1. Introduction

2. System Model

2.1. 3D WUSN Model

We propose a 3D WUSN system with M sensors [1-2-3-4]. These sensors are spread out evenly on the ground, each having a specific job based on where they are and what they do. This setup ensures full coverage and good data collection. Here are the main ideas guiding this setup:

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After setting them up, all sensors stay put. Even though they have different jobs, they all start with the same amount of power and keep it consistent.

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There are an equal number of sensors on the ground and below it. In each group or “cluster” (let’s call C), there’s an even mix of M sensors.

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Sensors on the ground can either be leaders (Cluster Heads or CH) or just regular members. Only the leader sensor collects data from both on-the-ground and underground sensors, sending it to the Base Station (BS) every hour.

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Within each group, some sensors, marked as k, focus mainly on collecting data. They then send this data to their leader sensors above the ground.

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For communication, sensors talk directly to their leader sensors. These leader sensors then send data to the main control point without any stops in between.

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The main control point, crucial for putting all data together, is always fixed in one central spot.

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At the end of each hour of communication, we check how much power each sensor has left. This helps us decide if we need to change the leader sensors.

In short, this setup, based on the points above, helps us collect data in the best way while saving as much power as possible in our sensor network.

Figure 1. The WUSN Model.

Figure 1. The WUSN Model.

2.2. Mathematical Model

We assume a 3D WUSN model where M sensor nodes are in the field. They are distributed on the ground to collect information. They are fixed in locations and uniformly distributed in configuration and function. Sensors are placed in coverage with each other. Some assumptions are made as follows:

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All sensor nodes are fixed after being deployed, and the energy of the nodes is the same at the beginning, although in the model the sensors are used for different purposes, and their energy is the same at initialization.

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M sensor nodes will be assigned to a given cluster C. The number of nodes above the ground and underground was divided based on the ratio of the datasets.

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The number of members in clusters is not the same quantity.

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Nodes marked as CH will be on the ground, and a cluster will form between the above-the-ground and underground nodes. The nodes that are above the ground, can be designated as member nodes or can be cluster heads if it is a cluster head, then it will collect information from underground and above the ground nodes which transmits the information to the BS. The transmission schedule will be 1 hour each.

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In a cluster, there will be k underground nodes, the underground nodes are responsible for collecting and transmitting information to the CH node on the ground.

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Single-hop communication is assumed in the sense that member clusters to CH directly and CHs forward data to BS in the communication range.

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The BS will be permanently assigned in a central location.

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After each round, all nodes will be calculated with the remaining energy so that BS would consider updating the array nodes for the next rounds based on the remaining energy.

Figure 2. The overview of the proposed network model.

Figure 2. The overview of the proposed network model.

This study is motivated based on the model, then we deliver the new model combining those above assumptions. Sensors have limited power (mainly battery-powered) and processing capabilities. Sensing and processing require energy-intensive processes, specifically including environmental sensing components, data aggregation, and data transmission and reception [4-9]. The energy consumed in the transmission of l-bit data [9] at the distance (d) is represented [3] by E_CM−CH and it is calculated in Equation (1). If d ≤ d^th in which d^th is the constant of the distance, energy consumption is calculated based on the free space model with d². Conversely, it is the multi-path fading model with d⁴ energy dissipation. The energy consumption of a sensor in the cluster is given by the formula below.

$$E_{CM-CH}=l{E}_{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{c}}+{\mathit{lϵ}}_{\mathit{fs}}{d_{toCH}}^2=l{E}_{elec}+{lϵ}_{fs}\sum _{i=1}^M\sum _{j=1}^C{‖X_i-V_j‖}^2$$

(1)

where $l$ is the number of data bits and $E_{elect}$ be energy consumption per bit to run the transmitter or the receiver data. ${ϵ}_{fs}$ be the energy required for amplifying mode, and $d_{toCH}$ be the distance from CM to CH [4]. $V_j\left(j=1,\dots ,C\right)$be the location of CH of $j^{th}$ clusters, and $X_i$ be the location of $i^{th}$ sensor nodes. The formula for calculating the energy consumption of a CH for assembling data and transferring it to BS is as follows:

$$E_{CH-BS}=l{E}_{\mathrm{D}\mathrm{A}}+{\mathit{lϵ}}_{\mathit{mp}}{d_{toBS}}^4=l{E}_{elec}+{lϵ}_{mp}\sum _{i=1}^M\sum _{j=1}^C{‖V_j-X_{BS}‖}^4$$

(2)

where $E_{DA}$ be the energy consumption for data aggregation, ${ϵ}_{mp}$ be the energy loss by the amplifier mode [4], and $d_{toBS}$ be the distance from CH to BS [9] at a certain terrain [4-9]. $X_{BS}$ be the location of BS. Hence the entire network has energy consumption [4] is as Equation (3):

$$\begin{array}{c}{E}_{CM-CH-BS}=C(l{E}_{\mathrm{D}\mathrm{A}}+{\mathit{lϵ}}_{\mathit{mp}}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{‖V_j-X_{BS}‖}^4)\\ +(M-C)(l{E}_{elec}+{lϵ}_{fs}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{‖X_i-V_j‖}^2)\end{array}$$

(3)

The energy path loss from the underground nodes to above-the-ground CH is calculated according to the formula. When transmitting data from underground sensors to CH, the Friis model [5] is adopted to calculate the path loss, it is derived from the Friis equation in free space and is modified when electromagnetic waves are transmitted in sediments. The path loss quantifies the EM wave attenuation with the transmission distance. When the antenna gains are not included, then the path loss in free space can be simplified to Equation (4):

$$E_{{pathlossCM}_u-CH}=6+20\log d+20\log \beta +8.69\alpha d$$

(4)

where $d$ be the distance between the sender sensor as the underground sensor and the received sensor as CH placed on the terrain. The distance could be calculated via the 3D Euclidean. Accordingly, $\mathrm{z}$ will be the depth of node ${\mathrm{X}}_{{\mathrm{i}}_{\mathrm{u}}}$ and ${\mathrm{z}}_{\mathrm{v}}$ be the height of node CH above the ground. Variants for underground are $\mathsf{\alpha }$ and $\mathsf{\beta }$ are the constant of phase shifting and attenuation, respectively. Equation (4) is used to compute the energy of the path loss.

$$E_{{CM}_u-CH}=6+20\log \beta +20\log ‖{X_i}_u-V_j‖+8.69\mathsf{\alpha }‖{X_i}_u-V_j‖$$

(5)

The Taylor series is used to expand the log function in Equation (5) into a polynomial function.

$$\mathit{log}x=\frac{1}{ln10}\left(x-1\right)-\frac{1}{2ln10}{\left(x-1\right)}^2+\frac{1}{3ln10}{\left(x-1\right)}^3+\dots$$

(6)

$$E_{{CM}_u-CH}=6-\frac{20}{ln10}+20\log \beta +(8.69\mathsf{\alpha }+\frac{20}{\ln 10})‖{X_i}_u-V_j‖$$

(7)

As per the above assumption, the network consists of k underground nodes. The CM-CH-BS nodes’ energy transmissions and the energy lost by the k nodes buried underground will make up the network’s total energy as Equation (8):

$$E_{total}=E_{CM-CH-BS}+k{E}_{{pathloss}_{{CM}_u-CH}}$$

(8)

Combining Equations (3), and (8), we have the total energy model [4] of the network as follows:

$$\begin{array}{c}{E}_{total}=C(l{E}_{DA}+{\mathit{lϵ}}_{\mathit{mp}}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{‖V_j-X_{BS}‖}^4)\\ +\left(M-C\right)\left(l{E}_{elec}+{lϵ}_{fs}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{‖X_i-V_j‖}^2\right)\\ +k\left(6-\frac{20}{ln10}+20\mathit{log}\beta +(8.69\alpha +\frac{20}{\mathit{ln}10})‖{X_i}_u-V_j‖\right)\end{array}$$

(9)

From the energy model in Equation (9), we fuzzed them and proposed the mathematical model to minimize the total energy of the network as the Equation (10) below.

$$\begin{array}{c}{J}_{\left(u_{ij},V\right)}=C(l{E}_{elec}+l{E}_{DA}+{\mathit{lϵ}}_{\mathit{mp}}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u_{ij}^m‖V_j-X_{BS}‖}^4)\\ +\left(M-C\right)\left(l{E}_{elec}+{lϵ}_{fs}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{ij}^m{‖X_i-V_j‖}^2\right)\\ +k\left(6-\frac{20}{ln10}+20\mathit{log}\beta +(8.69\alpha +\frac{20}{\mathit{ln}10}){\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{\mathit{ij}}^m‖{X_i}_u-V_j‖\right)\to \mathrm{M}\mathrm{i}\mathrm{n}\end{array}$$

(10)

The clustering problem’s objective function is represented by Equation (10). The degree of belonging ${\mathrm{u}}_{\mathrm{i}\mathrm{j}}$ and the cluster center ${\mathrm{V}}_{\mathrm{j}}$ are the two primary factors that need to be assigned to cluster in this kind of situation. The problem’s objective is to distribute sensors among clusters [9], and then choose the cluster header. The data is passed directly from CM to CH and CH to BS. The BS would take into account re-clustering and re-selecting CH, depending on the remaining energy of the network’s sensors after each loop. Regarding the limitation of communication constraints, the following conditions are incorporated:

$$\sum _{i=1}^M\sum _{j=1}^C{u}_{ij}^m=1,u_{ij}ϵ\left[0,1\right]$$

(11)

$$‖X_i-V_j‖\le 2Tr$$

(12)

$$‖V_j-X_{BS}‖\le 9Tr$$

(13)

We have the new clustering model in equations (10-13). Before we state the new algorithm, we solve the optimization problem by the Lagrange multiplier method.

Where $m$ is the fuzzification coefficient (usually $m=2)$, $C$ is the number of clusters, $M$ is the number of sensors, and $l$ is the number of bits in each data packet. $T_r$ is the communication range, and is calculated by 3D Euclidean. The objective Equation (11) is a nonlinear optimization problem. Constraints (12) and (13) are given in the Wireless Sensor Networks problem to simulate the communication range between two normal sensor nodes, as well as from normal sensors to BS. Constraint (11) means that the membership of each sensor node in each cluster is represented to a degree as the membership function. The result of equations to calculate $u$ and $V$ is provided in Equations (14), and (15) below.

$$L\left(u,V_j\right)=C\left({\mathit{lϵ}}_{\mathit{mp}}{u}_{ij}^m\sum _{i=1}^M\sum _{j=1}^C{‖V_j-X_{BS}‖}^4\right)+\left(M-C\right)\left({lϵ}_{fs}\sum _{i=1}^M\sum _{j=1}^C{u}_{ij}^m{‖X_i-V_j‖}^2\right)$$

(14)

$$\begin{array}{c}\frac{\partial J}{{\partial V}_i}=C4{lϵ}_{mp}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u_{ij}^m‖V_j-X_{BS}‖}^3+\left(M-C\right)\left(2{lϵ}_{fs}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{ij}^m‖X_{ij}-V_j‖\right)\\ -k\left(8.69\alpha +\frac{20}{\mathit{ln}10}\right){\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{ij}^m\end{array}$$

(15)

Since $\frac{\partial \mathrm{J}}{\partial {\mathrm{V}}_{\mathrm{i}}}=0$there is:

$$\begin{array}{c}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{\left(V_j-X_{BS}\right)}^3\left(C4{lϵ}_{mp}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{ij}^m\right)+\left(M-C\right)\left(2{lϵ}_{fs}{\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{ij}^m\left(X_{ij}-V_j\right)\right)\\ -k\left(8.69\alpha +\frac{20}{\mathit{ln}10}\right){\displaystyle \sum _{i=1}^M}{\displaystyle \sum _{j=1}^C}{u}_{ij}^m=0(\mathrm{i}=1\dots \mathrm{M},\mathrm{j}=1\dots \mathrm{C})\end{array}$$

(16)

Denote from:

$$A=C4l{\epsilon }_{mp}\sum _{i=1}^M\sum _{j=1}^C{u}_{ij}^m$$

$$B=2l{\epsilon }_{fs}(M-C)\sum _{i=1}^M\sum _{j=1}^C{u}_{ij}^m$$

(17)

$$C=k\left(8.69\alpha +\frac{20}{ln10}\right)\sum _{i=1}^M\sum _{j=1}^C{u}_{ij}^m$$

$$u_{ij}=\frac{1}{\sum _{i=1}^C{\left(\frac{{ϵ}_{mp}{‖V_j-X_{BS}‖}^4+\left(M-C\right){ϵ}_{fs}{‖X_i-V_j‖}^2+k\left(\frac{20}{ln10}+8.69\alpha \right)‖X_{iu}-V_j‖}{{ϵ}_{mp}{‖V_j-X_{BS}‖}^4+\left(M-C\right){ϵ}_{fs}{‖X_i-V_j‖}^2+k\left(\frac{20}{ln10}+8.69\alpha \right)‖X_{iu}-V_j‖}\right)}^{\frac{1}{m-1}}}$$

(18)

$$\begin{array}{c}{V}_i=\frac{\frac{B}{A}}{\sqrt[3]{\frac{B\left(X_{BS}-X_i\right)+C}{2A}}+\sqrt{\frac{1}{4}{\left(\frac{B\left(X_{BS}-X_i\right)+C}{A}\right)}^2+\frac{B^3}{27A^3}}}\\ -\sqrt[3]{\frac{B\left(X_{BS}-X_i\right)+C}{2A}+\sqrt{\frac{1}{4}{\left(\frac{B\left(X_{BS}-X_i\right)+C}{A}\right)}^2+\frac{B^3}{27A^3}}}+X_{BS}\end{array}$$

(19)

Figure 3. The general process algorithm.

Figure 3. The general process algorithm.

3. The FCM-WSN algorithm

4. Experiment

5. Conclusions

References

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