# Efficient Wireless Charger Deployment for Wireless Reshapeable Sensor Networks
## 3 Modeling and Problem Definition  
Sensor nodes in the WRSN are assumed to be distributed in a cuboid with the length \(L\), width \(W\) and height \(H\); they have different energy requirements and can be located on the floor plane or object surfaces. The energy requirements and locations are summed to be known in advance and fixed. Wireless chargers equipped with directional antennas are assumed to be deployed at grid points on the grid plane or the deployment plane that is with height \(H\) and parallel to the floor plane. Moreover, the grid points are with the separation \(S\); that is, the nearest grid points are of the distance \(S\). Please refer to Figure 2 for the illustration of the distribution of sensor nodes and the deployment of wireless chargers. Note that we assume no more than \(Q\), \(Q\geq 1\), chargers can be deployed at a same grid point.  
The effective charging space of wireless chargers is modeled as a cone, called a charger cone. Note that the two terms \"charger\" and \"cone\" are used interchangeably later in this paper. The modeling is based on the fact that direction antennas usually have specific beamwidth due to vertical and horizontal polarization. For example, the direction antenna of the Powercast TX91501-3W-ID charger has \(60^{\circ}\) beamwidth in both the horizontal direction and the vertical direction. As shown in Figure 3, every charger cone is characterized by an apex \(X\), a normal vector \(\overrightarrow{N}\) whose direction is parallel to the symmetrical axis of the cone, an effective charging distance (i.e., the cone generatrix length) \(D\), and an effective charging angle (i.e., the opening angle) \(A\). When a sensor node is within the charger cone of a charger, we assume the sensor node can be charged effectively by the charger; otherwise, the sensor node cannot be charged effectively. Note that the point \(E\) is regarded as an extreme point within the charger cone, where the distance between \(E\) and \(X\) is \(D\) and the angle between the normal vector \(\overrightarrow{N}\) and the vector going from \(X\) to \(E\) is \(A/2\).  
Figure 3: A charger cone and its parameters.  
Figure 2: The schematic view of the WRSN deployment cuboid.  
The famous Friis transmission equation [23], shown in Equation (1), is used to model the energy received by the harvester. By the equation, we can see that the received energy \(P_{\mathrm{r}}\) is inversely proportional to \(D^{2}\), that is, \(P_{\mathrm{r}}=\alpha D^{-2}\), where \(\alpha\) is a constant and \(D\) is the distance between the charger and the harvester. This paper utilizes a more general empirical model \(P_{\mathrm{r}}=\alpha D^{-\beta}\), where \(\beta>0\), to estimate the energy received by the harvester. Later in Section 5, we will show how to apply the power regression analysis to derive appropriate \(\alpha\) and \(\beta\) values for the model to fit experimental data accurately.  
With all the assumptions mentioned above, the WCDO problem deals with how to deploy the minimum number of wireless chargers to cover all sensor nodes and fulfil their energy demands. To be more precise, given a set of sensor nodes distributed in a cuboid and a set of grid points, the WCDO problem addresses how to deploy as few as possible chargers at grid points to charge sensor nodes to make them sustainable under the assumption that the effective charging space of chargers is a cone. A special case of the WCDO problem in which every charger's charging space is a half sphere (instead of a cone) and every sensor node needs to be covered by only one charger can be solved by the reduction to the NP-hard set covering (SC) problem [24]. It is believed the WCDO problem is also NP-hard. Nevertheless, the NP-hardness of the WCDO problem has not yet proven.