1. Introduction

2. Related works

3. Proposing method

3.3. Fuzzy calculation

This fuzzy method selects the appropriate next step with the remaining energy, less than the sink node, less than the water level and the neighbour in the current cluster head, which are considered as the inputs of the fuzzy system:

• Network residual energy

Network residual energy is an essential factor in calculating the lifetime of a node in the network. The energy the node releases is used to communicate with other nodes or in idle mode; the remaining energy is calculated using formula 1.

$$RemainingEnergy=TotalEnergy-EnergyConsumption$$

(1)

• Distance between nodes

The distance between 2 connected nodes is calculated at a particular time. Distance is essential, as the energy consumed in transmitting a packet depends on the distance between two nodes. The distance is calculated by the Euclidean geometry formula 2:

$$Distance= \sqrt{{\left(X_1-X_2\right)}^2+{\left(Y_1-Y_2\right)}^2+{\left(Z_1-Z_2\right)}^2}$$

(2)

• Depth: In underwater sensor nodes, by default, all nodes have a depth gauge, and Thay have not applied any additional costs to the network.
• Number of Neighbours: Each node constructs a list of accessible nodes in its vicinity based on the packets received from neighbouring nodes.

Enhancing network efficiency and minimising congestion requires optimising the fitness function, as outlined in Equation 3.

$$FF\left(f_i\right)={{\displaystyle \sum }}_{i=1}^N\left(w_1\times E_i)+(w_2\times D_{j,i})+(w_3\times D{E}_i)+(w_4\times N{N}_i\right)$$

(3)

Where i means iteration, which can range from 1 to N (the total number of nodes).

• And w₁, w₂, w₃, w₄ is the load (or weight) prepared on the algorithm.

• $E_i$ is the residual energy of each node in the network.
• $D_{j,i}$ is the distance between nodes i and j.
• $D{E}_i$ is the Depth of node.
• NN is the number of accessible neighbours.

We have used the triangular fuzzification model and state levelling method (Figure 4 and Table 2):

Figure 3. Fuzzification of Remaining Energy.

Figure 3. Fuzzification of Remaining Energy.

Finally, the cluster head node with the highest FF value should be selected as the most suitable next step. For example, a table is created in each cluster head node according to Table 3, which will be updated after different time intervals.

In the proposed method, after the interval of t seconds, a fuzzy value update will be done based on the network traffic. This update ensures that a node known as a suitable node by the initial fuzzy computation is not placed in the top traffic of the network. For this purpose, another fuzzy system has been proposed, which accepts as input the fuzzy amount obtained from the first stage, known as FF, and the traffic rate in the recent period, resulting in an updated fuzzy value.

4. Simulation

4.3. Energy efficiency and fairness

In underwater sensor networks, performance and data transfer are significantly affected by various challenges, especially the limited energy capacity of nodes. Considering the limited energy resources available, the main goal in designing such networks is to ensure that energy is used evenly across all nodes. Using energy more efficiently and within acceptable limits leads to a longer lifespan for individual nodes and the entire network, thereby preventing early node failures. The network’s energy efficiency is assessed using three principal metrics: the average balance of energy, the fairness in energy consumption among nodes, and the lifespan of individual nodes. However, before assessing energy consumption fairly, this study calculates standard energy consumption based on Equation 7.

$$E_{tx}\left(m,d\right)=(E_{con}\times m)+)t_{amp}\times m\times l^2)$$

(7)

In this context, $E_{con}$ represents energy consumption, $l^2$ denotes energy loss, and d is the distance between a node and its parent. The amplification factor for node transmission is represented by $t_{amp},$ and m signifies the bits transferred between the child and parent nodes. According to Equation 7, the average energy level is calculated as the ratio of the total remaining energy in all nodes at the end of the simulation to their total initial energy. This proposed method demonstrates several benefits over other approaches, leading to promising outcomes in this experiment. The variance in energy consumption is determined using Equation 7, while the fairness of energy consumption across nodes is calculated with Equation 8. The deviation in energy consumption is given by:

$$Dev={\displaystyle \sum }_{i=1}^n{\left(energ{y}_i-Average\right)}^2$$

(8)

Table 7. Node energy consumption variance in different methods.

Table 7. Node energy consumption variance in different methods.

Network Size	[16]	[17]	[19]	FuzzyCR
100	35.22	36.34	37.11	42.17
150	36.87	37.18	39.83	45.03
200	37.07	38.55	41.28	47.8

When the value derived from Equation 8 approaches 100%, it indicates a balanced energy consumption among nodes. Here, Dev_worst represents the most unfavourable scenario in which half of the nodes have depleted their energy while the remaining nodes have not consumed it. The fairness is calculated as follows:

$$Fairness=\frac{1-Dev}{De{v}_{worst}}$$

(9)

Table 8 shows that in the FuzzyCR protocol, fairness in energy consumption among network nodes improves with more nodes but worsens with increased traffic. Higher node counts provide more options for fair distribution, while heavy traffic reduces the ability to maintain energy equity. The table illustrates the direct and inverse relationships of node count and traffic intensity, respectively, with fairness.

5. Conclusion

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