## 3 Proposed Methodology
### Adaptive Power Control with Adaptive Clustering Routing Model  
The GRACE [33] principle of gradient cost field establishment, which is inspired by the natural phenomenon of water flowing down a valley from top to bottom, is the foundation of our adaptive power control routing model. Data propagates in the same way, in the direction where it finds the cheapest path. The gradient for the cost field establishment scheme and the factors are described as follows:  
* \(C_{E,i}\): The energy cost of the ith node is given by \[C_{E,i}=\frac{F_{i}^{0}}{E_{i}}\] (8)
* \(C_{L,u}\to v\): Link Cost: The cost of a link between a transmitting node u and a receiving node v is calculated as follows: \[C_{L,u\to v}=\frac{P_{L,u}}{P_{v}v}\] (9) where \(P_{L,u}\) denotes node u's transmitted power and \(P_{r,v}\) denotes node v's received power. * Route Cost, \(C_{N}\to N\) : Route cost between the path- starting node N1 and the path-terminating node N2 is given by \[C_{N_{1}\to N_{2}}=\sum_{i=N_{1}}^{N_{2}-1}(\Omega_{E}C_{E,i}+\Omega_{L}C_{ L,i\to i+1})\] (10)
* The following are some helpful concepts for establishing a cost field: \(\Omega_{L}\) Information about the advertising sent from node 'I' to its neighbors. CI S the cheapest route from node 'I' to sink 'S' This diagram depicts the only low-cost option. CI JS The routing path from node 'I' to sink 'S', using node 'J'. This reflects a single special path originating from node 'I' and ending at Sink 'S', out of many possible paths. Where \(\Omega_{E}\) and \(\Omega_{L}\) are the energy and link cost weighting factors, respectively. The cost field is defined using the APCEER protocol [9] notion, which is illustrated in Figure 2 where nodes J, K, and L are the Ith node's immediate neighbors. The cost fields and advertising information of the nodes can be defined as follows:  
\[ \begin{split}\alpha_{\text{K}}&=C_{K\to S}+C_{ \text{E,K}}\\ \alpha_{\text{J}}&=C_{J\to S}+C_{\text{E,J}}\\ C_{I\to J\to S}&=\alpha_{\text{J}}+C_{ \text{L,I}\to J}\\ \alpha_{\text{L}}&=C_{L\to S}+C_{\text{E,L}}\\ C_{I\to L\to S}&=\alpha_{\text{L}}+C_{ \text{L,I}\to L}\\ C_{I\to K\to S}&=\alpha_{\text{K}}+C_{ \text{L,I}\to K}\\ C_{I\to S}&=\min\bigl{(}C_{I\to J\to S},C_{ \text{L}\to K\to S},C_{\text{L}\to L\to S},C_{\text{L}\to L\to S}\bigr{)} \end{split}  \tag{11}\]  
Herein, we opted for the concept of adaptive clustering multi-hop routing scheme [34]. Here, the node can be CH, CM, or CR in the first, second, third round, respectively. In comparison to its neighbors, each node will choose to become a cluster head based on the selection conditions [32]. To form a cluster, the CH will relay the \"JOIN\" message to its neighbors. Otherwise, the node will wait for the \"JOIN\" message; if it only receives one message, it will become a cluster member. When a node receives a significant number of \"JOIN\" messages, it selects the sender node with the greatest received power or the shortest distance. Figure 2: Cost Field Setup. The cluster formed between nearby nodes, we think, to conserve energy and reduce transmission time. We assume that some nodes will become the CR and forward data to the sink due to the restricted transmission range of the nodes at depths D2 and D3. If a relay node receives the \"JOIN\" message from both the upper and lower depth nodes (DL and DL-1), the node decides to become a CR, which acts as a route forwarder. The nodes send a \"RESPONSE\" message to the CH to validate the cluster-ID. The CHs at the same level simply send them a \"JOIN\" message. Node Si tests Sk's depth if the CH Si receives a \"RELAY\" message from Sk, and if Sk's depth is near the surface, node Si or Dm(Sk) Dm(Si), and node Si adds Sk as a relay node. In a cluster scheme, the relay node may gather data from the sensor node or control the network topology. As a result of the node's energy conservation, the nodes' energy consumption has been reduced.