## 5 Case Study: HFECS in Neighborhood Area Networks  
In this paper, we have described the opportunistic clustering mechanism based on hesitant fuzzy entropy using several parameters, such as the time-frequency parameter, residual energy, LQR, distance from the BS and Nsupport. [PERSON] et al. in [26] proposed an energy-efficient multi-disjoint path opportunistic node connection routing protocol for smart grids in which the OCRG was utilized to calculate the link and path connectivity between sensor nodes for data forwarding purposes. Likewise, we have considered the neighborhood area networks (NANs) of smart grids in our case study to prove the significance of hesitant fuzzy entropy in heterogeneous clustering of WSNs. If we make an analogy between heterogeneous clustering and NANs of SGs, we can consider the NAN gateway as the CH and the several Home Area Network (HAN) gateways connected to the NAN gateway as CMs connected to their CH, as shown in Figure 2. Furthermore, the Access gateway in NANs can be considered as the BS, which receives the aggregated data from multiple NAN gateways and forwards it to the control center [46]. In this case study, we can see how the efficient use of hesitant fuzzy entropy can help us in making the CH election decision and also perform reliable data fusion. We have selected the measured values of six attributes from five sensor nodes placed randomly in a cluster. It is important to mention here that for this case study we considered the network area as centralized NAN for SGs communication. As each of the attributes have different [min max] values, so we needed to perform the data standardization in order to bring all the attributes on the same scale and to convert our measured data values into hesitant fuzzy set. So, we divided the attribute values into 10 parts according to their [min max] values and then assign values corresponding to 0-1.0. Table 2 represents the interval for each attribute used for data standardization whereas Table 3 represents the measured value of attributes for all sensor nodes. Different monitoring values for each attribute indicate that data acquisition cycle for each sensor node is different due to the asynchronous working-sleeping cycle strategy. For example, for the attribute time-frequency parameter in Table 3, its measured values are normalized to (([0,10] \(\rightarrow\) [0,0.1]), ([10,20] \(\rightarrow\) [0.1,0.2]),... ([60,70] \(\rightarrow\) [0.6,0.7]),... ([90,100] \(\rightarrow\) [0.9,1])). Similarly, for residual energy, the standardized attribute values will be (([0,0.25] \(\rightarrow\) [0,0.1]), ([0.25,0.5] \(\rightarrow\) [0.1,0.2]),... ([1.5,1.75] \(\rightarrow\) [0.6,0.7]),... ([2.25,2.5] \(\rightarrow\) [0.9,1])). The procedure of data standardization is the same for all other attributes mentioned in Table 3, along with their intervals, but for attribute 'Distance to BS', the standardized values are (([150,136] \(\rightarrow\) [0.0,0.1]), ([136,122]\(\rightarrow\) [0.1,0.2]),...... ([66,52] \(\rightarrow\) [0.6,0.5])... ([24,10] \(\rightarrow\) [0.9,1.0])] because if 'Distance to BS' is minimum, then we can have better connectivity with BS. Table 4 shows the corresponding hesitant fuzzy sets after the data standardization process. We utilized the hesitant fuzzy sets to compute hesitant fuzzy entropy using Equation (12) to acquire the statistical measure of uncertainty in our decision analysis for the CH election procedure and data fusion by CH. Table 5 represents the hesitant fuzzy entropy of each attribute used in our case study. After generating the hesitant fuzzy entropy matrix, we determined the entropy weight coefficients using Equation (13). Table 6 represents the entropy weight coefficients of all sensor nodes, i.e., \(W_{i}=[W_{1},W_{2},\ldots W_{r-1},W_{r}]^{T}\)