# Swarm Intelligence Techniques for Mobile Wireless Charging
## 3 The Network Model
### Redarge by Adaptive Network Partitioning and Clustering  
In order to effectively cover the wide geographical regions of the whole network, multiple MCs are employed to charge the sensor-nodes and to collect data. To solve the complex problem of scheduling multiple MCs in LS-WSNs, we propose to adaptively partition the whole network into several clusters and distribute an MC in each region. Some advantages of a distributed clustering approach in real-world scenarios are provided in [42], which include improving the route connection, minimizing the energy consumption, and increasing the scalability while reducing the traffic. To calculate the minimum number of MCs needed to achieve perpetual network operation, we use Equation (6) [43]. That is,  
\[n=\left[\frac{\beta^{-1}(\in)\sqrt{\overline{E_{c}(T)}}+\overline{E_{c}(T)}- \mathrm{E_{0}}\left(\frac{\sqrt{2L}}{\mathrm{v}}+\Gamma\right)}{\psi_{s}(T)}\right] \tag{6}\]  
where \(n\) denotes the number of MCs, \(\beta^{-1}\) is the inverse cumulative distribution function of normal distribution, \(\in\) is a value approaching 1 but is not equal to 1, T is a large time frame, \(E_{c}(T)\) is the total energy consumption of the network up to T, E0 is the initial energy of the sensor-nodes, L is the length of the sensor field, \(\mathrm{v}\) is the speed of the MCs, \(\Gamma\) is the recharge time of the sensor-node's battery, and \(\psi_{s}\) is battery capacity of the sensor-nodes. The probability for the energy-neutral condition [43] is modelled as:  
\[P_{r}=\beta\left(\frac{E_{r}(T)\;+\;\left[\varepsilon\right]_{0}-\;\overline{E _{c}(T)}}{\sqrt{\overline{E_{c}(T)}}}\right) \tag{7}\]  
where \(E_{r}\) is the total energy replenished into the network by the MCs, up to \(T\). Since \(\beta^{-1}(1)\rightarrow\infty\), the network operation is considered perpetual at a high probability of occurrence of \(\in\), if \(\in\to 1\) but is not equal to 1, e.g., \(\in\)\(=0.99\), \(\beta^{-1}(0.99)=2.33\). Though the rate of data generation is dynamic, the process can be modeled as a Poisson process to determine the number of MCs. Since the number of MCs \(n\) in the network is known, the number of network partitions \(\varepsilon\) can get through \(\varepsilon=n\). Hence, each sub-region of the network always has MCs to ensure wireless energy transfer and data collection. The process of network partitioning and clustering is as follows:  
* The first partition is conducted. This method is like that in [9] and follows a uniform division of the network into \(c\) parts. Unlike [9], our method uses a dynamic selection of the cluster centroids, where CHs can be in each region of the network. These nodes are located at the center of each region. Besides, we evaluate the distance \(r^{\prime}\) and the shortest hop routing \(h^{\prime}\) from each node to the cluster centroids. \(r^{\prime}_{jk}\) denotes the distance between node \(j\) and cluster centroid \(k\), where \(j\in N,\;k\in c\). Figure 3: Partitioning the network into slices and sectors. (**a**) Slices. (**b**) Sectors. * The second partition is conducted. Firstly, we evaluate the weight \(W_{jk}\) of each node \(j\) to \(k\). Here, \(W_{jk}=\alpha R_{jk}+\psi H_{jk}\); \(\alpha+\psi=1\), \(0\leq\alpha\), \(\psi\leq 1,\forall\ \psi\in N,\forall k\ \in[1,c]\). We define \(\alpha\) and \(\psi\) as part of the distance and routing hop priorities, respectively, and \(R_{jk}\) is the serial number of the distance \(r^{\prime}_{jk}\) from node \(j\) to cluster centroid \(k\). If \(R_{jk}=1\) denotes the minimum value of \(r^{\prime}_{jk}\), \(R_{jk}=c\) denotes the maximum distance \(r^{\prime}_{jk}\). We denote \(\
abla\!\!=\frac{\alpha}{\psi}\) as the ratio of \(\alpha\) and \(\psi\). If \(\
abla\!\!\to\infty\), our algorithm only considers a routing hop. On the contrary, our algorithm can also consider distance when \(\
abla\!\!\to 0\). In this case, we jointly consider the routing hop and distance to make \(\
abla\!\!=1\).