## 4 Energy Balanced Strategies for Sparsely Deployed UWA-SNs
### DIB: differential initial battery assignment strategy  
In this section, we try to pre-assign differential initial battery power according to workloads in a two-tier hierarchical UWA-SNs to achieve balanced energy consumption.  
The network architecture is depicted in Figure 5. There are two types of nodes in the network: basic nodes and super nodes. Both basic nodes and super nodes are evenly deployed along the line, and the internode distance is \(r\).  
The basic nodes have identical battery power and shorter transmit distance than super nodes. The super nodes have higher battery power and longer transmit distance.  
In our model, two basic nodes and a super node form a cluster. The basic nodes send sensed data directly to the super node, then the super node relays the data including its sensed data to the nearest super node, which delivers the data via a multi-hop path with super nodes until the data reaches the sink. To avoid interference of data transmission, TDMA or CDMA can be used [29, 30].  
Figure 5: Two-tier hierarchical UWA-SNs.  
The energy consumption of individual nodes  
In each round, each basic node in a cluster transmits a data packet to its super node, so the total energy consumption of basic node is the same, as shown below:  
\[E_{basic}=E_{TX}(k,r)+E_{SEN} \tag{16}\]  
where \(E_{basic}\) is in \(Joule\).  
As for a super node, it transmits three data packets and the transmission distance is \(3r\). Thus, the total energy consumption of super node \(S_{i}\) in each round can be expressed as  
\[E_{i}=3(n-i+1)E_{TX}(k,3r)+(3n-3i+2)E_{RX}(k)+E_{SEN} \tag{17}\]  
where \(E_{i}\) is in \(Joule\).  
Battery assignment analysis  
To achieve balanced energy consumption among nodes in the chain and maximize the lifetime of the network, we can assign differential battery power according to node's workloads. It is ideal that the nodes consume all battery power allocated to them when the targeted monitoring duration is achieved.  
We denote the targeted monitoring time duration is \(T\) rounds, and the assigned battery power of node \(S_{i}\) as \(E_{assign-ideal}(i)\), then we obtain \(E_{assign-ideal}(i)=T\times E_{i}\).  
Let's analyze the ideal battery assignment. Because the transmit power is often 100 times more expensive than that of the receive power underwater [10; 11], we omit the receiving power consumption in Equation (17). As the sensing energy consumption is the same for the nodes, we also omit this part in the equation.  
Thus, we obtain the battery assignment ration of super node \(S_{i}\) to \(S_{n}\) approximately equals to:  
\[\frac{E_{assign-ideal}(i)}{E_{assign-ideal}(n)}=(n-i+1). \tag{18}\]  
We also obtain the battery assignment ration of basic node to super node \(S_{n}\) approximately equals to:  
\[\frac{E_{assign-ideal}(n)}{E_{basic}}=\frac{3E_{Tx}(k,3r)}{E_{Tx}(k,r)}. \tag{19}\]  
Differential initial battery assignment to minimize the total battery budget  
When we design the network, we should take this into consideration: we only have a limited options of battery power levels to choose, because we must use manufactured batteries. Batteries may be connected in a parallel combination, which increases the batteries capacity, but the battery power levels are still limited.  
On the other hand, the volume of batteries is times of one unit, thus we get  
\[E_{assign-real}(i)=K\times E_{unit},K=1,2,......,W\]where \(E_{unit}\) is the unit of battery, \(W\) is the maximum number of battery levels which we can choose.  
Thus, the total battery budget is  
\[E_{total}=\sum_{i=1}^{n}E_{assign-real}(i).\]  
To achieve the targeted life of \(T\) rounds, we should satisfy \(E_{assign-real}>E_{assign-ideal}(i),i=1,2,......,n\).  
In addition, to minimize the total energy budget, the battery assignment should minimize  
\[\sigma^{2}(E_{total})=\frac{1}{n}\sum_{i=1}^{n}(E_{assign-real}(i)-E_{assign -ideal}(i))^{2}.\]