# Analytical Model for the Duty Cycle in Solar-Based EH-WSN for Environmental Monitoring
## 7 Duty Cycle for Energy Neutral Operation
We can now introduce the linear approximation into Equation (39) and isolate \(DC\) to obtain the following result:  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\]0  
Note that, in fact, this equation defines a threshold value for \(DC\): if the real \(DC\) is larger, then the energy available at the node, \(E(t,X)\), exhibits a decreasing trend until the node eventually reaches a blocking status; in contrast, if it is smaller, the trend is increasing but evolving towards a flat as the node energy approaches the capacity of the energy buffer. Though this latter situation is feasible from an operational point of view, it reflects a downward dimensioning of the duty cycle that entails some unnecessary performance degradation. The three situations are described in Figure 20, for which we have reused the data of Figure 16 (except the duty cycle). Specifically, for such dataset the duty cycle obtained by applying Equation (42) is \(DC\cong 46\%\). Accordingly, Figure 20 shows the evolution of \(E(t,X)\) corresponding to \(DC=46\%\), \(DC=40\%\) and \(DC=50\%\). Note that the evolution of \(E(t,X)\) has been extended to 10 days in the three curves, in order to clearly show their trends and the effects of the energy capacity in the case of Figure 20 located on the center. The last part of the analysis consists of determining a minimum value for the initial energy level, namely \(E(0,X)\), such that condition Equation (37a) is satisfied for the one-day period. Note that, in our modelling approach, if this condition holds and the duty cycle is less than or equal to the value obtained from Equation (42), then sustained operation is guaranteed. Note also that condition Equation (37a) is fulfilled as long as the minimum of \(E(t,X)\) along the one-day period is greater than or equal to zero. Thus, the next step is to determine such minimum by setting \(\frac{dE(t,X)}{t}=0\). In fact, this process leads to the calculation of both the maximum and minimum of \(E(t,X)\) with the help of the second derivative. Let \(T_{min}\) and \(T_{max}\) denote, respectively, the time instants within a one-day period for which the minimum and maximum of \(E(t,X)\) take place--recall \[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\]1 The results are as follows:  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\]2  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\]3  
For the calculation of \(E_{round}(X)\), either Equation (40) or its linear approximation Equation (41) can be used. Next, by imposing \(E(T_{min},X)\geq 0\) and performing some manipulations, we can end up with the following result for the initial energy:  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\]4  
\[E_{round}(X) = \sum_{i=1}^{CH(X)}(\sigma(c_{i}(X))+1)E_{R,i}(X)+(\sigma(X)+1)E_{T }(X) \tag{40}\]5 Evolution of \(E(t,X)\) for \(DC=46\%\) (**left**), for \(DC=40\%\) (**center**) and for \(DC=50\%\) (**right**). Equations (42) and (45) represent the final results of this paper for the modelling approach and assumptions that have been adopted. Certainly, real weather conditions often deviate from the periodic behavior exhibited by the prediction model proposed in Section 6, and in addition other unexpected changes may also take place (for instance, topological updates). However, we believe that those equations can be used as starting points to develop adaptive schemes that also rely on the capabilities of nodes of measuring their own energy level. We emphasize this task as an issue for further research in the next section.