## 4 Numerical Results
### Influence of System Parameters
#### 4.1.3 Effect of Recharge and Unequal Initial Energies  
Without loss of generality, the effect of recharge and unequal initial energies can be separately analyzed from the single realization of the consumptions used before and an optimization span \(F=1\). Again, we have used \(N=10\), \(K=400\), \(B_{\min}=0.001\), \(B_{\max}=1\) and \(\rho=0.98\). Figure 9 shows some results assuming normalized (with respect to \(S_{\max}\)) initial energies equally spaced in \([0.8,1]\) and a single recharge at \(t=t_{\rm r}=200\), with normalized recharge values equally distributed in \([0,0.2]\). These equally-spaced energies were adopted to facilitate visualization in the graphs. From this figure, the expected lifetime reduction produced by initial energies smaller than the maximum can be noticed, combined with the expected lifetime improvement achieved due to recharge, no matter the weights of the objective function. Nonetheless, the most important observation comes from the equalization of the consumptions departing from unequal initial energies and the reestablishment of the normal operation of the optimization process after the disruption in the residual energies caused by recharge. Observe that the sensor nodes with less initial energies are kept silent over larger intervals compared to those with larger initial energies, up to the moment when all consumptions  
Figure 8: Residual energies obtained by the proposed optimization method with \((w_{1},w_{2})=(1,0)\), for \(F=5\) and \(\rho=0.98\) (**a**); \(F=10\) and \(\rho=0.98\) (**b**); \(F=20\) and \(\rho=0.98\) (**c**); and \(F=5,10,20\) and \(\rho=0.999999\) (**d**). The non-optimized network is also considered for reference. are equalized in the case of \((w_{1},w_{2})=(1,0)\) or the characteristic operation is reestablished in the case of \((w_{1},w_{2})=(0,1)\). A similar behavior occurs regarding the recharge. It is worth highlighting that the effect of recharge only starts to be taken into account by the optimization during the block of frames subsequent to the one in which the recharge event occurred. This is not evident in Figure 9 because \(F=1\). In Figure 10, this specific character is demonstrated in terms of the residual energies and corresponding activity levels. To plot this figure, we have adopted \(F=20\) and the same previously considered realization of the consumptions with \(\rho=0.99999\). The set of initial energies and recharge values was the same as before, i.e., normalized initial energies and recharge values equally spaced in \([0.8,1]\) and \([0,0.2]\), respectively. For convenience, the recharge instant is now at \(t=t_{\rm r}=184\), that is it occurs sixteen frames before the beginning of the eleventh block of \(F=20\) frames. From the graph (**a**) of Figure 10, a behavior similar to the one identified in Figure 9 can be noticed regarding the unequal initial energies. However, the effect of recharge is now different: notice that after recharge, the sensor nodes maintain their energy expenditures, up to \(t=200\), as defined by the previous optimization event, that is the discharge rates are the same irrespective of the new energy levels in the sensor nodes' batteries. Only at \(t=200\) the optimization is aware of the recharge and controls the activity levels so that the consumptions start to be equalized again. The dynamics of the corresponding activity levels are shown in the graph (**b**) of Figure 10, from where the large variabilities associated with the adaptation to the unequal initial energies at the beginning of the analysis interval can be noticed and with the re-adaption due to recharge after \(t=200\). The spikes at each integer multiple of \(F=20\) in the activity levels are due to the compensation of small variations in the consumptions during the previous block. These variations exist since \(\rho\) is not exactly one and tend to diminish as \(\rho\to 1\).