# Energy-Aware Adaptive Weighted Grid Clustering Algorithm for Renewable Wireless Sensor Networks
## 3 The Preliminary Description of the Problem
### System Model  
To handle the data flow rate and energy consumption for each node of the renewable wireless sensor network, the following energy model has been used. This model represents a routing scheme with the following flow balance constraints:  
\[\sum\limits_{k\in\mathrm{N}}^{k\
eq i}g_{ki}(t)+R_{i}=\sum\limits_{j\in \mathrm{N}}^{j\
eq i}g_{ij}(t)+g_{iB}\ (t)\ (\mathrm{i}\in\mathrm{N},t\geq 0) \tag{1}\]  
Here \(g_{ij}\) and \(g_{iB}(t)\) are the data flow rate from sensor node \(i\) to \(j\) and base station. And, \(g_{ki}(t)\) is the rate of gathering data in a time \(t\). Equation (1) implies the rate of generating data at node \(i\) plus the rate of receiving data from any other node \(k\) is equal to the rate of sending data from node \(i\) to \(j\) and the base station. Here, the energy consumption of a node during a renewable cycle must meet two significant requirements:  
1. The energy of each node remains equal both at the beginning and at the end of one renewable cycle during the time of \(\tau\).
2. To make a node operational, the energy of the node is never less than the \(\mathrm{E_{min}}\). We handle the above issues by the following energy consumption constraints: \[p_{i}(t)=\rho\sum\limits_{k\in\mathrm{N}}^{k\
eq i}g_{ki}(t)+\sum\limits_{j \in\mathrm{N}}^{j\
eq i}C_{ij}g_{ij}(t)+C_{iB}g_{iB}(t)\ (\mathrm{i}\in\mathrm{N},t\geq 0)\] (2)  
There is some energy consumption for data sending and data gathering. Equation (2) is responsible for energy consumption in the network, as \(p_{i}(t)\) is the rate of energy consumption for node \(i\) over a time \(t\), whereas \(\rho\) is denoted as the energy consumption coefficient for receiving data measured by the rate of generating data (\(\mathrm{R_{i}}\)). Accordingly, in this energy model, \(\sum\limits_{k\in\mathrm{N}}^{k\
eq i}g_{ki}(t)\) represents the energy consumption rate for gathering data. \(\sum\limits_{j\in\mathrm{N}}^{j\
eq i}C_{ij}g_{ij}(t)+C_{iB}g_{iB}(t)\), is the rate of energy consumption for sending data from the sensor nodes.  
As \(\mathrm{E_{min}}\) is a threshold value, the proposed model assumes that the energy of each node is not less than this threshold value. A charge cycle is divided into three main parts:  
1. The WPCD starts its journey from a RS and arrives at node \(i\) on the prescribed path using the shortest path algorithm.
2. During time \(\tau_{i}\), the WPCD charges the battery of node \(i\) remotely via wireless power transfer technology.
3. Then, the WPCD revolves towards the next node to supply power. Finally, it takes a vacation at the RS until the start of the next cycle. Vacation time denoted as \(\tau_{vac}\), helps the WPCD to energize its battery for the next charge cycle whereas \(\tau\) is the complete time required for one charge cycle.  
To cope with the performance limitations on a network; the layout is transferring the power to each node intermittently so that its energy never falls below \(\mathrm{E_{min}}\). To increase \(\tau_{vac}\) for the WPCD at the RS, we need to maximize the objective function (i.e., \(\frac{T_{\text{succ}}}{\tau}\)) as presented in [36]. As per this objective function, we can either maximize the value of the numerator (i.e., \(\tau_{\text{succ}}\)) as a vacation time or minimize the value of the denominator (i.e., \(\tau\)) as overall traveling time.