periods (one or several minutes). For the same reason, we can undoubtedly assume that the energy consumed by the sensor node is uniformly distributed over the round duration, implying that power consumption is independent of time: \(P_{c}(t)=P_{c}=\frac{E_{round}}{T_{md}}\). Under these assumptions, ENO can be reformulated for a given node \(X\) by imposing that the energy at the beginning of an energy-harvesting period is equal to the energy at the beginning of the previous energy-harvesting period:  
\[E(T_{S},X)=E(0,X)+\int_{0}^{T_{S}}P_{out}(u,X)du-\int_{0}^{T_{S}}\frac{E_{ round}(X)}{T_{md}}du=E(0,X). \tag{12}\]  
Note that, in order to guarantee that \(E(t,X)\geq 0\ \forall t\), a condition on \(E(0,X)\) (initial energy) must also be fulfilled. Moreover, sufficiently large values of \(E(0,X)\) release nodes from the need to enter a recharging state, even if significant irregularities occur during the energy-harvesting process (for instance, cloudy days in the case of solar-based sensor networks). However, despite the importance of \(E(0,X)\) as an energy repository that mitigates the irregularities of the energy sources considered in this paper, its mathematical formulation has been omitted here as it is not relevant to the subsequent analysis. Then, by combining Equations (10) and (12), we can end up with the following expression for the duty cycle of node \(X\) in terms of its energy harvesting capability and traffic load:  
\[\frac{DC(X)}{100}\cong\frac{E_{out}(T_{S},X)}{E_{l}}\frac{T_{l}}{T_{S}}-\sigma (X).\frac{E_{xrpkt}+E_{xrpkt}+E_{trigger}(X)}{E_{l}}\frac{T_{l}}{T_{md}}. \tag{13}\]  
Here, \(E_{out}(T_{S},X)=\int_{0}^{T_{S}}P_{out}(u,X)du\). Equation (13) extends the ENO condition obtained in [14] to regular energy sources and generic duty-cycled MAC protocols. It makes it explicit the dependence of the duty cycle on the specific operating conditions of each node in the network, namely energy harvesting capability and traffic load. In particular, the presence of the energy harvesting term contributes to achieving much larger duty cycles in EH-WSN than those obtained in battery-powered WSN.