# Efficient Wireless Charger Deployment for Wireless Reshapeable Sensor Networks
## 2 Related Work  
In this section, we review the research results [3; 4; 5; 6; 7; 8; 13; 14; 15; 16; 17] most related to our work. [PERSON] et al. [3] investigated various ambient energy-harvesting technologies using different power sources, such  
Figure 1: A wireless rechargeable sensor network (WRSN) sensor node equipped with an energy harvester to harvest energy. RF: radio frequency. as solar power, thermal power, piezoelectric power, and wireless RF power, in the aspect of their applicability for self-sustainable wireless sensor platforms. They stated that although ambient RF energy has low power density values of 0.2 nW/cm\({}^{2}\)-1 uW/cm\({}^{2}\) and low radio frequency-direct current (RF-DC) conversion values of 10%-30%, a larger amount of total available power can be harvested by utilizing high gain antennas. Some antennas designed by the authors were shown [3]. For example, a folded dual-bad antenna at 915 MHz and 2.45 GHz was designed to harvest RF power from cellular and WiFi sources with power densities about 1 uW/cm\({}^{2}\). Moreover, a folded dipole antenna was designed to harvest RF power from a two-way radio at the UHF band (464 and 468 MHz). They also implemented a prototype of an embedded microcontroller-enabled sensor platform successfully powered by an ambient UHF digital TV signal (512-566 MHz) from a broadcasting antenna that is 6.3 km away. [PERSON] et al. [4] proposed a moving and stopping plan for mobile chargers to guarantee that the time to charge all given stationary sensor nodes is minimized. An optimal solution using linear programming for the plan is proposed; however, the computational complexity of the solution is very high. Therefore, a heuristic solution discretizing the charging power on a two-dimension space was proposed to reduce the time complexity. The heuristic solution has been proved to have a bounded approximation ratio. [PERSON] et al. [5] proposed two types of energy provisioning problems for WRSNs: point provisioning and path provisioning. The former addresses how to use the least number of chargers to ensure that a static sensor node placed in any position of the network will receive a sufficient recharge rate to ensure WRSN sustainability. The latter addresses the same problem for mobile sensor nodes under the assumption that sensor nodes can harvest excess energy in power-rich areas and store the energy for later use in power-deficient areas. The authors utilized the triangular deployment of chargers to solve the two problems. They also proposed a modified Friis free space transmission equation adjusted for short distance transmission for evaluating the deployment. The original Friis equation and the modified version are shown below in Equations (1) and (2), respectively:  
\[P_{\mathrm{r}}=G_{\mathrm{t}}G_{\mathrm{r}}\left(\frac{\lambda}{4\pi D} \right)^{2}P_{\mathrm{t}} \tag{1}\]  
\[P_{\mathrm{r}}=\frac{G_{\mathrm{t}}G_{\mathrm{r}}\eta}{L_{\mathrm{P}}}\left( \frac{\lambda}{4\pi\left(D+\beta\right)}\right)^{2}P_{\mathrm{t}} \tag{2}\]  
In Equations (1) and (2), \(P_{\mathrm{t}}\) and \(G_{\mathrm{t}}\) denote the power and the antenna gain of the transmitter (i.e., charger), respectively, and \(P_{\mathrm{r}}\) and \(G_{\mathrm{r}}\) denote the power and the antenna gain of the receiver (i.e., harvester), respectively. Furthermore, \(\lambda\) is the wave length, and \(D\) is the distance between the antennas of the charger and the harvester. \(\eta\) is the rectifier efficiency, \(L_{\mathrm{P}}\) is polarization loss, and \(\beta\) is a parameter to adjust the original Friis equation for short distance transmission. [PERSON] et al. [6] stated that if the constraints of sensor node speed and battery capacity are considered, the sustainable operation of sensor nodes may not be guaranteed. It proposed the concept of quality of energy provisioning (QoEP) to characterize the expected portion of time that a sensor node can sustain operation by taking into account node speed and battery capacity.