## 4 Proposed Protocol
### The Set-up Period
**Intra and Inter cell distance (\(CD(i)\)):** The objective of this criterion is to minimize intra-cell communication cost between MNs and respective CHN in an MST as well as minimize inter-cell communication cost from CHNs to the sink device in a chain that consumes less energy and balances the workload between CHNs. To achieve this objective, the \(CD(i)\) is defined as the total geographic distance of the candidate CHNs within their cell, which is calculated as  
\[CD(i)=\frac{h\sum_{j=1}^{nc}d(i,j)}{min_{j\in SN(i)}(d(i,j))} \tag{16}\]  
where \(h\) and \(SN(i)\) are the numbers of neighbor nodes and set of neighbor nodes of the candidate node \(i\)-th, respectively. **Cost function:** All the appropriate parameters introduced are combined in order to select suitable CHN for each cell, whose residual energy is higher than \(EC_{avg}(i,j)\) and has a maximum cost function value as Equation (17) follows:  
\[cost(i)=\left(c_{1}E_{res}(i)\right)\times\left(c_{2}CD(i)\right)\times\frac{c _{3}}{d_{toSink}(i,S)} \tag{17}\]  
The user establishes the coefficient parameters within cost function \(c_{j}\)\((j=1,2,3)\) for the heterogeneous and homogeneous network. _Step 3:_ Super-CHN selection  
In EEGT, only several super-CHNs are responsible for forwarding aggregated data packets to the sink to preserve energy in other nodes, so the distance from them to the sink device is as short as possible. In addition, the higher the residual energy of the candidate super-CHN is, the better the energy efficiency of the network achieves. Therefore, if the distance from the candidate super-CHN to the sink device is smaller than that of the average distance \(DCH_{seg}\) between CHNs within the list of CHNs to the sink device or if its position is in the \"row-1\" region (Figure 5). If it has the highest \(P\) values, then CHN will be selected as super-CH in the current round. The \(DCH_{seg}\) and \(P\) can be calculated as Equations (18) and (19) follow:  
\[DCH_{seg}=\frac{1}{N_{cell}}\sum_{i=1}^{N_{cell}}d(CH_{i},Sink) \tag{18}\]  
\[P(i)=\frac{c_{4}E_{res}(i)}{d_{toSink}(i,S)} \tag{19}\]  
where \(P(i)\) represents the ratio between the remaining energy of sensor node \(i\)-th and the distance from sensor node \(i\)-th to the sink. The process of cell head selection is illustrated in Algorithm 2 below:  
```
0:\(N_{cell}\) number of cells, \(N\) sensors distributed in \(N_{cell}\) cells, and their position of them
0:List of CHNs in each cell and one or more super-CHNs. 1:for\(i=1\) to \(N_{cell}\)do
2: Compute the residual average energy of nodes in each cell as in Equation (14)
3: Select CHN for cell \(i\)-th, which has a maximum cost function value as Equation (17)
4: Append CHN to List of the CHNs
5:endfor
6:\(minD\) = Maximum value;
7:for each CH in [List of the CHNs] do
8:\(disSink\leftarrow\) Compute the average distance from CHNs to the sink as Equation (18)
9:\(P_{i}\) = Compute the parameter criterion \(P\) as Equation (19);
10:if (\(disSink<minD\)) and (\(P_{i}\) is the highest) then
11:\(minD\) = \(disSink\);
12:\(super-CH\) = \(CHN_{i}\);
13:endif
14:endfor
15:return: [List of the CHNs and super-CHNs];
```  
**Algorithm 2** Cell Head Selection