# Swarm Intelligence Techniques for Mobile Wireless Charging
## 3 The Network Model
### The Deployment and Energy Model  
Three kinds of devices can be identified with the proposed scheme, namely, the static sensors, the mobile chargers (MCs), and the static sink, as shown in Figure 2. Our model assumes that there are \(\mathcal{N}\) sensors of the wireless communication distance \(\mathrm{d}\)randomly distributed in a circular sensing field with the area \(\mathrm{\SIUnitSymbolAngstrom}\) and the radius \(\Re\) (we investigate both uniform and non-uniform distributions) and \(\mathrm{n}\) MCs are initially deployed at the center of a circular area along the x-y coordinates in Equation (1):  
\[(m,n)=\left\{\frac{\Re}{2}\cos\Bigl{(}\frac{\pi}{\mathrm{n}}(2p-1)\Bigr{)}, \ \frac{\Re}{2}\sin\Bigl{(}\frac{\pi}{\mathrm{n}}(2p-1)\Bigr{)}\right\} \tag{1}\]  
where \(p\) = {1, 2,... n}. If we assume that the \(\mathcal{N}\) sensors are uniformly and randomly distributed in a circular area of the radius \(\Re\), then the network density is given by:  
\[\hat{\$}=\frac{\mathcal{N}}{\pi\mathcal{R}^{2}} \tag{2}\]  
The communication range of the sensors varies according to requirements of the underlying routing protocols. Because the algorithms can run for an extended time with only one initialization, the initial position of the MCs may not be important as it could also be located at the center of the network. The base station or sink can lie anywhere in the  
Figure 1: Network model and components. Figure 2: The network architecture. network and is used to replenish the MCs and aggregate data from the MCs. This model relies on both wireless power transfer and data collection. For simplicity, we assume that all sensors can generate data packets at the same rate of \(\delta\) per unit of time. This assumption is typical of the monitoring and reporting applications in a WSN. If we denote \(\hat{\textbf{\acute{E}}}_{total}\) as the total available energy in the network, then initially,  
\[\hat{\textbf{\acute{E}}}_{total}=\hat{\textbf{\acute{E}}}_{sensors}+\ \hat{ \textbf{\acute{E}}}_{MC}^{init} \tag{3}\]  
where \(\hat{\textbf{\acute{E}}}_{sensors}\) is the amount of energy distributed among the sensor-nodes and \(\hat{\textbf{\acute{E}}}_{MC}^{init}\) is the total energy initially possessed by the MCs that can be delivered to the network to charge the sensors. Each sensor and each MC, respectively, have a maximum amount of energy representing their initial energy, given as:  
\[\hat{\textbf{\acute{E}}}_{sensor}^{max}=\frac{\hat{\textbf{\acute{E}}}_{sensors }}{\mathcal{N}}\ \text{and}\ \hat{\textbf{\acute{E}}}_{MC}^{max}=\frac{\hat{\textbf{ \acute{E}}}_{\text{MCs}}}{\text{n}} \tag{4}\]  
For message transmission and reception, our model assumes that the amount of energy dissipated by the radio module is proportional to the message size. For a \(\gamma\)-bit transmitted message, the radio expends \(\text{I}\text{I}\text{C}_{T}(\gamma)=\epsilon_{trans\cdot\text{\text{Y}}}\) and to receive the \(\gamma\)-bit message. It expends \(\text{I}\text{C}_{R}(\gamma)=\epsilon_{rec\cdot\text{\text{Y}}}\) where \(\epsilon_{trans}\) and \(\epsilon_{rec\cdot\text{\text{Y}}}\) are constants which depend on the transmission range \(\text{dof}\) the sensors and the radio module. Generally, the power needed for transmitting a message at a distance \(\text{djs}\) roughly \(d^{\beta}\) where \(2\leq\beta\leq 6\) is a constant; for simplicity, we take \(\beta=2\). We investigate both cases of uniform and non-uniform deployments of the \(\mathcal{N}\) sensor-nodes in the network. We show, in Figure 3, a network that is virtually partitioned into \(\frac{2\pi}{\beta}\) slices and \(\frac{2\pi}{d}\) co-centric rings of small sectors. We define a sector as the intersection between a slice and a ring. Figure 3a shows that there are 8 slices (\(\varnothing=\pi/4\)), while Figure 3b shows the different partitions of the slice sectors. Here, each slice contains 5 sectors, thus giving a total of 40 sectors in the network. We assume a case of random non-uniform deployment of the sensor-nodes.