# A New Centralized Clustering Algorithm for Wireless Sensor Networks
## 3 Proposed Method
### Computation of the Skip Value

Table 2: Parameters for the losses in the first order radio model and data aggregation. To effectively calculate the value of the parameter \(skip\), it is necessary to estimate the energy spent by normal nodes and by the CHs during that period. Consequently, two equations were developed to model both processes. The energy consumed by a normal node (not elected as CH) during \(skip\) rounds is modelled by Equation (7):  
\[E_{sensor}=E_{Rx}(l_{control})+E_{Tx}(l_{data},d)\times skip; \tag{7}\]  
It includes the energy cost due to the reception of a control packet (in which the BS informs which nodes are CHs) and the energy cost of sending a data packet multiplied by the number of times that this process must be repeated (the value of the \(skip\) parameter). After the node joins a CH, it adjusts the transmission power according to the distance to the CH. In this case, we suppose that the distance \(d\) is small because normal nodes send the information to their closest CH, not to the BS. The energy consumed by a node selected as CH is higher as can be noticed in Equation (8). To compute this consumption, we consider that the distance from the CH to the BS is usually greater than the distance to its contributing nodes. \[E_{CH}=E_{Rx}(l_{control})+(E_{Rx-DA}(l_{data})\times X+E_{Tx}(l_{data},d_{SB} ))\times skip; \tag{8}\]  
Apart from the energy cost of the control message from the BS as in Equation (7), it includes the cost of reception and aggregation of \(X\) data packets from its contributing nodes, plus the message that must send to the BS each round with the aggregated data. It must be taken into account that the distance to the BS from the CH could be considerable, which converts that factor into the main energy cost. It is evident that for high values of \(skip\), a CH could spend enough energy to deplete its battery, even in the first rounds. This would eventually lead to a really low FND. Therefore, the present approach is based on the best balance between the number of CHs and the \(skip\) parameter because nodes that are not CH spend much less energy during skipped rounds. The evolution of the average energy spent by the nodes, acting as a normal node and as a CH can be studied in Figure 6. As can be noticed, the effect of being CH is clearly worse in terms of energy consumption when compared with a normal node. Consequently, the election of the value for the \(skip\) parameter is of paramount importance in CSBA. As was commented above, to estimate the most appropriate value for the \(skip\) parameter, an empirical method, based on the simulation of three different scenarios of a wireless sensor network, was applied. All the scenarios are composed of 150 nodes randomly deployed over a deployment field of 100 m \(\times\) 100 m and the initial energy of each node is set to 0.5 J. The difference among them is the location of the BS which was deployed as follows:  
* Scenario 1. The BS is in position (100, 0) m, that means that it is in the border of the sensing area. * Scenario 2. The BS is in position (150, 50) m, that means that it is outside the sensing area. * Scenario 3. The BS is in position (50, 50) m, that means that it is in the centre of the sensing area. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & \(\mathbf{k}\) & \(\mathbf{RnoCH_{th}}\) \\ \hline Value & 7 (150 \(\times\)\(p\)) & 20 (1/\(p\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter configuration for \(skip\) estimation. To achieve a statistically consistent result for the \(skip\) value a set of 30 different networks were simulated for any of the scenarios above. The average value of \(FND\), \(HND\) and \(LND\) obtained for \(skip\) values from 0 to 140 are shown in Figures 7-9. The values for \(FND\), shown in Figure 7 reveals that for the initial increments in the value of the \(skip\), an improvement in this metric is achieved. Hence, this behaviour validates the first hypothesis exposed in the introduction.