EOCGS: energy efficient optimum number of cluster head and grid head selection in wireless sensor networks

Abstract

Wireless Sensor Network (WSN) is widely used for collecting the information from the target region by deploying the sensor nodes in that region. In WSNs, energy saving is the prime task which depends upon the number of Cluster Heads (CH) and on their selection techniques. Thus, in order to optimize the energy consumption of the clusters, we are proposing an Energy-efficient technique for selection of Optimum Number of Cluster Head and Grid Head (EOCGS) which extends the network lifetime. Here, firstly we give the expression of optimum number of clusters, then propose a new technique for selecting the optimum number of CHs in an energy efficient manner. For saving the energy of the CHs, the concept of Grid Head (GH) is being added in an efficient way, which works in dynamic mode. When the number of CHs are greater than threshold limit, then few of the CHs work as GHs and they are selected by using the proposed fitness function that depends on residual energy, Euclidean distances, and location of the grid-centroid of CHs. We have depicted that the proposed work saves the network energy efficiently, extends the network lifetime and coverage as compared to similar clustering algorithms e.g., LEACH, TL-LEACH, R-LEACH, RCH-LEACH, UCRA-GSO & CCA-GWO. It also stabilizes the cluster formation in the network.

Appendix

1.1 Optimum number of clusters

Here, we are deriving the expression for the optimum number of clusters based on the energy consumption of CMs, CHs and GHs of the network, where the energy consumption by each CM & CH individually can be expressed by Eqs. 6 & 7 respectively, and the energy consumption by GHs is expressed by Eq. 8. We know that in the three layer structure, CMs and CHs follow $d^{2}$ energy model (Eq. 5) because their distances from their destination are less than the threshold distance ($d_{th}$), while each GH directly communicate with the BS, thus it may follow $d^{4}$ energy model due to their large distances from the BS [15]. Now, the energy consumption of the whole network is given as,

$$\begin{aligned} E_{Network}=N_{GH}N_{G_{C}}E_{cluster}+E_{GHs} \end{aligned}$$

(24)

where $E_{cluster}$ describes the energy consumption of the cluster, $N_{G_{C}}$ denotes the total number of clusters in a grid and $E_{GHs}$ denotes the energy consumption by among GHs (Eq. 8). As we know that the $N_{CH}$ is distributed into $N_{GH}$ grids, thus $N_{G_{C}}$ can be defined as,

$$\begin{aligned} N_{G_{C}}\!=\!\frac{N_{CH}}{N_{GH}} \end{aligned}$$

(25)

Here, we are evaluating the value of $E_{cluster}$ using the following expression,

$$\begin{aligned} E_{cluster}=(n_{c}-1)E_{n}+E_{CH} \end{aligned}$$

(26)

Now, the value of $E_{n}$ & $E_{CH}$ is taken from Eqs. 6 & 7, and put in the above Eq. 26 as

$$\begin{aligned} E_{cluster}&=(n_{c}-1)(lE_{elec}+l\epsilon _{fs}d_{n2CH}^{2})\nonumber \\&\quad +ln_{c}(E_{elec}\!+\!E_{DA})\!+\!l\epsilon _{fs}d^{2}_{CH2GH} \end{aligned}$$

(27)

In the above equation, replace the value of $n_{c}$ from Eq. 8 as,

$$\begin{aligned} E_{cluster}\!&=\!l\left[ \left( \frac{N'\!-\!N{CH}}{N_{CH}}\right) (E_{elec}+\epsilon _{fs}d_{n2CH}^{2})\!\!\right. \nonumber \\&\quad \left. +\!\!\left( \frac{N'}{N_{CH}}\right) (E_{elec}\!+\!E_{DA})\!+\!\epsilon _{fs}d^{2}_{CH2GH}\right] \end{aligned}$$

(28)

Now, we are taking the expressions of $E_{GHs}$, $N_{GH}$ & $E_{cluster}$ from Eqs. 8, 25 and 28, and place into Eq. 24 as,

$$\begin{aligned} E_{Network}&=N_{CH}\left( \frac{N'-N{CH}}{N_{CH}}\right) (lE_{elec}+l\epsilon _{fs}d_{n2CH}^{2})\nonumber \\&\quad +lN_{CH}\left( \frac{N'}{N_{CH}}\right) (E_{elec}\!+\!E_{DA})\nonumber \\&\quad +N_{CH}l\epsilon _{fs}d^{2}_{CH2GH}+lN_{CH}(E_{elec}\!\nonumber \\&\quad +\!E_{DA})+N_{GH}l\epsilon _{mp}d_{GH}^{4} \end{aligned}$$

(29)

Now, the above equation can be simplified as,

$$\begin{aligned} E_{Network}&=l\big [(N'-N{CH})(E_{elec}+\epsilon _{fs}d_{n2CH}^{2})\nonumber \\&\quad +N'(E_{elec}\!+\!E_{DA})\nonumber \\&\quad + N_{CH}(E_{elec}{+}E_{DA}{+}\epsilon _{fs}d^{2}_{CH2GH}{+}p_{g}\epsilon _{mp}d_{GH}^{4})\big ]\nonumber \\ E_{Network}&=l\big [N'(2E_{elec}+E_{DA}+\epsilon _{fs}d_{n2CH}^{2})\nonumber \\&\quad +N_{CH}(E_{DA}-\epsilon _{fs}d_{n2CH}^{2}+\epsilon _{fs}d_{CH2GH}^{2}\nonumber \\&\quad +p_{g}\epsilon _{mp}d_{GH}^{4})\big ] \end{aligned}$$

(30)

Here, we are giving the expression for the expected value of square of E.d. of CMs ($\bar{E}[d_{n2CH}^2]$) and CHs ($\bar{E}[d_{CH2GH}^2]$) from their destination. For this, we have assumed that the nodes are uniformly distributed and each cluster is having circular dimension. Thus, the radius ($R_{c}$) and node density ($\rho _{c}(\bar{r},\theta )$) of the clusters are same as described in [35], and can be expressed as,

$$\begin{aligned}&R_{c}=\frac{M}{\sqrt{\pi N_{CH}}} \end{aligned}$$

(31)

$$\begin{aligned}&\rho _{c}=\frac{N_{CH}}{M^{2}} \end{aligned}$$

(32)

Here, the expression of $\bar{E}[d_{n2CH}^2]$ can be given as,

$$\begin{aligned}&\bar{E}[d_{n2CH}^2]=\int _{\theta =0}^{2\pi }\int _{\bar{r}=0}^{R_{c}}\bar{r}^{3}\rho _{c}(\bar{r},\theta )d\bar{r}d_{\theta } \end{aligned}$$

(33)

$$\begin{aligned}&\bar{E}[d_{n2CH}^2]=\frac{M^{2}}{2\pi N_{CH}} \end{aligned}$$

(34)

Similarly, the expression of $\bar{E}[d_{CH2GH}^2]$ can be given as,

$$\begin{aligned}&\bar{E}[d_{CH2GH}^2]=\int _{\theta =0}^{2\pi }\int _{\bar{r}=0}^{R_{g}}\bar{r}^{3}\rho _{c}(\bar{r},\theta )d\bar{r}d_{\theta } \end{aligned}$$

(35)

$$\begin{aligned}&\bar{E}[d_{CH2GH}^2]=\frac{M^{2}}{2\pi N_{GH}} \end{aligned}$$

(36)

where $R_{g}\!=\!\frac{M}{\sqrt{\pi N_{GH}}}$ & $\rho _{g}\!=\!\frac{N_{GH}}{M^{2}}$ denote the radius and node density of the grid respectively. Now, the value of $\bar{E}[d_{n2CH}^2]$ & $\bar{E}[d_{CH2GH}^2]$ put in Eq. 30

$$\begin{aligned}&E_{Network}=l\left[ N'(2E_{elec}+\frac{\epsilon _{fs}M^{2}}{2\pi N_{CH}}+E_{DA})\right. \nonumber \\&\quad \left. + N_{CH}(E_{DA}-\frac{\epsilon _{fs}M^{2}}{2\pi N_{CH}}+\frac{\epsilon _{fs}M^{2}}{2\pi N_{GH}}+p_{g}\epsilon _{mp}d_{GH}^{4})\right] \end{aligned}$$

(37)

$$\begin{aligned}&E_{Network}=l\left[ N'(2E_{elec}+E_{DA})+\frac{\epsilon _{fs}M^{2}N'}{2\pi N_{CH}}+\frac{\epsilon _{fs}M^{2}}{2\pi }\right. \nonumber \\&\quad \left. \bigg (\frac{1}{p_{g}}-1\bigg ) + N_{CH}(E_{DA}+p_{g}\epsilon _{mp}d_{GH}^{4})\right] \end{aligned}$$

(38)

In order to get the optimum value of $N_{CH}$ i.e., ($N_{CH}^{opt}$), we differentiate Eq. 38 w.r.t. $N_{CH}$ and equate it to zero,

$$\begin{aligned}&\frac{\mathrm {d} E_{Network}}{\mathrm {d} N_{CH}}=0 \end{aligned}$$

(39)

$$\begin{aligned}&E_{DA}+p_{g}\epsilon _{mp}d_{GH}^{4}-N'\epsilon _{fs}\frac{M^2}{2N_{CH}^{2}\pi }=0 \end{aligned}$$

(40)

$$\begin{aligned}&N_{CH}^{opt}=M\sqrt{\frac{N'}{2\pi }}\sqrt{\frac{\epsilon _{fs}}{p_{g}\epsilon _{mp} d_{GH}^4+E_{DA}}} \end{aligned}$$

(41)