# An Enhanced Energy Balanced Data Transmission Protocol for Underwater Acoustic Sensor Networks
## 4 EBET and EEBET: Proposed Techniques
### Route Establishment
\[R_{RL^{\prime}}^{j}=\frac{E_{o}}{\beta}-\frac{\beta}{3}\ \ \ \ \forall(j\in Sr_{i}) \tag{13}\]  
\[R_{RL^{\prime}}^{k}=\frac{E_{o}}{\beta}-\beta\ \ \ \ \ \ \forall(k\in Sr_{i-1}) \tag{14}\]  
Figure 6: Variations in relay node selection based on energy levels. As the process evolves, the nodes in nearer regions to the sink consumes \(URL\) earlier than the far away nodes due to additional relaying of data packets. We assume that a node can transmit data of two more nodes along with its own data. Equations (13) and (14) relates our assumption and specifies the decrease in Energy Level of a node. As the node \(k\) belongs to \(Sr_{i-1}\) which is nearer to the sink than the Ring Sector \(Sr_{i}\), therefore, its \(URL\) drops and the decrease in energy level of node \(j\) is its one third. Before sending next data packet, node \(k\) checks for the following condition:  
\[R_{RL}^{k}<R_{RL}^{j} \tag{15}\]  
If the condition specified in Equation (15) is true, then the node informs its predecessor node by sending a control packet to stop transmitting its data to that node. Thus the node only transmits its own data to next hop neighbor node. This approach avoids the introduction of critical nodes in the network and hence prevents the network from partitioning [23]. However, this approach consumes surplus energy which leads to reduced network lifetime. The predecessor then looks for another relay node to restart its data transmission. It selects a node having high energy level than itself within its transmission range. In this way, the chain of linked nodes break into multiple chains and more data packets simultaneously arrive at sink. Also, energy consumption is balanced by dividing the load of data packets over multiple nodes. We opt to choose direct transmission over long distance. However, it consumes more than one Energy Level in transmitting data over long distance which decreases network lifetime. Lemma 1 proves the energy balancing in EBET by specifying the Energy Level difference between two nodes in two consecutive Ring Sectors is not more than \(URL\). Lemma 1: _If \(R_{Tx}(x,d)<\beta\) then \(RL_{j}-RL_{k}\leq URL\), where \((j\in Sr_{i})\) and \((k\in Sr_{i-1})\)_  
**Proof:** As \(R_{Tx}(x,d)<RL\), then in sending one data packet a node consumes less than unit Energy Level. Using Equations (11)-(14), we have:  
Figure 3:0  
Figure 3:1  
Figure 3:2  
Figure 3:3  
Hence, the difference of RLs between any two nodes in successive Ring Sectors is not more than \(URL\). By using Lemma 1, we calculate max. residual energy of a node in \(Sr_{i}\) as  
Figure 3:4  
Therefore, the total remaining energy of all nodes is calculated as follows,  
Figure 3:5In the above equation, \(2^{(i-1)}\) is the maximum number of nodes in each layer of the tree and \((i-1)\) specifies that the maximum \(RL\) difference between nodes in successive Ring Sectors is 1. As we assume that the complete energy depletion of first sensor node collapses network. Therefore, the energy consumption in sending control packets by all the network nodes is as follows,  
Figure 3:6  
**Theorem 1**. : _If \(R_{Tx}(x,d)\leq\beta\), we can calculate optimal Energy Levels as:_  
Figure 3:7  
**Proof**: We calculate the total energy wasted in the network using Equations (22) and (23)  
Figure 3:8  
Figure 3:9  
Taking derivative of Equation (22) with respect to \(L\), we have Equation (26). \[URL=\frac{E_{o}}{L} \tag{9}\]0  
\[URL=\frac{E_{o}}{L} \tag{9}\]1  
\[URL=\frac{E_{o}}{L} \tag{9}\]2  
Thus, we obtain optimum number of Energy Levels as follows,  
\[URL=\frac{E_{o}}{L} \tag{9}\]3  
The optimum RLs depend on the initial energy of sensor nodes \(E_{o}\), transmit power \(P_{o}\) and the Ring Sector numbers \(Sr_{max}\). Algorithm 1 shows the pseudo-code of EBET, where, \(EL_{s}\) and \(EL_{p}\) are energy levels of successor and predecessor, respectively. ```
1:procedure:INITIALIZATION.