Efficient Wireless Charger Deployment for

energies Article

Efficient Wireless Charger Deployment for Wireless Rechargeable Sensor Networks Jehn-Ruey Jiang * and Ji-Hau Liao Department of Computer Science and Information, National Central University, Taoyuan 32001, Taiwan; [email protected] * Correspondence: [email protected]; Tel.: +886-3-422-7151 Academic Editor: Chang Wu Yu Received: 11 June 2016; Accepted: 22 August 2016; Published: 31 August 2016

Abstract: A wireless rechargeable sensor network (WRSN) consists of sensor nodes that can harvest energy emitted from wireless chargers for refilling their batteries so that the WRSN can operate sustainably. This paper assumes wireless chargers are equipped with directional antennas, and are deployed on grid points of a fixed height to propose two heuristic algorithms solving the following wireless charger deployment optimization (WCDO) problem: how to deploy as few as possible chargers to make the WRSN sustainable. Both algorithms model the charging space of chargers as a cone and calculate charging efficiency according power regression expressions complying with the Friis transmission equation. The two algorithms are the greedy cone covering (GCC) algorithm and the adaptive cone covering (ACC) algorithm. The GCC (respectively, ACC) algorithm greedily (respectively, adaptively) generates candidate cones to cover as many as possible sensor nodes. Both algorithms then greedily select the fewest number of candidate cones, each of which corresponds to the deployment of a charger, to have approximate solutions to the WCDO problem. We perform experiments, conduct simulations and do analyses for the algorithms to compare them in terms of the time complexity, the number of chargers deployed, and the execution time. Keywords: wireless rechargeable sensor network (WRSN); energy harvesting; charger deployment; sustainability; greedy algorithms; directional antennas; Friis transmission equation

1. Introduction A wireless rechargeable sensor network (WRSN) [1] consists of many rechargeable sensor nodes and one or more sink nodes. Sensor nodes are powered by batteries and can sense physical phenomena (e.g., temperature, humidity, atmospheric pressure, light intensity, and electromagnetic field strength) and forward the sensed data to the sink node(s) via multi-hop wireless communications. WRSNs are sustainable since they utilize energy harvesting technologies to replenish the batteries of sensor nodes. The replenishing is achieved by equipping every sensor node with an energy harvester to harvest energy from different sources [2,3], such as solar power, thermal power, radio frequency (RF) waves, or air flow, as shown in Figure 1. Many recent studies [4–13] focus on the WRSNs due to their sustainability. Some other studies [14–20] investigate battery-less or battery-free wireless sensor networks (WSNs) in which sensor nodes are powered by capacitors storing energy harvested through wireless power transmission (WPT), especially microwave power transmission (MPT). Without loss of generality, such WSNs are regarded as WRSNs in this paper. Energy harvesting technologies are generally divided into two classes: (1) charger-based energy harvesting and (2) ambient energy harvesting. The former technology deploys specific devices, called chargers, to emit energy, and utilizes harvesters to capture the energy. The latter technology applies ambient energy harvesting devices to capture energy from the ambient environment. Typical devices include solar panels, thermoelectric generators, and ambient RF harvesters to capture energies from TV, Energies 2016, 9, 696; doi:10.3390/en9090696

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Wi-Fi, and 3G/4G stations. It is more difficult for the latter to control the amount of harvested energy Energies 2016, 9, 696  2 of 21  due to the unstable nature of ambient energy sources (ESs). This is problematic if sensor nodes have to TV, Wi‐Fi, and 3G/4G stations. It is more difficult for the latter to control the amount of harvested  meet specificenergy due to the unstable nature of ambient energy sources (ESs). This is problematic if sensor nodes  energy requirements. Therefore, this paper pays attention to WRSNs using charger-based have  to technologies. meet  specific  energy  requirements.  Therefore,  paper  pays  attention  to  WRSNs  energy harvesting It practically focuses onthis  indoor WRSNs using the RFusing  wireless energy charger‐based energy harvesting technologies. It practically focuses on indoor WRSNs using the RF  harvesting technology. wireless energy harvesting technology. 

  Figure  1.  A  wireless  rechargeable  sensor  network  (WRSN)  sensor  node  equipped  with  an  energy  wireless rechargeable sensor network (WRSN) sensor node equipped with harvester to harvest energy. RF: radio frequency. 

Figure 1. A harvester to harvest energy. RF: radio frequency.

an energy

The deployment of “RF wireless chargers” (or “chargers” for short) is challenging, since every  harvester needs to be covered by at least one charger and the charging space of chargers is usually  limited.  For  of example,  the  effective  charging  distance  is  3–5  m  and for the short) effective ischarging  angle  is   since every The deployment “RF wireless chargers” (or “chargers” challenging, 30°–45° between the Powercast TX91501‐3W‐ID charger (Powercast Corp., Pittsburgh, PA, USA) [21]  harvester needs to be covered by at least one charger and the charging space of chargers is usually and the Powercast P2110‐EVAL‐02 harvester (Powercast Corp.) [22]. It is also a cost‐consuming task,  limited. Foras  example, the effective charging distance is 3–5 TX91501‐3W‐ID  m and the effective charging angle is wireless  chargers  are  expensive.  For  example,  the  Powercast  charger  currently  costs about 1,000 US dollars. This motivated the authors to study the wireless charger deployment  30◦ –45◦ between the Powercast TX91501-3W-ID charger (Powercast Corp., Pittsburgh, PA, USA) [21] optimization (WCDO) problem defined below. Given a set of WRSN sensor nodes equipped with  and the Powercast P2110-EVAL-02 harvester (Powercast Corp.) [22]. It is also a cost-consuming task, harvesters, the WCDO addresses how to deploy as few as possible chargers to cover all sensor nodes  as wireless chargers aretheir  expensive. For example, thethe  Powercast TX91501-3W-ID currently for  satisfying  energy  requirements  to  make  WRSN  sustainable.  To  the  best  charger of  our  knowledge, no research has addressed the same problem up to now.  costs about 1,000 US dollars. This motivated the authors to study the wireless charger deployment This paper proposes two algorithms for solving the WCDO problem. It is assumed that wireless  optimization (WCDO) problem defined below. Given a set of WRSN sensor nodes equipped with chargers  are  equipped  with  directional  antennas  whose  charging  space  is  modeled  as  a  cone  and  harvesters, the WCDO addresses how to deploy as few as possible chargers to cover all sensor nodes charging efficiency complies with the Friis free space transmission equation [23]. It is also assumed  that chargers are deployed on grid points at a specific height and the sensor nodes equipped with  for satisfying their energy requirements to make the WRSN sustainable. To the best of our knowledge, wireless harvesters are distributed beneath the chargers on the floor or object surfaces. The proposed  no research has addressed the same problem up to now. algorithms  are  the  greedy  cone  covering  (GCC)  algorithm  and  the  adaptive  cone  covering  (ACC)  This paper proposes two algorithms for solving the WCDO problem. It is assumed that wireless algorithm. Both algorithms treat chargers as cones. The GCC (respectively, ACC) algorithm greedily  chargers are (respectively, adaptively) generates candidate cones that can cover as many as possible sensor nodes.  equipped with directional antennas whose charging space is modeled as a cone and The  two  algorithms  then  greedily  select  the  fewest  number  of  candidate  cones,  each  of  which  charging efficiency complies with the Friis free space transmission equation [23]. It is also assumed corresponds to the deployment of a charger, to derive approximate solutions to the WCDO problem.  that chargersWe  are deployed on experiments  grid points at a specific height We  andalso  theconduct  sensor nodes equipped with perform  extensive  to  justify  our  assumptions.  comprehensive  simulations and do analyses for the algorithms to compare them in terms of the time complexity, the  wireless harvesters are distributed beneath the chargers on the floor or object surfaces. The proposed number of chargers deployed, and the execution time.  algorithms are the greedy cone covering (GCC) algorithm and the adaptive cone covering (ACC) The remainder of the paper is organized as follows: some related studies are reviewed in Section  algorithm. Both algorithms treat chargersand  as the  cones. Thedefinition.  GCC (respectively, ACC) algorithm greedily 2.  Section  3  presents  the  modeling  problem  In  Section  4,  the  two  proposed  (respectively,algorithms to solve the problem are described. Section 5 shows algorithm time complexity analyses  adaptively) generates candidate cones that can cover as many as possible sensor and grid point separation analyses for deploying chargers. Experiments about the charging efficiency  nodes. The two algorithms then greedily select the fewest number of candidate cones, each of which and results of algorithm simulations are described in Section 6. Finally, Section 7 concludes the paper. 

corresponds to the deployment of a charger, to derive approximate solutions to the WCDO problem. We perform 2. Related Work  extensive experiments to justify our assumptions. We also conduct comprehensive simulations and In this section, we review the research results [3–8,13–17] most related to our work. Kim et al. [3]  do analyses for the algorithms to compare them in terms of the time complexity, the investigated various ambient energy‐harvesting technologies using different power sources, such as  number of chargers deployed, and the execution time. The remainder of the paper is organized as follows: some related studies are reviewed in Section 2. Section 3 presents the modeling and the problem definition. In Section 4, the two proposed algorithms to solve the problem are described. Section 5 shows algorithm time complexity analyses and grid point separation analyses for deploying chargers. Experiments about the charging efficiency and results of algorithm simulations are described in Section 6. Finally, Section 7 concludes the paper. 2. Related Work In this section, we review the research results [3–8,13–17] most related to our work. Kim et al. [3] investigated various ambient energy-harvesting technologies using different power sources, such

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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/cm2 –1 µW/cm2 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 µW/cm2 . 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. Fu 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. He 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:  Pr = Gt Gr

Pr =

Gt Gr η Lp



λ 4πD

2 Pt

λ 4π ( D + β)

(1)

2 Pt

(2)

In Equations (1) and (2), Pt and Gt denote the power and the antenna gain of the transmitter (i.e., charger), respectively, and Pr and Gr denote the power and the antenna gain of the receiver (i.e., harvester), respectively. Furthermore, λ is the wave length, and D is the distance between the antennas of the charger and the harvester. η is the rectifier efficiency, Lp is polarization loss, and β is a parameter to adjust the original Friis equation for short distance transmission. Dai 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. The upper and lower bounds of QoEP in 1D and 2D sensor node mobility cases are derived. Xu et al. [7] considered the scheduling problem of arranging multiple mobile chargers to collaboratively charge sensor nodes with the goal of minimizing the total traveling distance of all chargers. This can be formulated as finding multiple travelling salesperson problem (TSP) closed tours covering all sensor nodes such that the total length of the tours is minimized. The authors proposed a 2-approximation algorithm to solve the problem under the assumption that the energy consumption rates of all sensor nodes are fixed. Liang et al. [8] considered a scheduling problem similar to that addresses by Xu et al. in [6]. The problem is how to use the minimum number of mobile chargers to charge sensor nodes so that none of sensor nodes runs out of energy, subject to the energy capacity of mobile chargers. The authors designed a 5-approximation algorithm to find multiple closed tours. Afterwards, by bounding the

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total distance of each tour, the authors designed another approximation algorithm invoking the 5-approximation algorithm to provide approximate solutions to the problem. Ye et al. [9] introduced a new metric, the charging utility, to measure the charging quality of the charger with the considerations of both the fairness of charging sensor nodes and the number of sensor nodes charged by mobile chargers traveling along closed tours. It is assumed that each mobile charger can only travel a limited length per tour, due to its limited energy capacity. The authors formulated an optimization problem of scheduling a mobile charger to charge sensor nodes, with an objective of maximizing the charging utility, subject to the total traveling distance of the mobile charger per tour and the deadline of each sensor node to be charged. An approximation algorithm with a guaranteed approximation ratio was proposed to solve the problem. Maehara et al. [14] proposed a WPT scheme called multi-point carrier shift diversity (MPCSD), in which the center frequency of the carrier of each power transmitting point (or transmitter) is slightly shifted with a predefined amount to create artificial fading. Suppose there are N different transmitters, the carrier frequency f n of the nth transmitter is defined in Equation (3), shown below: fn = fc −

B (2n − 1) B + 2 2N

(3)

In Equation (3), f c is the center frequency, B is the channel bandwidth, and n = 1, . . . , N. The proposed scheme is shown to be capable of mitigating power attenuation caused by the path loss effect and the standing-wave effect created by multipath and interference between multiple power transmitters so that the coverage of energy supply field is improved. Maehara et al. [15] investigated the energy transmission coverage of the MPCSD scheme through theoretical discussions and practical experiments. It is shown that the MPCSD scheme can achieve 100% of coverage in an indoor environment for the sensor node duty factor of 1/100 (i.e., the sensor node is in the transmitting mode for the duration of 10 ms per second). Imoto et al. [16] conducted experiments in an IEEE 802.11g-based wireless local area network (WLAN) in which an ES transmits microwave power to a sensor station (SS) using the carrier sense multiple access with collision avoidance (CSMA/CA) protocol to transmit data to an access point (AP). They derived the following two requirements for coexistence of MPT and the WLAN wireless data transmission (WDT). First, MPT and WDT should not proceed simultaneously. Therefore, the ES should share the timing information of MPT with the SS so that the SS attempts to transmit data frames only when the ES does not transmit microwave power. Second, the ES should not transmit microwave power to the SS when the AP broadcasts beacon frames so that the SS is not disassociated from the AP. This can also prevent data throughput degradation due to WLAN automatic data rate selection. Yamashita et al. [17] assumed the same environment as assumed by Imoto et al. [16] and designed and implemented a scheduling scheme to solve the disassociation problem and the throughput degradation problem caused by the coexistence of MPT and WDT. In the scheduling scheme, the SS goes to the active mode to transmit data towards and to receive delivery traffic indication map (DTIM) beacons from the AP periodically. It goes to the power-saving (sleep) mode for most of the time to save energy. As long as the SS is in the sleep mode, the ES can transmit microwave power. On the basis of link budget analysis, the authors also clarified the maximum distance between the EC and the SS to conclude that the scheduling scheme can be applied to a closed space or room. Yoshida et al. [18] designed an aerospace WSN that allow the coexistence of high-power MPT and WDT. The aerospace WSN is used in the aircraft application of the reusable vehicle test (RVT) or space capsule for health and safety monitoring inside the RVT. The WSN is based on the time-division operation and the frequency-division operation. The former operation uses a control function to schedule MPT and WDT so that MPT is not activated when WDT is activated. The latter operation applies different frequency bands for WPT and WDT to avoid the interference between them. Through a practical experiment lasting 120 min, it is shown that the aerospace WSN can achieve sustainable balance between harvested power and consumed power.

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Energies 2016, 9, 696  3. Modeling and Problem Definition

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3. Modeling and Problem Definition  Sensor nodes in the WRSN are assumed to be distributed in a cuboid with the length L, width Energies 2016, 9, 696  5 of 21  W and height Sensor nodes in the WRSN are assumed to be distributed in a cuboid with the length L, width  H; they have different energy requirements and can be located on the floor plane or 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. 3. Modeling and Problem Definition  object  surfaces.  The  energy  requirements  and  locations  are  summed  to  be  known  in  advance  and  Wireless chargers equipped with directional antennas are assumed to be deployed at grid points on the Sensor nodes in the WRSN are assumed to be distributed in a cuboid with the length L, width  fixed.  Wireless  chargers  equipped  with  directional  antennas  are  assumed  to  be  deployed  at  grid  grid plane or the deployment plane that is with height H and parallel to the floor plane. Moreover, the points on the grid plane or the deployment plane that is with height H and parallel to the floor plane.  W and height H; they have different energy requirements and can be located on the floor plane or  grid points aresurfaces.  with the separation S; that is, the grid areto of distance S. Please Moreover, the grid points are with the separation S; that is, the nearest grid points are of the distance  object  The  energy  requirements  and nearest locations  are points summed  be the known  in  advance  and refer to S. Please refer to Figure 2 for the illustration of the distribution of sensor nodes and the deployment  Wireless  chargers  equipped  with  directional  are  assumed  to  be  deployed  at  grid  Figure 2fixed.  for the illustration of the distribution of sensorantennas  nodes and the deployment of wireless chargers. of wireless chargers. Note that we assume no more than Q, Q ≥ 1, chargers can be deployed at a same  points on the grid plane or the deployment plane that is with height H and parallel to the floor plane.  Note that we assume no more than Q, Q ≥ 1, chargers can be deployed at a same grid point. grid point.  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 ≥ 1, chargers can be deployed at a same  grid point. 

  Figure 2. The schematic view of the WRSN deployment cuboid. 

Figure 2. The schematic view of the WRSN deployment cuboid. 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  The effective charging space of wireless chargers is modeled as a  cone, called a charger modeling is based on the fact that direction antennas usually have specific beamwidth due to vertical  Figure 2. The schematic view of the WRSN deployment cuboid.  cone. Note that the two terms “charger” and usedof interchangeably later in this paper. and  horizontal  polarization.  For  example,  the “cone” direction are antenna  the  Powercast  TX91501‐3W‐ID  charger has 60° beamwidth in both the horizontal direction and the vertical direction. As shown in  The modeling is based on the fact that direction antennas usually have specific beamwidth due The effective charging space of wireless chargers is modeled as a cone, called a charger cone.    whose direction is  Figure 3, every charger cone is characterized by an apex X, a normal vector  to vertical polarization. For example, the direction later  antenna the The  Powercast Note and that  horizontal the  two  terms  “charger”  and  “cone”  are  used  interchangeably  in  this ofpaper.  parallel to the symmetrical axis of the cone, an effective charging distance (i.e., the cone generatrix  ◦ TX91501-3W-ID charger has 60 beamwidth in both the horizontal direction and the vertical direction. modeling is based on the fact that direction antennas usually have specific beamwidth due to vertical  length) D, and an effective charging angle (i.e., the opening angle) A. When a sensor node is within  * and  For  example,  direction  antenna  of apex the  Powercast  TX91501‐3W‐ID  the charger cone of a charger, we assume the sensor node can be charged effectively by the charger;  As shown inhorizontal  Figure 3,polarization.  every charger cone is the  characterized by an X, a normal vector N whose charger has 60° beamwidth in both the horizontal direction and the vertical direction. As shown in  node  cannot  be  charged  effectively.  that  the charging point  E  is distance regarded  as  an  the cone direction isotherwise,  parallelthe  to sensor  the symmetrical axis of the cone, anNote  effective (i.e., whose direction is  Figure 3, every charger cone is characterized by an apex X, a normal vector  extreme  point  within  the  charger  cone,  where  the  distance  between  E  and  X  is   D  and  the  angle  generatrix length) D, and an effective charging angle (i.e., the opening angle) A. When a sensor node parallel to the symmetrical axis of the cone, an effective charging distance (i.e., the cone generatrix  between the normal vector    and the vector going from X to E is A/2. 

is withinlength) 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;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 otherwise,  the within sensor  node  cannot  be  charged  effectively.  Note  that  the  point  E  is  X regarded  as  an  as an extreme point the charger cone, where the distance between E and is D and the angle * charger  cone,  where  the  distance  between  E  and  X  is  D  and  the  angle  extreme  point  within  the  between the normal vector    and the vector going from X to E is A/2. 

between the normal vector N and the vector going from X to E is A/2. A

A

Figure 3. A charger cone and its parameters. 

Figure 3. A charger cone and its parameters. 

Figure 3. A charger cone and its parameters.

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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 Pr is inversely proportional to D2 , that is, Pr = αD−2 , where α is a constant and D is the distance between the charger and the harvester. This paper utilizes a more general empirical model Pr = αD −β , where β > 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 α and β values for the model to fit experimental data accurately. Energies 2016, 9, 696  With all the assumptions mentioned above, the WCDO problem deals with how 6 of 21  to deploy the minimum number of wireless chargers to cover all sensor nodes and fulfil their energy demands. The famous Friis transmission equation [23], shown in Equation (1), is used to model the energy  To be morereceived  precise, a set ofBy  sensor nodeswe  distributed inthe  a cuboid a set  is  ofinversely  grid points, the by given the  harvester.  the  equation,  can  see  that  received  and energy  2, that is,  WCDO problem addresses how to deploy as few as possible chargers at grid points to charge sensor proportional to D α , where α is a constant and D is the distance between the charger  and  the  harvester.  This  paper  utilizes  more  general that empirical  model  α  ,  where  β  > of 0,  to  nodes to make them sustainable under the a assumption the effective charging space chargers is estimate  the  energy  received  by  the  harvester.  Later  in  Section  5,  we  will  show  how  to  apply  the  a cone. A special case of the WCDO problem in which every charger’s charging space is a half sphere power  regression  analysis  to  derive  appropriate  α  and  β  values  for  the  model  to  fit  experimental    (instead of data accurately.  a cone) and every sensor node needs to be covered by only one charger can be solved by the reduction toWith all the assumptions mentioned above, the WCDO problem deals with how to deploy the  the NP-hard set covering (SC) problem [24]. It is believed the WCDO problem is also minimum number of wireless chargers to cover all sensor nodes and fulfil their energy demands. To  NP-hard. Nevertheless, the NP-hardness of the WCDO problem has not yet proven. 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  4. Proposed Algorithms 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  Two algorithms, namely the GCC algorithm and the ACC algorithm, are proposed to solve the (instead of a cone) and every sensor node needs to be covered by only one charger can be solved by  WCDO problem efficiently. The algorithms have different time complexity and achieve different the reduction to the NP‐hard set covering (SC) problem [24]. It is believed the WCDO problem is also  approximate solutions to the WCDO problem. The details of the two algorithms are described below. NP‐hard. Nevertheless, the NP‐hardness of the WCDO problem has not yet proven. 

For a WSRN with n sensor nodes and with chargers being deployed on some of m grid points, 4. Proposed Algorithms  the set of sensor nodes is denoted by NS j k= {s1 , s2 ,j . . .k, sn } and the set of grid points is denoted by

GS = { g1 , g2 , Two algorithms, namely the GCC algorithm and the ACC algorithm, are proposed to solve the  . . . , gm }, where m = ( SL + 1) × ( W S + 1), S is the separation of grid points, and WCDO  problem  efficiently.  The  algorithms  have  different  time  complexity  and  achieve  different  L, W and Happroximate solutions to the WCDO problem. The details of the two algorithms are described below.  are the length, width, and height of the sensor node deployment cuboid, respectively. Note that we assume each sensor node si in NS = {s1 , s2 , . . . , sn } has energy requirement ei in For a WSRN with n sensor nodes and with chargers being deployed on some of m grid points,  the  set  of  sensor  nodes  is  denoted  by  of energy , , …requirements: ,   and  the  set coverage of  grid  points  is  denoted  by  ES = {e1 , e2 , . . . , en }. There are two forms requirement and power , , … , , where  1 1 , S is the separation of grid points, and L, W and  requirement. For the coverage requirement, ei is a positive integer 1, 2, . . . , indicating sensor node si needs theH are the length, width, and height of the sensor node deployment cuboid, respectively. Note that  coverage of ei chargers to fulfill its energy demand. For the power requirement, ei is of we  assume  each  sensor  node    in  , ,…,   has  energy  requirement    in  the unit mW,, indicating si needs to capture ei mW requirements:  of power to coverage  make itself sustainable. The coverage .  There  are  two  forms  of  energy  requirement  and  power  ,…, requirement is actually about extreme cases. Specifically, a node’s coverage requirement can be requirement. For the coverage requirement,    is a positive integer 1, 2,…, indicating sensor node    derived needs the coverage of    chargers to fulfill its energy demand. For the power requirement,    is of  by calculating the ratio of the node’s power requirement over the power harvested by the extreme unit  mW,  point E, as the  indicated in indicating  Figure 3.   needs  to  capture    mW  of  power  to  make  itself  sustainable.  The  coverage requirement is actually about extreme cases. Specifically, a node’s coverage requirement  Both algorithms have two phases. The first one is the cone generation phase to generate candidate can be derived by calculating the ratio of the node’s power requirement over the power harvested by  cones to bethe extreme point E, as indicated in Figure 3.  selected; the second one is the cone selection phase to select some candidate cones for Both algorithms have two phases. The first one is the cone generation phase to generate candidate  deploying chargers. In the first phase, the algorithms generate cones by taking every grid point as the cones to be selected; the second one is the cone selection phase to select some candidate cones for  apex of cones. As shown in Figure 4, for every grid point, the algorithms first generate a half sphere deploying chargers. In the first phase, the algorithms generate cones by taking every grid point as  (HS) centered at the point and calculate the sensor set (SS) of sensor nodes covered by HS (i.e., within the  apex  of  cones. As  shown  in  Figure  4, for every grid  point,  the  algorithms first  generate  a  half  HS). Afterwards, the first algorithm generates candidate cones on a node-pair-by-node-pair basis (i.e., sphere (HS) centered at the point and calculate the sensor set (SS) of sensor nodes covered by HS (i.e.,  within HS). Afterwards, the first algorithm generates candidate cones on a node‐pair‐by‐node‐pair  for every pair of nodes in SS), while the second algorithm, on a node-by-node basis (i.e., for every basis (i.e., for every pair of nodes in SS), while the second algorithm, on a node‐by‐node basis (i.e.,  node in SS). For generating better candidate cones, the algorithms greedily adjust cones so that every for every node in SS). For generating better candidate cones, the algorithms greedily adjust cones so  single conethat every single cone can cover as many as sensor nodes or can charge sensor nodes more efficiently.  can cover as many as sensor nodes or can charge sensor nodes more efficiently.

Figure 4. The illustration of the lower half sphere HS centered at    with radius D. 

Figure 4. The illustration of the lower half sphere HS centered at g with radius D.

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Energies 2016, 9, 696  In the second phase, both algorithms first unmark all sensor nodes. Note that a sensor 7 of 21  node is unmarked if its energy requirement is not satisfied; otherwise, it will be marked. The algorithms then In the second phase, both algorithms first unmark all sensor nodes. Note that a sensor node is  run iteration by iteration, and greedily select the qualified cone covering most unmarked sensor nodes, unmarked if its energy requirement is not satisfied; otherwise, it will be marked. The algorithms then  where a cone is said to be qualified if it takes a grid point g as its apex and g is totally taken by no more run  iteration  by  iteration,  and  greedily  select  the  qualified  cone  covering  most  unmarked  sensor  than Q selected cones as their apex. A selected cone corresponds to a deployed charger. For every nodes, where a cone is said to be qualified if it takes a grid point    as its apex and    is totally taken  newly selected cone, the algorithms marks every sensor node whose energy requirement is satisfied by no more than Q selected cones as their apex. A selected cone corresponds to a deployed charger.  by the cone. And the algorithms continue until all sensor nodes are marked. In this way, the WCDO For every newly selected cone, the algorithms marks every sensor node whose energy requirement  problem can be solved near-optimally. is satisfied by the cone. And the algorithms continue until all sensor nodes are marked. In this way,  * The pseudo code of the GCC algorithm is shown in Figure 5. In the code, u x stands for a vector the WCDO problem can be solved near‐optimally.  * going from grid point g to sensor node s ax , and Cone(u x ) stands for a cone which takes g as the The pseudo code of the GCC algorithm is shown in Figure 5. In the code,    stands for a vector  * apex, takes u x as the symmetrical axis (or the , and Cone( direction), and) stands for a cone which takes  has an effective charging distance D and going from grid point    to sensor node    as the   *  an apex, takes  effective charging angle A. Also note that Cone u x stands for the number of unmarked sensor   as the symmetrical axis (or the direction), and has an effective charging distance D and    * an effective charging angle A. Also note that  | * * |  stands for the number of unmarked sensor  nodes in NS covered by Cone (u x ). Moreover, ψ u x , uy stands for the angle between the two vectors nodes in NS covered by Cone ( ). Moreover,  ψ ,   stands for the angle between the two vectors  * * u x and uy . .    and 

  Figure 5. The greedy cone covering (GCC) algorithm.  Figure 5. The greedy cone covering (GCC) algorithm.

In the GCC algorithm, for every grid point  , a lower half sphere HS is obtained with    as the  In the GCC algorithm, for every grid point g, a lower half sphere HS is obtained with g as the center and D as the radius. If HS covers only one sensor node  , then the algorithm generates one  center and D as the radius. If HS covers only one node s a1 , then the algorithm generates one candidate cone taking the vector going from    to sensor   as the direction. However, if HS covers k, k >  candidate cone taking the vector going from g to s as the direction. However, covers k, k >  1, a1 1, sensor nodes  , , … , , then the algorithm generates k unit vectors  , , …if, HS   going from  * * * sensor a2 , . . . , s ak , then the algorithm generates k unit vectors u1 , u2 , . . . , uk going from g to , , …s a, 1 , s. Afterwards, the GCC algorithm compares the opening angle A with the angle B (i.e.,  to  nodes s a1 ψ , s a2 , . . . ), between any two vectors  s ak . Afterwards, the GCC algorithm compares the opening angle A with the angle  and  B (i.e., and   associated with the two distinct sensor nodes  . There are three cases for the comparison: B = A, B > A, and B  A, and B < A. The GCC algorithm performs the aforementioned angle comparison by projecting every vector  The GCC algorithm performs the aforementioned angle comparison by projecting every vector  The GCC algorithm performs the aforementioned angle comparison. As shown in Figure 6, the  by projecting every vector and every cone onto the surface of a unit sphere centered at grid point  and every cone onto the surface of a unit sphere centered at grid point  and every cone onto the surface of a unit sphere centered at grid point g.. As shown in Figure 6, the  As shown in Figure 6, the projection of a vector is a point, while the projection of a cone is a circle of diameter d. Referring to  projection of a vector is a point, while the projection of a cone is a circle of diameter d. Referring to  projection of a vector is a point, while the projection and of a cone is a circle of Referring to Figure 7a–c, for any pair of distinct sensor nodes  , where, 1 , diameter , let od.x and o y be the  Figure 7a–c, for any pair of distinct sensor nodes  and , where, 1 , , let o x  and o y  be the  Figure 7a–c, for any pair  of distinct sensor nodes s ax and s ay , where, 1 ≤ x, y ≤ , and let d k, let ox and oy be and    on the unit sphere O centered at grid point  xy be the  projection points of  * *   and    on the unit sphere O centered at grid point  , and let d xy be the  projection points of  gs gs the projection points of and on the unit sphere O centered at grid point g, and let d be the ax ay xy xy and  Euclidean distance between the two points. There are three cases of the relationship between d Euclidean distance between the two points. There are three cases of the relationship between d xy and  Euclidean distance between the two points. There are three cases of the relationship between dxy and d, which correspond to the three relationship cases of A and B. The three cases are elaborated below.  d, which correspond to the three relationship cases of A and B. The three cases are elaborated below.  d, which correspond to the three relationship cases of A and B. The three cases are elaborated below. Case (a) dxy = d. This case corresponds to the case of B = A. For such a case, one candidate cone is  Case (a) d Case (a) dxy = d. This case corresponds to the case of B = A. For such a case, one candidate cone is xy = d. This case corresponds to the case of B = A. For such a case, one candidate cone is  generated. Note that o x and o y should be on the diameter of the projection circle of the cone. Case (b)  generated. Note that o generated. Note that oxx and o and oyy should be on the diameter of the projection circle of the cone. Case (b)  should be on the diameter of the projection circle of the cone. Case (b)  > d. This case corresponds to the case of B > A. For such a case, two candidate cones taking  * dand  > d. This case corresponds to the case of B > A. For such a case, two candidate cones taking gs x and    as directions are generated. Case (c)   A. For such a case, two candidate cones taking  * and    as directions are generated. Case (c)   1, sensor nodes  *

*

*

algorithm u1 , ,u2 ,,.…. ., , uk going  going from  from g to The ACC  ACC algorithm   to s a1 , ,s a2 , ,.…. ., , s ak. . The  algorithm  algorithm generates generates kk unit unit vectors vectors  constructs cones onon  a node by node basis. basis.  In practice, the ACCthe  algorithm initially constructs constructs candidate candidate  cones  a  node  by  node  In  practice,  ACC  algorithm  initially  * aconstructs a candidate cone  candidate cone Cone(u x ) toCone( cover )sensor node s for 1 ≤ x ≤ k. Besides covering node s , the ACC,   to cover sensor node    for 1 ≤ x ≤ k. Besides covering node  ax ax * * u x )Cone( algorithm adaptively and greedily adjusts u x so that Cone( can cover as many as possible other   so that  ) can cover as many as possible  the ACC algorithm adaptively and greedily adjusts  * * * sensor nodes. This is achieved by adjusting the direction of u by composing u x and x y for every other sensor nodes. This is achieved by adjusting the direction of    by composing   uand    for  *

vector uy , where, where 1 ≤ y ≤ k and x ≠ y (Figure 9). The direction adjustment (Lines 12–14) proceeds  1 ≤ y ≤ k and x 6= y (Figure 9). The direction adjustment (Lines 12–14) proceeds every vector  on a node-by-node basis. Certainly, the direction adjustment should guarantee that the number of on a node‐by‐node basis. Certainly, the direction adjustment should guarantee that the number of  * * sensor byby  Cone( u x ) is) increasing andand  that that  the sensor node snode  by Cone(uby  sensor nodes nodes covered covered  Cone( is  increasing  the  sensor  is  still  covered  a x is still  covered x) * whatever the adjusted u x is.   is.  Cone( ) whatever the adjusted 

  Figure 8. The adaptive cone covering (ACC) algorithm.  Figure 8. The adaptive cone covering (ACC) algorithm.

 

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Figure 8. The adaptive cone covering (ACC) algorithm. 

  Figure 9. Illustration of candidate cone direction adjustment by composing two vectors.  Figure 9. Illustration of candidate cone direction adjustment by composing two vectors.

In the GCC and the ACC algorithms, US stands for the set of unmarked sensor nodes with unsatisfied energy requirement. For most cases, the two algorithm stop and return CS* and US, with US is empty. For such cases, the cones in CS* can be used to deploy chargers to satisfy the energy requirements of all sensor nodes. On the contrary, when the algorithms terminate and return a non-empty set of US, they should re-run by setting NS = US, and the re-running should proceed until US is empty. The re-running is to satisfy the pending energy requirements of unmarked nodes. When US is empty, the requirements of all sensor nodes are satisfied and all the cones ever reported in CS* by running and re-running the algorithms are the final selected cones for deploying chargers. However, the re-running algorithms will report an error if it finds that no available grid point exists, where an available grid point is a grid point that no more than Q selected cones take as their apex. For such an error condition, we may need to decrease S, the separation of grid points, to increase the total number of grid points for more possible locations for deploying chargers to satisfy sensor nodes’ energy requirements. 5. Analysis 5.1. Algorithm Time Complexity Analysis In this subsection, we analyze the time complexity of the two proposed algorithms. In the GCC algorithm, for every grid point g, a half sphere HS centered at g of radius D is formed (Lines 3–4). If S covers k sensor nodes, then candidate cones are generated on a node-pair-by-node-pair basis (Lines 9–17). There are up to 4C2k candidate cones that can be generated, where k ≤ n. Since there are m grid points, the number of candidate cones generated is then 4mC2k ≤ mn2 = O(mn2 ) and the time complexity to generate the candidate cones is also O(mn2 ). On each iteration in the repeat-until loop (Lines 18–24), the GCC algorithm first selects the cone c from CS that covers most unmarked nodes in NS (Line 19). Since there are at most mn2 cones in CS, and the algorithm should check if a cone can cover all nodes, the selection can be done in O(mn3 ) time complexity. The GCC algorithm then checks if every unmarked node covered by cone c can be marked after updating its energy requirement (Lines 21–23). The checking takes O(n) time complexity. A cone can be selected only once in the repeat-until loop, so there are at most O(mn2 ) iterations in the loop. In summary, there are O(mn2 ) iterations in the repeat-until loop, and each iteration’s time complexity is O(mn3 ) (Line 19). Thus, the time complexity of the loop is O(m2 n5 ). Since the repeat-until loop has the highest time complexity in the algorithm, we thus have that the time complexity of the GCC algorithm is O (m2 n5 ). The ACC algorithm forms a half sphere HS centered at g of radius D for every grid point g (Lines 3–4). If HS covers k sensor nodes, then k candidate cones will be generated, k ≤ n (Line 8). Every candidate cone is tested for possible direction adjustment for k times (Lines 9–15), so there are k2 = O(n2 ) adjustment tests. Since there are m grid points, the ACC algorithm forms m half spheres. Let the half spheres cover k1 , k2 , . . . , k m sensor nodes, respectively. Thus, the total number of candidate cones is ∑im=1 k i ≤ mn = O(mn) and the total number of adjustment tests is of the time complexity O(mn3 ). On each iteration in the repeat-until loop (Lines 16–22), the ACC algorithm first selects the cone c from CS that covers the most unmarked nodes in NS (Line 17). Since there are at most mn cones

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in CS, and the algorithm should check if a cone can cover all nodes, the selection (Line 17) can be done in O(mn2 ) time complexity. The ACC algorithm then checks if every unmarked node covered by cone c can be marked after updating its energy requirement (Lines 19–21). The checking takes O(n) time complexity. A cone can be selected only once in the repeat-until loop, so there are at most O(mn) iterations in the loop. Thus, the time complexity of the repeat-until loop is O(m2 n3 ). Since the repeat-until loop has the highest time complexity in the ACC algorithm, the time complexity of the ACC algorithm is O(m2 n3 ). 5.2. Grid Point Separation Analysis Energies 2016, 9, 696  In this subsection,

11 of 21  we analyze the relationship of S, D and H, which are the grid point separation, the effective charging distance and high of the deployment plane, respectively. For an arbitrary sensor charging distance). The intersection of O and the charger deployment plan (i.e., the grid plane) P is a  node located at position v = (X, Y, Z), we can construct a sphere O with radius D (i.e., the charging circle C with radius R distance). The intersection of O and the  centered at u (Figure 10). For any point u’ within the area of  charger deployment plan (i.e., the grid plane) P is a circle q . So, if there is at least one grid point    within C, the sensor node located at v can  C, we have  Energies 2016, 9, x  ′ 0 2 11 of 21  C with radius R = D2 − ( H − Z ) centered at u (Figure 10). For any point u within the area of C, be covered by a charger deployed at  . When Z = 0, v is on the floor plane and the circle C has the  0 we have vu charging distance). The intersection of O and the charger deployment plan (i.e., the grid plane) P is a  ≤ D. So, if there . As shown in Figure 11, if  is at least one grid point g within C,, then there may be no grid point  the sensor node located at v can smallest radius  √ √2 circle C with radius R   centered at u (Figure 10). For any point u’ within the area of  be covered by a charger at g. When Z = 0, v is on the floor plane and the circle C has the √ ′ deployed within the area of circle C. On the contrary, if  2 √ , there is at least one grid point  √2 =  within C, the sensor node located at v can  C, we have  . So, if there is at least one grid point  2 − H 2 . As shown in Figure 11, if S > smallest radius r = D 2 r, then there may be no grid point be covered by a charger deployed at  . When Z = 0, v is on the floor plane and the circle C has the  p √   ensures that every sensor node  within the area of circle C. In summary, In summary, S ≤ 2 2 within the area of circle C. On if S ≤ 2 r= p2(√2 D , then there may be no grid point  − H 2 ), there is at least one grid point . As shown in Figure 11, if  smallest radius  √ the contrary, is covered by at least one charger deployed at one of the grid points.  within the area of circle C. On the contrary, if  = 2 2( D 2 − , there is at least one grid point  within the area of circle C. In summary, In summary,√2 S≤ H 2 ) ensures that every sensor node   ensures that every sensor node  within the area of circle C. In summary, In summary, S ≤ 2 is covered by at least one charger deployed at one of the grid points. is covered by at least one charger deployed at one of the grid points. 

 

 

Figure 10. Illustration of the intersection of the grid plane with an upper half sphere of radius D (the  Figure 10. Illustration of the intersection of the grid plane with an upper half sphere of radius D (the charging distance) centered at a sensor node.  Figure 10. Illustration of the intersection of the grid plane with an upper half sphere of radius D (the 

charging distance) centered at a sensor node. charging distance) centered at a sensor node. 

  Figure 11. Illustration of no grid point within the area of circle C with radius r. 

Below we show that if  , then there is at least two grid points within the area  √ of circle C centered at u with radius r.    implies  √2 , which in turns implies there is at    least one grid point  within C by the fact just mentioned in the last paragraph. Let C’ be the circle  centered at  with radius r and let    and b be the intersection of C and C’ (Figure 12). Since C and  Figure 11. Illustration of no grid point within the area of circle C with radius r.  Figure 11. Illustration of no grid point ∠within the area of circle C with radius r. C’ have the same radius, we have that the angel    associated with the minor arc    should be  larger than 120°; otherwise,  will not be within C. For neighboring grid points  ,  ’ and ′′ that  Below we show that if  , then there is at least two grid points within the area  √ √ ∠ ′ ′′  is 90°. We have that either  are apart from each other with separation S, their associated angle  Below we′  or  show that if S ≤ r = D2 − H  2implies  , then there is at least two grid points within the area of circle C centered at u with radius r.  √ . Therefore, there are at least two grid  √2 , which in turns implies there is at  ′′  is within the sector associated with the minor arc  of circle C centered at u within C by the fact just mentioned in the last paragraph. Let C’ be the circle  with radius r. S ≤ r implies S ≤ 2 r, which in turns implies there is at least points within the area of C. That is to say, there are at least two chargers covering a sensor node. In  least one grid point  0 summary,  S  ≤ C by the fact   ensures mentioned that  every  sensor  node  is  paragraph. covered  by  at  Let least C two  one grid point g within in the last bechargers  the circle centered centered at  deployed at two of the grid points.  with radius r and let  just   and b be the intersection of C and C’ (Figure 12). Since C and  0 0

at g with radius r and let a and b be the intersection and C (Figure 12). Since C and C  should be  have the C’ have the same radius, we have that the angel  ∠ of  Cassociated with the minor arc  ◦ a b same radius, we have that the angel ∠agb associated with the minor arc ab should be, larger larger than 120°; otherwise,  will not be within C. For neighboring grid points  ’ andthan′′ 120 that ; 0 00 g' otherwise, g will not be within C. For neighboring that are apart from each r grid points g, g and g are apart from each other with separation S, their associated angle  ∠ ′ ′′  is 90°. We have that either  Cʹ 0Sgg00 is 90◦ . We other with separation S, their associated angle ∠ g have that either g0 or g00 is within the ′  or  ′′  is within the sector associated with the minor arc  . Therefore, there are at least two grid  g b sector associated with the minor arc ab.C Therefore, there are at least two grid points within the area of u points within the area of C. That is to say, there are at least two chargers covering a sensor node. In  S

r

summary,  S  ≤    ensures  that  every  g''sensor  node  is  covered  by  at  least  two  chargers  deployed at two of the grid points.  b   Figure 12. Illustration of two circles both of the radius r intersecting at two points a and b.  a

least one grid point  within C by the fact just mentioned in the last paragraph. Let C’ be the circle  centered at  with radius r and let    and b be the intersection of C and C’ (Figure 12). Since C and  C’ have the same radius, we have that the angel  ∠   associated with the minor arc    should be  larger than 120°; otherwise,  will not be within C. For neighboring grid points  ,  ’ and ′′ that  are apart from each other with separation S, their associated angle  ∠ ′ ′′  is 90°. We have that either  Energies 2016, 9, 696 12 of 22 ′  or  ′′  is within the sector associated with the minor arc  . Therefore, there are at least two grid  points within the area of C. That is to say, there are at least two chargers covering a sensor node. In  p 2 − H2 ) C. That is toS say, two chargers covering sensoris node. In summary, S ≤two  ( Dchargers  summary,  ≤  there are at  least ensures  that  every  sensor a node  covered  by  at  least  ensures that every sensor node is covered by at least two chargers deployed at two of the grid points. deployed at two of the grid points.  a

g'

r Cʹ

S

g C

u S

r

g'' b

 

Figure 12. Illustration of two circles both of the radius r intersecting at two points a and b.  Figure 12. Illustration of two circles both of the radius r intersecting at two points a and b.

For even smaller separation, we can use the “dies per wafer” equation in [25] to estimate √ 2 approximately the number of grid points within circle C with radius r = D − H 2 . The equation is shown below: π × (Wafer Ridus)2 2π × (Wafer Ridus) Dies per wafer = − p (4) Die area 2 × (Die area) If we take the grid square as the die area and the wafer radius as r, the radius of circle C, then we have the following equation for NGS , the number of grid squares in circle C: NGS =

2π × r π × r2 − √ S2 S 2

(5)

One grid square has four grid points. However, adjacent grid square share grid points. When we count the number of grid points, a grid square is associated with two non-redundant grid points. Therefore, we have the following equation for NGP , the number of grid points in circle C:  2π × r π × r2 − √ NGP = 2 × NGS = 2 × (6) S2 S 2 √ In summary, for grid separation S much smaller than r = D2 − H 2 , each sensor node is covered by  

at least 2 ×

π×r 2 S2

−

2π√×r S 2

chargers approximately.

6. Experiments and Simulations 6.1. Experiments In this subsection, we show results of extensive experiments in indoor environments for modeling the RF charging efficiency (i.e., the received energy or received power). The devices used in the experiments are Powercast TX91501-3W-ID wireless charger [21] and P2110-EVAL-02 power harvester [22], as shown in Figure 13. The former is a wireless charger with a patch directional antenna with 8 dBi gain and 60◦ beamwidth in both horizontal and vertical directions. It is a 915 MHz transmitter with 3 Watts equivalent isotropically radiated power (EIRP). The latter is a power harvester with a dipole omni-directional antenna with 1 dBi gain. It can harvest energy from RF waves, provide intermittent/pulsed power output up to 4.21 V, and connect to rechargeable batteries.

in the experiments are Powercast TX91501‐3W‐ID wireless charger [21] and P2110‐EVAL‐02 power  in the experiments are Powercast TX91501-3W-ID wireless charger6.1. [21]Experiments and P2110-EVAL-02 power harvester  [22],  as  shown  in  Figure  13.  The  former  is  a  wireless  charger  with  a  patch  directional  harvester [22], as shown in Figure 13. The former is a wireless charger with a patch directional In this subsection, we show results of extensive exp antenna with 8 dBi gain and 60° beamwidth in both horizontal and vertical directions. It is a 915 MHz  antenna with 8 dBi gain and 60° beamwidth in both horizontal and vertical directions. It is a 915efficiency MHz modeling the RF charging (i.e., the received energy with equivalent 3  Watts  equivalent  isotropically  radiated  power  (EIRP).  The  latter  is  a  power  transmittertransmitter  with 3 Watts isotropically radiated power (EIRP). The latter is a power in the experiments are Powercast TX91501-3W-ID wireless c harvester  with  Energies 9, 696a  dipole  omni‐directional  antenna  with  1  dBi  gain.  It  can  harvest  energy  from  13 ofRF  22 harvester with a 2016, dipole omni-directional antenna with 1 dBi gain.harvester It can harvest RF [22], asenergy shownfrom in Figure 13. The former is a wir waves, provide intermittent/pulsed power output up to 4.21 V, and connect to rechargeable batteries.  waves, provide intermittent/pulsed power output up to 4.21 V, and antenna connect to rechargeable with 8 dBi gain batteries. and 60° beamwidth in both horizonta

transmitter with 3 Watts equivalent isotropically radiated harvester with a dipole omni-directional antenna with 1 dB waves, provide intermittent/pulsed power output up to 4.21 V

  (a)

(a) 

(b)

(b)

Figure 13. (a) Powercast TX91501‐3W‐ID wireless charger; and (b) P2110‐EVAL‐02 power harvester.  13. (a)TX91501-3W-ID Powercast TX91501-3W-ID wireless and (b) P2110-EVAL-02 power harvester. Figure 13. (a)Figure Powercast wireless charger; andcharger; (b) P2110-EVAL-02 power harvester.

As  shown  in  Figure  14,  a  wireless  charger  and  a  power are harvester  are  set  up.  The  distance  D  (a)Ddistance As shown As in Figure a wireless and a power set up.are The shown 14, in Figure 14, acharger wireless charger andharvester a power harvester setdistance up. The D between the two devices is set to be 0.5 m, 1.0 m, …, or 4.5 m for experiments. As shown in Figure 15,  between the two devices is set to be 0.5 m, 1.0 m, …, or 4.5 m for experiments. As shown in Figure 15, between the two devices is set to be 0.5 m, 1.0 m, . . . , or 4.5 m for experiments. As shown in Figure 15, 13. angles,  (a) Powercast TX91501-3W-ID wireless charger; and the  orientation  of  the  charger’s  patch  antenna  is  specified Figure by  two  the  horizontal  angle  the orientation of the charger’s patch antenna is specified by two angles, thethehorizontal the orientation of the charger’s patch antenna is specified by two angles, horizontalangle angle (azimuth) (azimuth) ф and the vertical angle (elevation) θ. Either angle is set to be 0°, 15°, …, or 90°. Figure 16  ◦ , 15 ◦ , . …, (azimuth) ф and and the the vertical vertical angle angle (elevation) (elevation)θ. θ.Either Eitherangle angleisisset settotobe be0As 0°, 15°, or 90°. . . ,in orFigure 90◦ . Figure Figure 16 illustrates shown 14, a16 wireless charger and a powe illustrates the distance and the angle settings. In Figure 16, the charger is assumed to be at the origin  illustrates the distance and the angle settings. In Figure 16, the charger is assumed to be at the origin the distance and the angle settings. In Figure 16, the charger is assumed to be at is the andm, the between the two devices setorigin to be 0.5 1.0 m, …, or 4.5 m f and  the  harvester  is  assume  to  be  on  the  x‐axis  and  be  D  apart  from  the  charger.  Moreover,  the  and the harvester the the harvesteris isassume assumetotobebeon onthe thex-axis x-axisand andbe beDD apart apart from fromorientation the charger. charger. Moreover, the patch charger’s the of Moreover, the charger’s antenna is specified charger’s  antenna  direction  is  indicated  by  vector    with  the  horizontal  angle  ф  and  the  vertical  ⃑ with charger’s antenna direction is indicated by vector* 𝑁 the horizontal angleand ф and the vertical (azimuth) the vertical vertical angleθ(elevation) antenna direction is indicated by vector N with the horizontal angle ф and the angle relative θ. Either angle angle θ relative to the x‐axis.  angle θ relative to the x-axis. illustrates the distance and the angle settings. In Figure 16, th to theEnergies 2016, 9, 696  x-axis. 13 of 21  and the harvester is assume to be on the x-axis and be D a ⃑ with t charger’s antenna direction is indicated by vector 𝑁 Energies 2016, 9, 696  13 of 21  angle θ relative to the x-axis.

  Figure 14. The experimental setting of a wireless charger and a power harvester.    

Figure 14. The experimental setting of a wireless charger and a power harvester. Figure 14. The experimental setting of a wireless charger and a power harvester. 

(a)  (a) 

(b)(b)

 

 

  (c) 

(d)

 

Figure 15. Experimental settings for (a,b) different horizontal angles (azimuths); and (c,d) different  (c)  (d) 15.vertical angles (elevations).  Experimental settings for (a,b) different horizontal angles (azimuths); and (c,d)

Figure different Figure 15. Experimental settings for (a,b) different horizontal angles (azimuths); and (c,d) different  vertical angles (elevations). vertical angles (elevations). 

antenna with 8 dBi gain and 60° beamwidth in both h

harvester transmitter with a dipolewith omni-directional antenna with 1 dBi gain.r 3 Watts equivalent isotropically waves, provide intermittent/pulsed power output up to 4.21 V, and c harvester with a dipole omni-directional antenna w   waves, provide intermittent/pulsed power output up (d)

(c) 

Figure 15. Experimental settings for (a,b) different horizontal angles (azimuths); and (c,d) different  Energies 2016, 9, 696 14 of 22 vertical angles (elevations). 

(a)

Figure 13. (a) Powercast TX91501-3W-ID wireless charger; and (b) P21

(a)

As shown in Figure 14, a wireless charger and a power harve between the two devices is set to be 0.5 m, 1.0 m, …, or 4.5 m for expe   Figure 13. (a) Powercast TX91501-3W-ID wireless cha the orientation of the charger’s patch antenna is specified by tw Figure 16. The illustration of distance D, horizontal angle ф, and vertical angle θ.  (azimuth) vertical θ. Eithercharger angle is set Asthe shown inangle Figure 14, a wireless andto Figure 16. The illustration of distance D, horizontal angle ф, and and vertical angle θ. (elevation) illustratesbetween the distance angle settings. theand twothe devices is set to In beFigure 0.5 m,16, 1.0the m,charg …, o By a multimeter, we measure the voltage (V) and current (I) of the output of the harvester 10  and the harvester is assume to be on the x-axis and be D apart fr the charger’s patch antenna is s By a multimeter, we measure the voltage (V) and current (I) ofthe the orientation output of theofharvester 10 times ⃑ with the hori charger’s antenna direction is indicated by vector 𝑁 times for every distance D = 0.5 m, 1.0 m, …, and 4.5 m, every horizontal angle ф = 0°, 15°, …, and  ◦ ◦ ◦ (azimuth) ф and the vertical angle (elevation) θ. Eith for every distance D = 0.5 m, 1.0 m, . . . , and 4.5 m, every horizontal angle = 0 , 15 , . . . , and 90 , angle θ relative to the x-axis. 90°, and every vertical angle θ = 0°, 15°, …, and 90°. We average the measured data and then apply 

distance and the angle and every vertical angle θ = 0◦ , 15◦ , . . . , and 90◦ . We average theillustrates measuredthe data and then apply the settings. In Figu the average data to derive the received power (i.e., received energy). Table 1 (respectively, Table 2)  and the harvester is assume to be on the x-axis an average data to derive the received power (i.e., received energy). Table 1 (respectively, Table 2) shows shows the experimental results of the received power in the unit of mW for different distances and  ⃑ charger’s antenna direction is indicated by vector 𝑁 the experimental results of the received power in the unit of mW for different distances and horizontal horizontal (respectively, vertical) angles with the vertical (respectively, horizontal) angle is 0. Note  angleangle θ relative to the x-axis. (respectively, vertical) angles with the vertical (respectively, horizontal) is 0. Note that 0.01 V and that 0.01 V and 0.01 mA are the smallest measurable voltage and current, respectively. Therefore, the  0.01 mA are the smallest measurable voltage and current, respectively. Therefore, the voltage values voltage values smaller than 0.01 V or the current value smaller than 0.01 mA are not measurable and  smaller than 0.01 V or the current value smaller than 0.01 mA are not measurable and such cases lead such cases lead to “not available” (i.e., “‐”) symbols shown in the Tables.  to “not available” (i.e., “-”) symbols shown in the Tables. Energies 2016, 9, x 

14 of 21 

Table 1. Received power (mW) for different distances and horizontal angles.

Table 1. Received power (mW) for different distances and horizontal angles. 

D\ф  0.5 m  0.5 m 1.0 m  1.0 m 1.5 m  1.5 m 2.0 m  2.0 m 2.5 m  2.5 m 3.0 m  3.0 m 3.5 m  3.5 m 4.0 m  4.0 m 4.5 m 

4.5 m

0◦ 0° 

15°  30◦ 17.63  16.38 17.63 16.38 13.89 6.3  5.72  6.3 5.72 3.96 1.93  1.26  1.93 1.26 1 1.39  1.04  1.39 1.04 0.8 0.84  0.74  0.84 0.74 0.62 0.47  0.31  0.47 0.31 0.18 0.28  0.15  0.28 0.15 0.21  0.11  0.21 0.11 0.14  ‐ 

0.14

15◦

-

-

30°  45◦ 13.89  9.89 3.96  2.29 1  0.72 0.8  0.57 0.62  0.34 0.18  ‐  ‐  ‐ 

-

◦ 6045° 

75◦60° 

90◦ 75° 

9.89 3.96 2.29  1.14 0.72  0.42 0.57  0.25 0.34  ‐  ‐  ‐  ‐ 

3.96 2.15 1.14  0.58 0.42  0.25  ‐  ‐  ‐  ‐  ‐ 

2.15 0.37 0.58  ‐  ‐  ‐  ‐  ‐  ‐  ‐ 

-

-

-

90°  0.37 ‐  ‐  ‐  ‐  ‐  ‐  ‐  ‐ 

Table 2. Received power (mW) for different distances and vertical angles. 

Table 2. Received power (mW) for different distances and vertical angles.

D\θ  0.5 m  D\θ 1.0 m  1.5 m  0.5 m 2.0 m  1.0 m 2.5 m  1.5 m 3.0 m  2.0 m 3.5 m  2.5 m 4.0 m  3.0 m 4.5 m  3.5 m

0°  17.63  6.3  17.631.93  6.3 1.39  1.930.84  1.390.47  0.840.28  0.470.21  0.280.14 

0◦

15◦

15°  13.52 ◦ 5.23 30 1.52  9.07 0.95  2.65 0.61  1.02 0.42  0.87 0.21  0.76 0.140.36 0.12  0.19

30°  9.07 ◦ 45 2.65  1.02  5.55 0.87  1.26 0.76  0.56 0.36  0.34 0.19  0.28 ‐  0.15 ‐ -

45°  5.55 ◦ 60 1.26  0.56  2.78 0.34  0.79 0.28  0.29 0.15  0.18 -‐  -‐ -‐ 

60°  2.78 ◦ 750.79  0.29  0.92 0.18  0.41 - ‐  - ‐  - ‐  - ‐ - ‐ 

75°  0.92 0.41  0.26 ‐  - ‐  - ‐  - ‐  - ‐  - ‐ - ‐ 

90◦

90°  0.26 ‐  ‐  ‐  ‐  ‐  ‐  ‐ ‐ 

13.52 5.23 1.52 0.95 0.61 0.42 0.21 4.0 m 0.21 0.14 By experimental data in Table 1, Table 2 and the specification [13] stating that the charger has a  4.5 m 0.14 0.12 beamwidth  of  60°  in  both  horizontal  and  vertical  directions,  we  model  the  charging  space  of  a  wireless charger as a cone with the opening angle of 60° and the generatrix length of D. We perform  By experimental data in Table 1, Table 2 and the specification [13] stating that the charger has power regression on the data to obtain an exponential function taking D as the independent variable  ◦ in both horizontal and vertical directions, we model the charging space of a beamwidth of 60 (i.e., x) and P as the dependent variable (i.e., y) for every vertical and horizontal angle of 0°, 15° and  a wireless charger as a cone with the opening angle of 60◦ and the generatrix length of D. We perform 30°. For example, Figure 17 shows the functions derived by performing power regression analysis on  experimental data of received energy (or power) for horizontal angel ф = 0°, 15° and 30°, and vertical  power regression on the data to obtain an exponential function taking D as the independent variable angle  θ  =  0°.  Moreover,  Figure  18  shows  the every functions  derived  performing angle power  (i.e., x) and P as the dependent variable (i.e., y) for vertical andby horizontal ofregression  0◦ , 15◦ analysis on experimental data of received power for horizontal angel ф =0° and vertical angle θ = 0°,  15° and 30°. We can see that the functions, namely y = 5.019x−2.217, y = 3.8881x−2.225, y = 2.6394x−1.821, y =  1.3522x−1.939,  y  =  0.7134x−2.01,  fit  well  with  the  experimental  data,  as  they  all  have  coefficient  of  determination  (i.e.,  R2) [26] values larger than  0.958, which  is very close to the maximum value  1.  They  also  comply  with  the  Friis  equation,  since  the  function  exponents  are  approximately‐2.  Therefore, we may well estimate the power received by the harvester with the functions. In practice, 

Commented

4.5 m 

0.14 

equation for NGP‐ , the number‐ of grid points in circle C: 0.12  Therefore, ‐ we have the following ‐  ‐ 

π  𝑟2 2π  𝑟 (a) NGP = 2 × NGS = 2 × ( 2 − 12 of)21 Energies 2016, 9, 696 By experimental data in Table 1, Table 2 and the specification [13] stating that the charger has a  𝑆 √2 𝑆 Figure 13. (a) Powercast TX91501-3W-ID wirele beamwidth  of  60°  in  both  horizontal  and  vertical  directions,  we  model  the  charging  space  2 −a  2 , each sensor nod (a) for grid separation S much smaller 𝐻 √𝐷of  For even smaller separation, In wesummary, can use the “dies per wafer” equation in than [25] 𝑟to=estimate 2 wireless charger as a cone with the opening angle of 60° and the generatrix length of D. We perform  π𝑟 2π  𝑟 Energies 2016,approximately 9, 696 15 of 22 2 2 the number of grid r =As − 13. 𝐻 .(a) The equation isa wireless charge √𝐷 by atpoints least 2within × ( circle −C with ) radius chargers approximately. shown in Figure 14,TX91501-3W-ID Figure Powercast wirel 𝑆√2 𝑆2 power regression on the data to obtain an exponential function taking D as the independent variable  shown below: between the two devices is set to be 0.5 m, 1.0 m (i.e., x) and P as the dependent variable (i.e., y) for every vertical and horizontal angle of 0°, 15° and  2 6. Experiments andRidus) Simulations π (Wafer 2π(Wafer Ridus) As shown of in the Figure 14, a wireless charge the orientation charger’s patch antenn and 30◦ . For example, FigureDies 17 per shows the wafer = functions derived − by performing power regression (4) 30°. For example, Figure 17 shows the functions derived by performing power regression analysis on  Die area area) √2(Die ◦ ◦ ◦ between two devices 0.5 m, 1.0 θ m (azimuth) ф= and angle analysis on experimental data of received energy (or power) for horizontal angelthe 0 ,the 15 vertical and is 30set , to be(elevation) experimental data of received energy (or power) for horizontal angel ф = 0°, 15° and 30°, and vertical  6.1. Experiments ◦ illustrates the distance and the angle settings. In the orientation of the charger’s patch antenn If we take the grid square as the die area and the wafer radius as r, the radius of circle C, then and vertical angle θ = 0 . Figure  Moreover, Figurethe  18 functions  shows thederived  functions by performing power angle  θ  =  0°.  Moreover,  18  shows  by derived performing  power  regression  In this we show and results ofharvester experiments ◦ and we have the following equation forof NGS , thesubsection, number of grid squares in circle C:extensive the is assume toinbeindoor on theenviro x-ax (azimuth) ф =0 and the vertical angle (elevation) regression analysis on experimental data received power for horizontal angel vertical analysis on experimental data of received power for horizontal angel ф =0° and vertical angle θ = 0°,  (a) (b) modeling the RF charging efficiency (i.e., the received energy or received power). The dI −2.217 −2.225 illustrates the and the angle settings. charger’s antenna direction is, indicated by vec π  𝑟 2 namely 2π  𝑟 y = 5.019x angle θ = 0◦ , 15◦ and 30◦ . We can see that the functions, , y =distance 3.8881x −2.217 −2.225, y = 2.6394x −1.821, y =  15° and 30°. We can see that the functions, namely y = 5.019x , y = 3.8881x (5) N GS = − in the experiments are Powercast TX91501-3W-ID wireless charger [21] and P2110-EVA −1.821 , y = 1.3522x −1.939(a) − 2.01 , fit well and the harvester is all assume to be on the x-a angle θ relative the x-axis. 𝑆2 𝑆√2 Figure TX91501-3W-ID wireless charger; and (b) P2110-EVAL-02 power y1.3522x = 2.6394x , yPowercast = 0.7134x with the experimental data, asto they have −1.939,  y  =  −2.01,  13. 0.7134x fit  well  with  the  experimental  as 13. they  have  of  harvester. harvester [22], as shown indata,  Figure Theall  former iscoefficient  a wireless charger with a patch 2 antenna direction is indicated by ve coefficient of determination (i.e., R four ) [26] values larger than which issquare veryinclose thepoints. maximum One grid square has grid points. However, adjacent gridcharger’s share grid When 2) [26] values larger than 0.958, which is very close to the maximum value 1.  determination (i.e., R antenna with 8 dBi gain0.958, and 60° beamwidth bothto horizontal and vertical directions. It is As shown inFriis Figure wireless charger and atwo power harvester arex-axis. set up. The distance D angle θ relative to the we count the number ofthe grid points, a14, grida with square associated with non-redundant grid points. value They also comply with equation, since the function exponents are approximately-2. transmitter 3 isWatts equivalent isotropically radiated power (EIRP). The latter They 1.also  comply  with  the  Friis  equation,  since  the  function  exponents  are  approximately‐2.  between the two devices is set to be 0.5 m, 1.0 m, …, or 4.5 m for experiments. As shown in Figure 15 Therefore, we have the following equation for N GP , the number of grid points in circle C: Therefore, we may well estimate the power received by the harvester with the functions. In practice, harvester with a dipole omni-directional antenna with 1 dBi gain. It Therefore, we may well estimate the power received by the harvester with the functions. In practice,  can harvest energ orientation of the charger’s patch antenna is 𝑟power specified byup two angles, the horizontal angle waves, intermittent/pulsed to 4.21 V, and connect to rechargeab π  𝑟2 2π we interpolate thethe values of the functions to provide estimate the received power foroutput given parameters, like we interpolate the values of the functions to estimate the received power for given parameters, like  NGP = 2 × NGS = 2 × ( 2 − ) (6) √2 (azimuth) ф and the vertical angle (elevation) θ. Either angle is set to be 0°, 15°, …, or 90°. Figure 16 𝑆 𝑆 distance D, horizontal angle , and vertical angle θ. distance D, horizontal angle ф, and vertical angle θ.  illustrates the distance and the angle settings. In Figure 16, the charger is assumed to be at the origin In summary, for grid separation S much smaller than 𝑟 = √𝐷 2 − 𝐻 2 , each sensor node is covered and the πharvester  𝑟2 2π  𝑟is assume to be on the x-axis and be D apart from the charger. Moreover, the by at least 2 × ( 2 − ) chargers approximately. 𝑆√2direction is indicated by vector 𝑁 ⃑ with the horizontal angle ф and the vertica charger’s 𝑆antenna angle θ relative to the x-axis. 6. Experiments and Simulations

6.1. Experiments

In this subsection, we show results of extensive experiments in indoor environments for modeling the RF charging efficiency (i.e., the received energy or received power). The devices used (b) in the experiments are Powercast TX91501-3W-ID wireless (a) charger [21] and P2110-EVAL-02 power harvester [22], as shown in FigureFigure 13. The is a wireless charger with charger; a patch and directional 13. former (a) Powercast TX91501-3W-ID wireless (b) P2110-EVAL-02 power ha antenna with 8 dBi gain and 60° beamwidth in both horizontal and vertical directions. It is a 915 MHz transmitter with 3 Watts equivalent radiated is a harvester power As isotropically shown in Figure 14, a power wireless(EIRP). charger andlatter a power are set up. The   The harvester with a dipole omni-directional withis1set dBi from RF between theantenna two devices to gain. be 0.5Itm,can 1.0harvest m, …, orenergy 4.5 m for experiments. As shown i Figure 17. The functions derived by performing power regression analysis on experimental data (ED)  waves, provide intermittent/pulsed power output up to 4.21 V, and connect to rechargeable batteries. the orientation of the charger’s patch antenna is specified by two angles, the horiz Figure 17. The functions derived by performing power regression analysis on experimental data (ED) of the power received (PR) for horizontal angel ф = 0°, 15° and 30°, and vertical angle θ = 0°.  ◦ ◦ ◦ ◦ (azimuth) ф and the vertical angle (elevation) θ. Either angle is set to be 0°, 15°, …, or 90 of the power received (PR) for horizontal angel = 0 , 15 and 30 , and vertical angle θ = 0 . Energies 2016, 9, 696  illustrates the distance and the angle settings. In Figure 16, the15 of 21  charger is assumed to be a and the harvester is assume to be on the x-axis and be D apart from the charger. Mo ⃑ with the horizontal angle ф and charger’s antenna direction is indicated by vector 𝑁 angle θ relative to the x-axis.

(a)

(b)

Figure 13. (a) Powercast TX91501-3W-ID wireless charger; and (b) P2110-EVAL-02 power harvester.

As shown in Figure 14, a wireless charger and a power harvester are set up. The distance D   between the two devices is set to be 0.5 m, 1.0 m, …, or 4.5 m for experiments. As shown in Figure 15, Figure  18. orientation The  functions  derived  by  performing  power  analysis  on  ED the of  the  PR  for angle of the charger’s patch antenna is regression  specified two angles, Figurethe 18. The functions derived by performing power regressionbyanalysis on ED of horizontal the PR for horizontal angel ф = 0°, and vertical angle θ = 0°, 15° and 30°.  ◦ ◦ ◦ ◦ (azimuth) ф and the vertical angle (elevation) θ. Either angle is set to be 0°, 15°, …, or 90°. Figure 16 horizontal angel = 0 , and vertical angle θ = 0 , 15 and 30 . illustrates the distance and the angle settings. In Figure 16, the charger is assumed to be at the origin 6.2. Simulations  and the harvester is assume to be on the x-axis and be D apart from the charger. Moreover, the 6.2. Simulations ⃑ with the horizontal angle ф and the vertical charger’s antenna direction is indicated by vector 𝑁 This  subsection  shows  the  simulation  results  of  the  proposed  GCC  algorithm  and  ACC  This subsection shows the x-axis. simulation results of the proposed GCC algorithm and ACC algorithm. angle θ relative to the

algorithm. The simulator is implemented in C++ language, and the simulation parameters are shown  The simulator is implemented in C++ language, and the simulation parameters are shown in Table 3. in Table 3. The sensors are distributed randomly in a 20 m × 15 m × 2.3 m cuboid. The chargers can  The sensors are distributed randomly in a 20 m × 15 m × 2.3 m cuboid. The chargers can be deployed at be deployed at grid points on the grid plane at the height of 2.3 m, and the grid point separation is  grid points on the grid plane at the height of 2.3 m, and the grid point separation is 1.8 m. The charging 1.8 m. The charging distance is 3 m, and the charging cone opening angle is 30°. It is noted that 30  distance is 3 m, and the charging cone opening angle is 30◦ . It is noted that 30 experiments were experiments were executed per simulation case for averaging the simulation results. The parameter  executed per simulation case for averaging the simulation results. The parameter settings are shown settings are shown in Table 3.  in Table 3. We  first  perform  simulations  for  three  scenarios  of  coverage  requirements:  all  sensor  nodes  require 1‐, 2‐, or 3‐coveage of chargers. Figure 19 shows the comparisons of GCC and ACC algorithms  in terms of the number of the candidate cones generated. The simulation results are all the same for  the 1‐, 2‐ and 3‐coverage requirement scenarios, so only one diagram is shown in Figure 19. We can  observe that GCC generates much more candidate cones than ACC. This is because GCC has O(mn2)  of candidate cones, while ACC has O(mn) of candidate cones, where n is the number of sensor nodes  and m is the number of grid points. 

This  subsection  shows  the  simulation  results  of  the  proposed  GCC  algorithm  and  ACC  algorithm. The simulator is implemented in C++ language, and the simulation parameters are shown  in Table 3. The sensors are distributed randomly in a 20 m × 15 m × 2.3 m cuboid. The chargers can  be deployed at grid points on the grid plane at the height of 2.3 m, and the grid point separation is  1.8 m. The charging distance is 3 m, and the charging cone opening angle is 30°. It is noted that 30  Energies 2016, 9, 696 16 of 22 experiments were executed per simulation case for averaging the simulation results. The parameter  settings are shown in Table 3.  We first simulations forfor  three scenarios of coverage requirements: all sensor nodes require We  first perform perform  simulations  three  scenarios  of  coverage  requirements:  all  sensor  nodes  1-, 2-, or 3-coveage of chargers. Figure 19 shows the comparisons of GCC and ACC algorithms in terms require 1‐, 2‐, or 3‐coveage of chargers. Figure 19 shows the comparisons of GCC and ACC algorithms  of the number of the candidate cones generated. The simulation results are all the same for the 1-, 2in terms of the number of the candidate cones generated. The simulation results are all the same for  and 3-coverage requirement scenarios, so only one diagram is shown in Figure 19. We can observe that the 1‐, 2‐ and 3‐coverage requirement scenarios, so only one diagram is shown in Figure 19. We can  2)  GCC generates much more candidate cones than ACC. This is because GCC has O(mn2 ) of candidate observe that GCC generates much more candidate cones than ACC. This is because GCC has O(mn cones, while ACC has O(mn) of candidate cones, where n is the number of sensor nodes and m is the of candidate cones, while ACC has O(mn) of candidate cones, where n is the number of sensor nodes  number of grid points. and m is the number of grid points.  Table 3. Simulation settings. Table 3. Simulation settings. 

Parameter ItemItem  Parameter Sensor deployment cuboid  20 m (L) × 15 m (W) × 2.3 m (H)  Sensor deployment cuboid 20 m (L) × 15 m (W) × 2.3 m (H) 50, 100, 150, 200, 250  Number of sensors  Number of sensors 50, 100, 150, 200, 250 Effective charging distance  Effective charging distance 33 m (D)  m (D) Opening angle  30° (A)  ◦ (A) Opening angle 30 Height of grid points  2.3 m (H)  Height of grid points 2.3 m (H) 1.8 m (S)  GridGrid point separation  point separation 1.8 m (S) Coverage requirement of sensor nodes 1-, 2-, or 3-coverage Coverage requirement of sensor nodes  1‐, 2‐, or 3‐coverage  Energy requirement of sensor nodes 0.18, 0.54, 0.9 mW Energy requirement of sensor nodes  0.18, 0.54, 0.9 mW  Times of simulation per case 30 times/case Times of simulation per case  30 times/case 

  Figure 19. The comparisons of GCC and ACC in terms of the number of generated candidate cones.  Figure 19. The comparisons of GCC and ACC in terms of the number of generated candidate cones.

Figure 20 shows the comparisons of GCC and ACC in terms of the number of the chargers deployed for the 1-, 2- and 3-coverage requirement scenarios. By Figure 20, we can observe that GCC outperforms ACC in terms of the number of chargers required. This is because GCC greedily adjusts charger antenna directions to the extreme for some cases to cover more sensor nodes by one candidate cone, and GCC generates more candidate cones to be selected. On the contrary, ACC adaptively adjusts antenna directions to cover sensor nodes with smaller horizontal angels and vertical angels, and ACC generates fewer candidate cones. Figure 21 shows the comparisons of GCC and ACC in terms of execution time for scenarios of different coverage requirements. We can observe that ACC runs much faster than GCC, especially for the scenario of high energy consumption. This is because ACC has O(m2 n3 ) time complexity, while GCC has O(m2 n5 ) time complexity. However, the time complexity of the big O notation is just an upper bound. Some experimental cases may have sensor nodes whose coverage requirements are met very early. For such cases, the algorithm execution may not be too long. This accounts for why the execution times of algorithms do not grow according to the functions of m2 n3 or m2 n5 . We now consider the form of power requirements, in which every sensor node present its power in the unit of mW for compensating its power consumption. For setting simulation parameters, we derive the sensor node power consumption from the specification of TI eZ430 (Texas Instruments Inc., Dallas, TX, USA) [27], which a common off-the-shelf sensor node device. The working voltage of eZ430

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Figure  20  shows  the  comparisons  of  GCC  and  ACC  in  terms  of  the  number  of  the  chargers  is 1.8 V and the working currents are 270 µA, 0.1 µA, 11.1 mA and 16.6 mA for processor processing, deployed for the 1‐, 2‐ and 3‐coverage requirement scenarios. By Figure 20, we can observe that GCC  sleep mode, transmission mode, and reception mode, respectively. According to Reference [28], the outperforms ACC in terms of the number of chargers required. This is because GCC greedily adjusts  sensor node can use the energy most efficiently when the ratios for the processor processing, sleep charger antenna directions to the extreme for some cases to cover more sensor nodes by one candidate  mode, transmission mode,more  and reception 0.47%, 98.77%, and 0.28%, respectively. cone,  and  GCC  generates  candidate mode cones are to  be  selected.  On 0.48% the  contrary,  ACC  adaptively  According to the ratios, the current is about 0.1 mA for the eZ430-based sensor node to use the energy adjusts antenna directions to cover sensor nodes with smaller horizontal angels and vertical angels,  most efficiently, so the power consumption is thus about 0.1 mA × 1.8 V = l.8 mW for such ratios. and ACC generates fewer candidate cones. 

(a) 

(b) 

(c)  Figure 20. The comparisons of GCC and ACC in terms of the number of deployed chargers for (a)    Figure 20. The comparisons of GCC and ACC in terms of the number of deployed chargers for: 1‐coverage; (b) 2‐coverage; and (c) 3‐coverage requirement.  (a) 1-coverage; (b) 2-coverage; and (c) 3-coverage requirement.

Figure 21 shows the comparisons of GCC and ACC in terms of execution time for scenarios of  We perform simulations for three scenarios of energy requirements, namely the low, medium, different coverage requirements. We can observe that ACC runs much faster than GCC, especially  and high power requirement scenarios. For the low power requirement scenario, it is assumed 80% 2n3) time complexity, while  for the scenario of high energy consumption. This is because ACC has O(m of sensor nodes are of 1.8 mW power requirement, 10%, 5.4 (=1.8 × 3) mW and 10%, 9.0 (=1.8 ×an  5) GCC  has  O(m2n5)  time  complexity.  However,  the  time  complexity  of  the  big  O  notation  is  just  mW. For the medium energy requirement scenario, it is assumed 10% of sensor nodes are of 1.8 mW upper  bound.  Some  experimental  cases  may  have  sensor  nodes  whose  coverage  requirements  are  met very early. For such cases, the algorithm execution may not be too long. This accounts for why  the execution times of algorithms do not grow according to the functions of m2n3 or m2n5. 

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power requirement, 80%, 5.4 mW and 10%, 9.0 mW. And for the high power requirement scenario, it is assumed 10% of sensor nodes are of 1.8 mW power requirement, 10%, 5.4 mW and 80%, 9.0 mW. In the candidate cone generation phase, the GCC and the ACC algorithms generate the same number of candidate cones no matter what energy requirements (coverage requirements or power requirements) are considered. Please refer to Figure 19 for the comparisons of GCC and ACC in terms Energies 2016, 9, x  17 of 21  of the number of candidate cones.

(a) 

(b) 

(c)  Figure 21. The comparisons of GCC and ACC in terms of the execution time for: (a) 1‐coverage; (b) 2‐ Figure 21. The comparisons of GCC and ACC in terms of the execution time for: (a) 1-coverage; covearge; and (c) 3‐coverage requirements.  (b) 2-covearge; and (c) 3-coverage requirements.

We now consider the form of power requirements, in which every sensor node present its power  Figure 22 shows the comparisons of GCC and ACC in terms of the number of the chargers in the unit of mW for compensating its power consumption. For setting simulation parameters, we  deployed for different power requirement scenarios. We can observe that ACC outperforms GCC derive the sensor node power consumption from the specification  of TI eZ430 (Texas  Instruments  for almost all scenarios. The superiority of ACC is most significant in the high power requirement Inc., Dallas, TX, USA) [27], which a common off‐the‐shelf sensor node device. The working voltage  scenario, while is least significant in the low power requirement scenario. This is because GCC greedily of eZ430 is 1.8 V and the working currents are 270 μA, 0.1 μA, 11.1 mA and 16.6 mA for processor  adjust directions of cones to the extreme for some sensor node-pair cases that nodes can cover by one processing,  sleep  mode,  mode,  reception  mode,  respectively.  According  cone. Consequently, sometransmission  sensors are covered byand  chargers with larger horizontal and vertical angles.to  Reference [28], the sensor node can use the energy most efficiently when the ratios for the processor  processing, sleep mode, transmission mode, and reception mode are 0.47%, 98.77%, 0.48% and 0.28%,  respectively. According to the ratios, the current is about 0.1 mA for the eZ430‐based sensor node to  use the energy most efficiently, so the power consumption is thus about 0.1 mA × 1.8 V = l.8 mW for  such ratios. 

Comme

of the number of candidate cones.  Figure  22  shows  the  comparisons  of  GCC  and  ACC  in  terms  of  the  number  of  the  chargers  deployed for different power requirement scenarios. We can observe that ACC outperforms GCC for  almost  all  scenarios.  The  superiority  of  ACC  is  most  significant  in  the  high  power  requirement  scenario,  Energies 2016,while  9, 696 is  least  significant  in  the  low  power  requirement  scenario.  This  is  because  19GCC  of 22 greedily  adjust  directions  of  cones  to  the  extreme  for  some  sensor  node‐pair  cases  that  nodes  can  cover by one cone. Consequently, some sensors are covered by chargers with larger horizontal and  Hence, those sensor nodes may need to be covered by more cones (i.e., chargers) to make their power vertical angles. Hence, those sensor nodes may need to be covered by more cones (i.e., chargers) to  requirements satisfied. On the contrary, ACC greedily and adaptively adjust the direction of every make their power requirements satisfied. On the contrary, ACC greedily and adaptively adjust the  cone to cover manycone  as possible sensor composition. Consequently, most sensors are direction  of  as every  to  cover  as  nodes many by as vector possible  sensor  nodes  by  vector  composition.  covered by chargers with smaller horizontal and vertical angles. Hence, sensor nodes may need to be Consequently,  most  sensors  are  covered  by  chargers  with  smaller  horizontal  and  vertical  angles.  covered by fewer cones (i.e., chargers) to make their power requirements satisfied. The explanation Hence,  sensor  nodes  may  need  to  be  covered  by  fewer  cones  (i.e.,  chargers)  to  make  their  power  above also accounts for the reason why the superiority of ACC is more significant for scenarios of requirements satisfied. The explanation above also accounts for the reason why the superiority of  higher power requirements. ACC is more significant for scenarios of higher power requirements. 

(a) 

(b) 

(c)  Figure 22. The comparisons of GCC and ACC in terms of the number of deployed chargers for (a)  Figure 22. The comparisons of GCC and ACC in terms of the number of deployed chargers for: (a) low; low; (b) medium; and (c) high power requirement.  (b) medium; and (c) high power requirement.

Figure 23 shows the comparisons of GCC and ACC in terms of execution time for scenarios of  Figure 23 shows the comparisons of GCC and ACC in terms of execution time for scenarios of different power requirements. We can observe that ACC runs much faster than GCC, especially for  different power requirements. We can observe that ACC runs much faster than GCC, especially for the scenario of high energy consumption. This is because ACC has O(m2 n3 ) time complexity, while GCC has O(m2 n5 ) time complexity.

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2n3) time complexity, while  Energies 2016, 9, 696 20 of 22 the scenario of high energy consumption. This is because ACC has O(m 2 5 GCC has O(m n ) time complexity. 

(a) 

(b) 

(c)  Figure 23. The comparisons of GCC and ACC in terms of the execution time for: (a) low; (b) medium; and  Figure 23. The comparisons of GCC and ACC in terms of the execution time for: (a) low; (b) medium; (c) high power requirement.  and (c) high power requirement.

7. Conclusions  7. Conclusions In this paper, we study the WCDO problem concerning with how to deploy as few as possible  In this paper, we study the WCDO problem concerning with how to deploy as few as possible chargers to make a WRSN sustainable. We have proposed two heuristic algorithms, namely the GCC  chargers to make a WRSN sustainable. We have proposed two heuristic algorithms, namely the GCC algorithm and the ACC algorithm, to provide approximate solutions to the problem. We consider  algorithm and the ACC algorithm, to provide approximate solutions to the problem. We consider two forms of energy requirements: coverage requirement and power requirement. For the coverage  two forms of energy requirements: coverage requirement and power requirement. For the coverage requirement scenarios, GCC outperforms ACC in terms of the number of chargers deployed. On the  requirement scenarios, GCC outperforms ACC in terms of the number of chargers deployed. On the contrary,  for  the  power  requirement  scenarios,  ACC  outperforms  GCC  in  terms  of  the  number  of  contrary, for the power requirement scenarios, ACC outperforms GCC in terms of the number of chargers deployed. However, for both forms of requirements, ACC has shorter execution time than  chargers deployed. However, for both forms of requirements, ACC has shorter execution time than GCC, as ACC has O(m22n33) time complexity and GCC has O(m22n5) time complexity, where n is the  GCC, as ACC has O(m n ) time complexity and GCC has O(m n5 ) time complexity, where n is the number of sensor nodes and m is the number of grid points.  number of sensor nodes and m is the number of grid points. In the proposed GCC and ACC algorithms, the chargers are restricted to be deployed on grid  In the proposed GCC and ACC algorithms, the chargers are restricted to be deployed on grid points of a grid, which may limit the application of the proposed algorithms. Moreover, the charger’s  points of a grid, which may limit the application of the proposed algorithms. Moreover, the charger’s charging space is modeled as a cone with a specific generatrix length and a specific opening angle,  charging space is modeled as a cone with a specific generatrix length and a specific opening angle, which may not be accurate enough for calculating the energy provision offered by the charger. We  which may not be accurate enough for calculating the energy provision offered by the charger. We are are now designing more efficient and flexible algorithms, such as meta‐optimization algorithms, to  now designing more efficient and flexible algorithms, such as meta-optimization algorithms, to solve solve the problem without the charger deployment restriction or the cone modeling of the charging  the problem without the charger deployment restriction or the cone modeling of the charging space.

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Such enhanced algorithms are suitable to WSNs in a closed space or room like space capsule in which sensor nodes have known and fixed positions and energy requirements. Redundant sensor nodes may be applied to such WSNs to prolong the network life time; however, the weight of those redundant nodes may be a burden on the space capsule. The WRSNs that rely on WPT to power sensor nodes are thus suitable for the environment, to which the enhancements of the GCC and ACC algorithms can be applied. We have noticed a related solution [29] using Voronoi diagrams to help solve the directional coverage problem in WSNs whose sensor nodes are associated with rotatable directional sector-shaped sensing areas. However, the solution is for 2D environments and cannot be applied directly for solving the WCDO problem which considers 3D environments. We thus plan to study how to apply 3D Voronoi diagram algorithms to solve the WCDO problem in the future. Acknowledgments: This work was supported in part by the Ministry of Science and Technology (MOST), Taiwan, under Grant Nos. 104-2221-E-008-017-, 105-2221-E-008-078- and 105-2218-E-008-008-. Author Contributions: Ji-Hau Liao conceived, designed and performed the experiments; Jehn-Ruey Jiang and Ji-Hau Liao designed the algorithms, analyzed the data, and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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