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1 Combining Solar Energy Harvesting with2 Wireless Charging for Hybrid Wireless3 Sensor Networks4 Cong Wang, Ji Li, Yuanyuan Yang, Fellow, IEEE, and Fan Ye

5 Abstract—The applicationQ1 of wireless charging technology in traditional battery-powered wireless sensor networks (WSNs) grows

6 rapidly recently. Although previous studies indicate that the technology can deliver energy reliably, it still faces regulatory mandate to

7 provide high power density without incurring health risks. In particular, in clustered WSNs there exists a mismatch between the high

8 energy demands from cluster heads and the relatively low energy supplies from wireless chargers. Fortunately, solar energy harvesting

9 can provide high power density without health risks. However, its reliability is subject to weather dynamics. In this paper, we propose a

10 hybrid framework that combines the two technologies - cluster heads are equipped with solar panels to scavenge solar energy and the

11 rest of nodes are powered by wireless charging. We divide the network into three hierarchical levels. On the first level, we study a

12 discrete placement problem of how to deploy solar-powered cluster heads that can minimize overall cost and propose a distributed

13 1:61ð1þ �Þ2-approximation algorithm for the placement. Then, we extend the discrete problem into continuous space and develop an

14 iterative algorithm based on the Weiszfeld algorithm. On the second level, we establish an energy balance in the network and explore

15 how to maintain such balance for wireless-powered nodes when sunlight is unavailable. We also propose a distributed cluster head

16 re-selection algorithm. On the third level, we first consider the tour planning problem by combining wireless charging with mobile data

17 gathering in a joint tour. We then propose a polynomial-time scheduling algorithm to find appropriate hitting points on sensors’

18 transmission boundaries for data gathering. For wireless charging, we give the mobile chargers more flexibility by allowing partial

19 recharge when energy demands are high. The problem turns out to be a Linear Program. By exploiting its particular structure, we

20 propose an efficient algorithm that can achieve near-optimal solutions. Our extensive simulation results demonstrate that the hybrid

21 framework can reduce battery depletion by 20 percent and save vehicles’ moving cost by 25 percent compared to previous works. By

22 allowing partial recharge, battery depletion can be further reduced at a slightly increased cost. The results also suggest that we can

23 reduce the number of high-cost mobile chargers by deploying more low-cost solar-powered sensors.

24 Index Terms—Wireless sensor networks, solar energy harvesting, wireless charging, mobile data gathering, facility location problem,

25 partial recharge

Ç

26 1 INTRODUCTION

27 DRIVEN by the prevalent energy needs from mobile devi-28 ces yet relatively stagnant battery technology, wireless29 charging took off recently as a convenient way to power30 small electronics. Its application in wireless sensor networks31 (WSNs) has been investigated [5], [6], [7], [8], [9], [10], [11],32 [12], [13]. By instrumenting wireless energy transmitters on33 mobile chargers (MCs) [5], [6], [7], [8], [9], [10] or at strategic34 locations [11], [12], [13], sensors can be charged conveniently35 without wires or plugs. Although wireless charging is a36 promising technique that can power hundreds of nodes reli-37 ably, rising energy demands in the network also increase the38 risks of electromagnetic exposure [12]. As a result, energy39 transmittersmust complywith standards fromFederal Com-40 munication Commission and limit their emitting power to

41human safe power densities (< 1 mW=cm2[3]). Neverthe-42less, nodes at data aggregation points (such as cluster heads43in a clustered WSN) usually consume very high energy44(10� 100 mW) due to data traffic. Thus limiting transmission45power at wireless chargers can easily cause battery depletion46and network interruption on such nodes.47In the meanwhile, there is another competitive technique48for environmental energy harvesting that has low risk yet49much higher power density. As shown in [21], among a50variety of harvesting techniques, solar harvesting through51photovoltaic conversion enjoys the highest power density52(15 mW=cm2), which is renewable and risk-free. In practice,53a solar panel commensurate with sensor’s size is sufficient54to meet the energy demands of cluster heads. However,55availability of sunlight is subject to dynamics in the environ-56ment. Not only weather conditions would have a direct57impact on the harvesting rates, but also a series of spatial-58temporal factors such as sunrise, sunset times, locations and59their surroundings would affect deployment decisions of60harvesting sensors.61Realizing the pros and cons of both technologies, in this62paper, we propose a hybrid framework to make use of their63advantages and overcome their drawbacks. In the new64framework, a majority of nodes are wireless-powered nodes

� The authors are with the Department of Electrical and Computer Engineer-ing, Stony Brook University, Stony Brook, NY 11794.E-mail: {cong.wang, ji.li, yuanyuan.yang, fan.ye}@stonybrook.edu.

Manuscript received 23 Jan. 2016; revised 11 Mar. 2017; accepted 24 July2017. Date of publication 0 . 0000; date of current version 0 . 0000.(Corresponding author: Yuanyuan Yang.)For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference the Digital Object Identifier below.Digital Object Identifier no. 10.1109/TMC.2017.2732979

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65 (WNs) due to the low costs of charging coils. On the other66 hand, due to the relatively higher manufacturing and67 deploying costs, a small number of solar-powered nodes68 (SNs) are responsible for aggregating data. Normally, a fleet69 of MCs roam over the field to serve recharge requests from70 WNs and collect data from SNs. In contrast to WNs, SNs’71 energy from the ambient source is self-sufficient. This72 scheme provides effective energy replenishment at cluster73 heads so that they can complete high volume data transmis-74 sions. Meanwhile, the rest of WNs can be recharged by MCs75 on demand. The hybrid framework raises several new chal-76 lenges. First, how many SNs are needed and where should77 we deploy them such that the total cost is minimized? Sec-78 ond, how to guarantee robustness of the network when sun-79 light is unavailable (e.g., cloudy/raining days)? Third, how80 to schedule the MCs to complete wireless charging and data81 gathering in the same tour? Can we further optimize system82 cost and improve network performance compared to previ-83 ous approaches?84 To answer these questions, in this paper we first study a85 placement problem in discrete formwhere SNs are deployed86 among the knownWN locations. We formulate it into a facil-87 ity location problem [26], [27], [28], [29] to minimize the total88 cost of packet routing and node deployment. Due to its NP-89 hardness, we use the primal-dual method to develop a dis-90 tributed 1:61ð1þ �Þ2-factor algorithm suitable for WSN91 applications based on the centralized paradigm in [28]. Then92 we show the locations of SNs can be further optimized93 within a cluster in continuous space and propose an iterative94 mechanism based on the Weiszfeld algorithm [30]. We also95 demonstrate how our algorithms can adapt to seasonal var-96 iations of sunlight by adjusting their locations accordingly.97 Second, we theoretically analyze network energy balance98 and propose a method to maintain such balance during99 cloudy/raining days. We find that using a smaller cluster

100 size is effective to reduce energy consumptions and develop101 a distributed algorithm to appoint some selected WNs as102 temporary cluster heads until solar energy becomes avail-103 able. Finally, we optimize MCs’ routes for the joint wireless104 charging and mobile data gathering problem. Different from105 [5] in whichMCs visit exact node locations, we point out that106 for data gathering, it is only necessary for MCs to move into107 SN’s transmission range. Based on this observation, we give108 a polynomial-time route improvement algorithm that can109 take shortcuts through SN’s neighborhood to further reduce110 the cost. Since some nodes may require expedited recharge111 due to limited lifetime, we allow the MCs to perform partial112 recharge rather than to refill batteries to full capacity [5], [6],113 [7], [8], [9], [10]. Our objective is to simultaneously maximize114 the time MCs spend in recharging and prevent nodes from115 energy depletion, which is formulated into a Linear Pro-116 gramming problem. For easy implementation on MCs, we117 propose an efficient algorithm based on the particular struc-118 ture of the lifetime constraints and validate its near-optimal-119 ity by extensive simulations.120 The contribution of this paper is multi-fold. First, we pro-121 pose a hybrid framework to overcome the constraints of122 wireless charging and environmental harvesting techniques.123 To the best of our knowledge, this is the first work dealing124 with WSNs using such hybrid energy sources. Second, we125 formulate the SN placement problem into a facility location126 problem and propose the first distributed 1:61ð1þ �Þ2-factor127 approximation algorithm for sensor applications. Third, we128 propose a method to maintain network robustness by

129reducing energy consumption. Fourth, we give a route130improvement algorithm that saves an extra 25 percent mov-131ing energy on MCs and surpasses the algorithm in [40] by132additional 5 percent. The algorithm can also be used in a gen-133eral setting for the Traveling Salesmen Problem with Neigh-134borhood (TSPN) and provide solutions very close to the135exact solutions found by exhaustive search. We also give136MCsmore flexibility to only partially refill the battery in case137of high energy demand and propose an efficient algorithm138that yields solutions within 5 percent to optimality. Finally,139we conduct extensive simulations to evaluate the perfor-140mance of the framework compared to WSNs that are solely141wireless-powered [5], [6], [7], [8], [9], [10], [11], [12], [13].142The rest of the paper is organized as follows. Section 2143conducts a literature review of the related works. Section 3144presents the network model and assumptions. Section 4145studies the placement problem of SNs. Section 5 provides146theoretical analysis and discusses how to maintain energy147balance using WNs. Section 6 optimizes MC’s migration148routes and explores partial recharge for further improve-149ments. Section 7 evaluates the new framework and Section 9150concludes the paper.

1512 RELATED WORKS

1522.1 Wireless Charging153Wireless charging technology has developed at an unprece-154dented pace recently [1], [2] and its application has been155considered in battery-powered WSNs [5], [6], [7], [8], [9],156[10], [11], [12], [13]. In [5], optimization of wireless charging157and mobile data gathering is studied by combining the two158utilities on a single MC. Based on the products from Power-159cast [4], wireless charging is explored in [6] to evaluate the160new impact on deployment patterns and packet routing.161In [7], an Oðk2k!Þ greedy algorithm (where k is the number162of nodes) is developed to maximize network lifetime163whereas the moving cost of wireless chargers is not consid-164ered. Upon realizing this problem, minimization of165chargers’ moving cost is considered in [8], [9]. In [8], a dis-166tributed real-time energy information gathering protocol is167proposed first. Then based on the updated energy informa-168tion, a weighted-sum algorithm considering nodes’ life-169times and MCs’ moving costs is developed. The problem is170further extended to jointly consider MCs’ recharge capaci-171ties and sensors’ dynamic battery deadlines in [9]. In [10], a172joint routing and wireless charging scheme is proposed to173improve network utilization and prolong network lifetime.174Similarly, in [11], deployment problems of wireless chargers175are studied to extend network lifetime.176However, since many wireless charging systems are radi-177ation-based, exposure to radio-frequency energy implies178potential health risks. The negative biological effects include179increased possibility of tumor and other impairments.180Therefore, the regulation limits the maximum transmitter181output power to be under 1 W [3]. In practice, the omnidi-182rectional propagation of electromagnetic wave causes183health risks in all directions. Due to regulations of emitting184power level on a single charger, multiple wireless chargers185are considered in [12], [13]. In [12], the problem of how to186adjust the transmitting power of wireless chargers such that187overall electromagnetic exposure does not exceed a thresh-188old is studied. A charger placement problem is formulated189to guarantee all the locations to satisfy the safety require-190ments. In [13], optimization of “useful” energy transferred

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191 from chargers to nodes under safety concerns is considered.192 However, since the accumulative emitting power from mul-193 tiple chargers is still restricted, nodes cannot perform194 energy-consuming applications.

195 2.2 Environmental Energy Harvesting196 Environmental energy harvesting provides another alterna-197 tive to extend network lifespan. Renewable energy from the198 environment such as solar, wind, vibration and thermal can199 be used effectively to power sensor nodes. Due to the200 dynamics in environmental energy sources, a majority of201 previous efforts focus on energy management of sensors202 [14], [15], [16], [17]. In [14], a power management scheme to203 maximize sensors’ duty cycles is proposed. Energy is pro-204 filed based on moving average to predict future income and205 the duty cycles are adjusted accordingly. In [15], the problem206 of maximizing quality of coverage is investigated for solar-207 powered WSNs. A nonlinear optimal control algorithm is208 proposed to dynamically allocate solar energy during the209 day and preserve energy for the night. Harvesting energy210 from light sources indoors is investigated in [16]. Energy211 allocation algorithms based on experimental measurements212 are developed to optimize energy storage on sensors. Joint213 energy management and resource allocation is considered in214 [17] for optimizing network performance. However, an inev-215 itable drawback of these earlier works is that the network216 operations would be disrupted when those ambient energy217 sources are unavailable (e.g,. during cloudy/raining days in218 a solar harvesting network). In contrast, our proposed frame-219 work in this paper incorporates a combination of hybrid220 energy sources so that steady and productive network per-221 formance can be guaranteed.

222 2.3 Tour Planning of Mobile Vehicles223 Tour planning of mobile vehicles for data collection in224 WSNs has been studied, see, for example in [18], [19], [20].225 The problem shares similarities to the well-known Travel-226 ing Salesmen Problem with Neighborhood [37], [38], [40] in227 which the salesman aims to find the shortest tour through228 city neighborhoods of arbitrary shapes. In the context of229 WSNs, due to the omnidirectional propagation of electro-230 magnetic waves, the neighborhoods are usually assumed to231 be circles. In [18], the tour planning problem is formulated232 into a mixed-integer program and a spanning tree covering233 algorithm is proposed. However, the algorithm requires the234 vehicles to visit exact node locations for data gathering,235 which is usually unnecessary in practice. If the node’s trans-236 mission range is considered, the performance can be further237 improved. The method proposed in [19] attempts to238 improve current solutions for TSPN. It first determines the239 shortest TSP routes among sensors without considering240 their transmission ranges. Then it searches along transmis-241 sion boundaries to find the best hitting points such that the242 tour length is minimized. In [20], a progressive tour con-243 struction method is proposed. It exploits the overlaps of244 transmission ranges of neighboring nodes so that the mobile245 vehicle can take shortcuts for cost savings. This method246 works effectively when there exist overlaps among nodes’247 transmission ranges. Different from [20] where nodes’ trans-248 mission ranges may have overlaps, in this paper, multi-hop249 clusters are formed such that one-hop transmission neigh-250 borhoods of cluster heads (SNs) are disjoint. In addition,251 different from all the previous works, a migration tour may

252incorporate both WNs and SNs in our framework. There-253fore, we focus on how to minimize tour lengths by taking254advantage of SNs’ neighborhoods in a joint wireless charg-255ing and data gathering tour.

2563 NETWORK MODEL AND ASSUMPTIONS

257In this section, we give an overview of the network model258and assumptions of the new framework. Based on the259energy sources, there are two types of nodes in the frame-260work: wireless-powered nodes and solar-powered nodes.261For brevity, we denote them by “WNs” and “SNs” respec-262tively. Based on the functionality of network components,263we divide the network into three hierarchical levels as264shown in Fig. 1: wireless-powered sensor, solar-powered265sensor and mobile charger levels.266The bottom level (wireless-powered sensor level) has N267WNs uniformly randomly distributed on a square field of268side length L. Since charging coils can be cheaply manufac-269tured, WNs are deployed in high density to perform basic270sensing missions such as environmental readings, target271tracking, etc. In particular, to monitor location-dependent272solar radiation strength, each node has an illuminance sen-273sor and reports its reading with other data to the base sta-274tion. The energy consumption for transmitting a packet275follows the widely adopted model [23], et ¼ e0 þ e1d

ar ,

276where et is the transmitting energy, e0 and e1 are the energy277consumed in electronics and amplifiers, dr is the distance278between the transmitter and the receiver (dr � r, r is the279transmission range), and a is the path loss exponent. To per-280form sensing tasks, sensors also consume es energy for each281packet. For message exchange, we assume the network is282connected. We also assume an event-driven data generation283model [22]. Each WN generates packets independently fol-284lowing a Poisson process with average rate �. Each WN is285powered by a 780 mAh rechargeable NiMH battery and the286recharge time Tr is 78 minutes [24]. If the energy drops287below a threshold, e.g., 50 percent, it sends out a request to288the MCs for scheduling energy replenishment.289The middle level (solar-powered sensor level) is com-290prised of self-sustaining, energy harvesting nodes. Nor-291mally, when solar energy income is sufficient, SNs act as292cluster heads for aggregating sensed data. However, when293energy supply is not enough during cloudy/raining days,294the network re-selects WNs as cluster heads so they can rely295on consistent wireless energy supply from the MCs. To min-296imize routing and deploying costs, SNs should be deployed297at advantageous locations. Due to varying nature of sun’s298angles during a year, building obstructions and tree shades

Fig. 1. Overview of a three-level network hierarchy.

WANG ET AL.: COMBINING SOLAR ENERGY HARVESTING WITH WIRELESS CHARGING FOR HYBRID WIRELESS SENSOR NETWORKS 3

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299 may exhibit different spatial-temporal patterns. Therefore,300 SN locations should be re-calculated based on the updated301 data once in a while (e.g., several weeks). The energy har-302 vesting rates are modeled according to [33], which will be303 discussed in Section 5. We assume the size of solar panel is304 chosen to be large enough to harvest enough energy for305 aggregating all the data. We consider a commercially avail-306 able panel of 10� 10 cm2 size which is connected to a 3 V,307 2150 mAh lithium-ion battery.308 The top level (mobile charger level) manages a fleet of309 m MCs through the base station. Proposed in [5], MCs310 are equipped with high-capacity batteries and powerful311 antennas for energy replenishment and data collection. Coor-312 dination among theMCs is conducted via long range commu-313 nications to exchange status, position, energy request, etc.314 They also have positioning devices (e.g., GPS, gyroscope, etc)315 to locate sensor positions so that they can approach them in316 close proximity for wireless charging at high efficiency. We317 assume each node is only recharged by one MC at a time.318 Depending on updated geographical solar energy distribu-319 tion, MCs can deploy SNs at appropriate locations. In case320 some nodes in the recharge sequences are bound to deplete321 their energy, the MCs can expedite the process by partially322 refilling nodes’ batteries. Since SNs and MCs are the323 main components to sustain network operations and their324 manufacturing costs are much higher than WNs, we assume325 their monetary expenses are ps (for SNs) and pm (for MCs),326 respectively and ps < pm. In this paper, we have made addi-327 tional assumptions: 1) The duration of MC’s tour is domi-328 nated by battery recharging time; 2) Data is aggregated at the329 SNs by the shortest path routing (Dijsktra’s algorithm) so330 routing paths are fixed; 3) For scalability, WNs do not upload331 their data to the MC directly. Finally, we summarize the332 important notations in Table 1.

333 4 SOLAR-POWERED SENSOR LAYER: PLACEMENT

334 PROBLEM

335 In this section, we study the Solar-powered sensor Place-336 ment Problem (SPP). It determines where to place SNs such337 that the total cost is minimized. In SPP, there are two types338 of costs: packet routing cost and sensor deploying cost.339 According to the energy model on packet transmissions340 [23], the routing cost is proportional to the number of hops341 thus the distance to the cluster head. The deploying cost is

342related to expense ps and the strength of sunlight at a spe-343cific location. Since harvested energy exhibits slow variation344due to seasonal changes of sun’s angle, solar radiations at a345fixed location also change slowly during a year. According346to the illuminance readings from sensors, we denote the347average solar strength at sensor location i by li. The deploy-348ing cost can be defined as the ratio ps to li, which can be349explained as the price we pay to gather a unit of solar350energy from a specific location. If more SNs are deployed,351nodes would have less relaying distance to the SNs. Thus,352less routing cost can be achieved. On the other hand, more353SNs would increase the deploying cost so our objective is to354minimize the sum of routing cost and deploying cost.355These observations suggest that our problem is in close356analogy to the classic Facility Location Problem (FLP) [26],357[27], [28], [29], which is NP-hard. In FLP, a set of facilities358and cities are given. There is an opening cost associated359with each facility and a transportation cost between any360pair of facility and city. The goal is to connect each city to an361open facility while minimizing the sum of transportation362and opening cost. Due to NP-hardness, obtaining an opti-363mal solution in polynomial time is infeasible. In practice,364approximation algorithms that can achieve certain factors to365the optimal solution are always preferred. After the first366polynomial time 3.16-approximation algorithm [26], there367has been encouraging progress in improving the approxi-368mation ratio and running time. An Oðn2 lognÞ algorithm is369proposed in [27] with an approximation ratio of 3. This370bound is soon improved by [28] from 1.86 to 1.61 which is371very close to the upper limit 1.46-ratio that polynomial-time372algorithms can achieve [29].373However, the aforementioned efforts only focus on cen-374tralized algorithms whereas distributed implementation of375FLP is rare in the literature. In dynamic wireless environ-376ments, a centralized algorithm requires the collection of vari-377ables across multiple dimensions to form global knowledge,378which is usually time-consuming and not cost-effective. To379this end, we propose a distributed 1:61ð1þ �2Þ-approxima-380tion algorithm based on the centralized approach in [28].381Next, we first formalize SPP and illustrate the centralized3821.61-approximation algorithm. Then we propose a distrib-383uted version of the algorithm. Since the locations of SNs are384not necessarily constrained to WN locations, we further385improve the solution using the Weiszfeld algorithm [30].386Finally, we discuss how to re-deploy SNs in order to adapt387variations in solar strength at different times.

3884.1 Placement of Solar-Powered Sensors389In this section, we formalize the problem and describe both390the centralized and distributed versions of the algorithm.

3914.1.1 Centralized Placement Algorithm

392First, let us formalize SPP. We denote the sets of SNs and393WNs by S andN , respectively. For simplicity, we first study394discrete SPP by assuming SNs can only be co-located at395WNs’ locations, S � N . We consider a graph G ¼ ðV;EÞ396where vertices are sensor nodes and edges are connections.397cij is the routing cost between nodes i and j, which is the398energy consumed for transmitting packets. fi is the deploy-399ing cost of SN i, and fi ¼ ps=li, where li is the solar strength400at node i. Since the energy consumed by WNs for data401transmissions ultimately comes from the MCs, to convert402cij’s energy units into monetary cost, we scale cij by how

TABLE 1List of Notations

Notation Definition

N Number of wireless-powered sensorss Number of solar-powered sensors

(calculated by algorithm)m Number of mobile chargersL Side length of squared sensing fieldr Transmission range of sensor nodeset; er; es Energy consumption to transmit, receive,

generate a packet� Average packet rate of Poisson distributed trafficps; pm Monetary expenses of SNs and MCs, respectivelyCh Battery capacity of sensor nodesTr Recharge time of sensor’s battery from zero

to full capacityv Average moving speed of MCs


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roof403 much the base station has paid for consuming per watt of

404 energy to recharge MC’s battery. The decision variable xij is405 1 if WN j is assigned to SN i; otherwise, it is 0. yi is 1 if we406 place an SN at i; otherwise, it is 0. Initially, all WNs are can-407 didate locations for SNs. Our objective is to minimize the408 total cost by finding the locations for SNs

P1 : minXi2S

Xj2N

cijxij þXi2S

fi (1)

410410

411 Subject to Xi2S

xij ¼ 1; j 2 N (2)413413

414

xij � yi; i 2 S; j 2 N (3)416416

417

xij; yi 2 f0; 1g; i 2 S; j 2 N : (4)419419

420 Constraints (2) and (3) impose that each WN is only con-421 nected to one SN. A centralized 1.61-approximation algo-422 rithm is proposed in [28]. Since we will use it as a guideline423 for designing the distributed algorithm, we briefly describe424 the centralized algorithm below. For each SN i, we intro-425 duce a set Bi to represent its connected WNs (Bi � N ). In426 each step, the algorithm selects the node i� with the mini-427 mum average cost

i� ¼ argmini2N

Xj2Bi

cij þ fi �Xj2B0

i

ðci0j � cijÞ24

35=jBij: (5)

429429

430 Node i0 is a deployed SN thatWN j has already connected to.431 B0i is the set of these already connected WNs which would be432 benefited by altering their connections to the new SN i. Hence,433 a saving of routing cost

Pj2B0

iðci0j � cijÞ should be deducted

434 from the total cost. To find theminimum average cost for each435 candidate SN i, we can sort the cost in an ascending order and436 select the least one. This would result in jBijWNs being cho-437 sen each time. After i� is found, we deploy an SN at its loca-438 tion and update all the WNs in Bi

S B0i to connect with node439 i�. The iteration continues to add SNs until all WNs are con-440 nected to them. The centralized algorithm has OðN3Þ com-441 plexity and is summarized in Table 2.

442 4.1.2 (1:61þ �2)-Factor Distributed Algorithm

443 To understand the nature of the problem, we first formulate444 the dual problem of P1. The introduction of dual variables445 will help design the distributed problem

P2 : maxj2N

aj (6)

447447

448Subject to

aj � bij � cij; i 2 S; j 2 N (7) 450450

451Xj2N

bij � fi; i 2 S (8)453453

454

ai; bij 0; i 2 S; j 2 N : (9)

456456

457Here, we can think the dual variable aj as a monetary offer458from node j to the total expense for deploying an SN. Con-459straints (7) and (8) can be combined into

Pj2N maxðaj � cij;

4600Þ � fi for SN i 2 S. It means that if offer aj is raised for all461the WNs at the same pace, and at the moment the total offer462minus the total cost is equivalent to fi, an SN can be success-463fully deployed at i. This method is known as the dual ascent464procedure [28] and its 1.61-factor approximation is proved in465[28] following the centralized paradigm. Based on [28], we466propose a distributed 1:61ð1þ �Þ2-approximation algorithm467next, where � is a small fraction greater than zero.468First, WNs will send out their offers to SNs. If a WN is not469connected to any i 2 S, the value of the offer is set to470maxðaj � cij; 0Þ; otherwise, the value is set tomaxðci0j � cij; 0Þ.471At the other side, SNs receive the offering messages from472WNs. If an SN j is not yet deployed while its received total473offers

Pj2N maxðaj � cij; 0Þ are greater than or equal to fi, we

474can successfully deploy an SN at i. If j is deployed and the475offer value aj ¼ cij, we connect j to i by sending a connection476message to j. In the next round, WNs increase their offers aj477by a ratio of ð1þ �Þ. After the locations for SNs have been cal-478culated, the WNs send out deploying requests to the MCs.479Then an MC is dispatched from the base station to deploy480SNs at their designated locations. The distributed algorithm481onWNs and SNs is summarized in Tables 3 and 4 and has the482following properties.

483Property 1. In principle, the distributed and centralized algo-484rithms are equivalent.

485Proof. See Section 10.1. tu

TABLE 2Centralized 1.61-Factor SN Placement Algorithm

Input: Set of WNN .Output: Set of SN S and Bi, i 2 S.While (N 6¼ ;)Find i� ¼ argmini2N

�Pj2Bi cij þ fi �

Pj2B0

iðci0j � cijÞ

�=jBij.

Deploy i�, connect 8j 2 BiS B0i to i�,N N � Bi.

End While

TABLE 3Distributed 1:61ð1þ �Þ2-Factor Algorithm for WN j

If j is not connected to any SN,send a message to iwith offer aj maxðaj � cij; 0Þ.Else If j is connected to a deployed SN i0,send a message to iwith offer aj maxðci0j � cij; 0Þ.End IfRaise offer aj ð1þ �Þaj.

TABLE 4Distributed 1:61ð1þ �Þ2-Factor Algorithm for SN i

Receive offering messages fromWNs.If i is not yet deployed AND

Pj2N maxðaj � cij; 0Þ fi

deploy an SN at i’s location, S S þ i.8j 2 N , Bi Bi þ j. Connect j to SN i.Else If i has been deployed AND aj ¼ cijconnect j to i, Bi Bi þ j, B0i B0i � j.Send a connection request message to j.End If


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roof486 Property 2. The distributed algorithm terminates in

487 Oðlog 1þ�fmÞ rounds, where fm ¼ maxfi; i 2 N . The total488 message overhead is Oððlog 1þ�fmÞN2Þ.489 Proof. Clearly, when the offering amount aj increases at a490 rate ð1þ �Þ, reaching the maximum value of fi requires491 Oðlog 1þ�fmÞ. In each round, the message overhead is492 bounded by OðN2Þ so the overall message overhead is493 Oððlog 1þ�fmÞN2Þ. tu494 Property 3. The distributed algorithm achieves 1:61ð� þ 1Þ2-fac-495 tor approximation to the optimal solution.

496 Proof. See Section 10.2. tu

497 4.2 Placement of SNs in Continuous Space498 SinceMCs enjoy the freedom to place SNs at any feasible loca-499 tion in the field, in this section, we explore the placement of500 SNs in continuous space. Clearly, the continuous problem is501 much harder than its discrete version (SPP). Thus we start502 from the discrete results and relax them into the continuous503 domain. Intuitively, the asymptotic behavior of the discrete504 problem should approach the continuous problem when the505 number of nodes is infinite. However, in our problem, the506 number is limited. Thus discrete results from the SPP provide507 a feasible, sub-optimal solution to the continuous problem.508 Our objective is to re-locate SNs in the continuous509 domain so the total cost which consists of the intra-cluster510 routing cost and the deploying cost is minimal. We find511 that, in most cases, variations of geographical solar radia-512 tion are quite small in a cluster (unless some spots are cov-513 ered by shades) so the total cost is usually dominated by the514 routing cost. Changing SNs to new locations might reduce515 the routing cost (by DC) but lead to an increase of deploying516 cost (by DF ). If the reduction in routing cost is higher than517 the increment in deploying cost (DC � DF > 0), there is still518 an extra saving in the total cost. To minimize the total cost, a519 naive approach is to divide the field into grids and enumer-520 ate through all possible locations. This method obviously521 requires enormous computation power and its accuracy522 also depends on the density of the grid.523 We shift our focus tominimizing intra-cluster routing cost,524 and at the same time, we look for locations that offer the larg-525 est overall savings DC � DF . From SPP, a set S is obtained526 (S � N ) with the corresponding cluster set Bi, 8i 2 S. We527 observe that the problem resembles the well-known Weber528 problem [30], [31] which finds the geometricmean for the set of529 Bi nodes. A well-known algorithm to solve the problem is to530 use an iterative procedure due to Weiszfeld [30]. In order to531 consider the deploying cost also, we introduce an additional532 step to the originalWeiszfeld algorithm.533 The algorithm is illustrated below. First, we initialize534 SN’s location x0 at node i’s coordinates (from the output of

535SPP). The Weiszfeld algorithm uses a recursive function536WðÞ to find the optimal cost. The computation of the kth537iteration is,

WðxkÞ ¼�XjBij

j¼1

aj

cðxk; jÞ��XjBi j

j¼1

1

cðxk; jÞ�; (10)

539539

540where cðxk; jÞ is the routing cost of transmitting a packet541from location xk to node j, and aj are the coordinates of542node j 2 Bi. The iteration continues by executing xkþ1 ¼543WðxkÞ until location changes are less than a small error544bound ". Since it has been proved in [31] that the Weiszfeld545algorithm can converge to an optimum, the intra-cluster

546routing cost Ck ¼PjBij

j¼11

cðxk;jÞ for the kth iteration is larger

547than Ckþ1 in the ðkþ 1Þth iteration. However, the corre-

sponding deploying cost may not share the same property

due to the relative randomness in geographical solar energydistribution. Thus, an additional step of finding the minimal

deploying cost km ¼ arg min1�i�n

ðCi þ FiÞ is needed when the

548iteration is over. If the initial location has the minimal cost,the SN does not need to change its location. Otherwise, we

re-locate the SN to xkm as the final solution. The algorithm is

summarized in Table 5.

549Remarks. The approximation ratio for the discrete problem550does not hold for the general continuous facility location551problem. Unlike the discrete problem which we can for-552mulate into an integer programming problem, the contin-553uous problem usually involves nonlinear optimizations554and complex iterative algorithms [32]. Our solution first555utilizes the discrete solution for allocation of SNs (i.e.,556clustering WNs with SNs/determining the number of557SNs). In this way, we can narrow down the search space558of SNs within established clusters. Although this method559may not be optimal, it offers a reasonable and fast solu-560tion for such an NP-hard problem.

5614.3 Spatial-Temporal Solar Variations562We now demonstrate spatial-temporal solar variations and563explain how our proposed algorithm adapts to these changes.564During different seasons of a year, the sun’s angle towards565earth surface varies slowly. Consequently, the harvested566energy at different locations reflects such changes due to567building obstructions, tree shades, etc. Fig. 2 shows the heat-568maps of solar energy distributions on our campus gathered at569different locations in February and May (Longitude at North57040�).We can see that the areas falling into tree shades are quite571different. This has a direct impact on the deploying cost fi at a572location. Second, the strength of solar radiation also varies

TABLE 5Extended Weiszfeld Algorithm for a Cluster

Initialize SN’s location x0 to i’s coordinates obtained by SPP.While xkþ1 � xk > "

WðxkÞ ¼�PjBij

j¼1aj

cðxk;jÞ�=�PjBij

j¼11

cðxk;jÞ�; xkþ1 ¼WðxkÞ.

Record deploying cost Fk at xk.End Whilen k is the number of iterations until convergence.Find km ¼ arg min

1�i�nðCi þ FiÞ and place SN at location xkm .

Fig. 2. Heatmaps of geographic solar energy distribution in differentmonths. (a) February. (b) May.


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573 dramatically. In February, the maximum level is 3 mW=cm2

574 and an increase of 13 percent is observed inMay. These obser-575 vations suggest that SN locations should be re-calculated after576 some time Tc. Otherwise, they might be covered in shades577 with limited harvesting capabilities.578 Our algorithm fully exploits the distributed nature of579 WSNs. During operation, each node records solar strength580 at its location periodically and maintains a trailing average581 for the past Tc time. fi is updated every Tc time accordingly.582 Once a new deployment is initiated, SNs’ locations are cal-583 culated using the distributed algorithm. An example of the584 algorithm is shown in Fig. 5a. After their locations are585 found, an MC is dispatched to re-locate the corresponding586 SNs to designated locations. Then data is first gathered via587 multi-hop forwarding at the SNs and then collected by the588 MC during recharge.

589 5 WIRELESS-POWERED SENSOR LEVEL:590 MAINTAINING ENERGY BALANCE

591 In this section, we study the wireless-powered sensor level.592 Our main objective is to maintain network energy balance593 on WNs in different scenarios. First, we derive energy bal-594 ance when SNs operate during sunny days. To facilitate our595 analysis, we denote the number of SNs obtained by the SPP596 algorithm as s ¼ jSj and the maximum hop count from WN597 to its assigned SN as h. We assume that a total budget B for598 s SNs and m MCs, sps þmpm � B. We explore the relation-599 ship between s and m given their manufacturing costs ps600 and pm (ps < pm). Second, during cloudy/raining days,601 energy balance might be broken. In this case, we further602 study how to regain such balance by refilling the energy603 gap. We propose to utilize several WNs to act as temporary604 cluster heads for aggregating data. A numerical range of605 WN cluster heads is first derived followed by a distributed606 algorithm to determine which WNs should be selected.

607 5.1 Energy Balance608 First, let us consider energy consumptions in the network.609 For s shortest path routing trees rooted at SNs, the total610 energy consumption is

Ec ¼Xj2N

��ðet þ esÞ þ

Xi2Cj

�ðet þ erÞ�T

�Xhi¼1

�Niðet þ esÞ þ

Xhj¼iþ1;i 6¼h

Njðet þ erÞ��sT

¼�

2

3h3 � 1

2h2 � 1

6h

ðet þ erÞ þ h2ðet þ esÞ

�pr2r�sT;

(11)612612

613 where Cj is the set of child nodes of j 2 N ,Ni ¼ ð2i� 1Þpr2r.614 The inequality holds because 1) a cluster can be estimated as615 a circle of radius R ¼ hrwhich consists of h concentric rings616 [25]; 2) summation of consumptions from all circle-shaped617 clusters has overlapping areas between neighboring clusters.618 The harvested solar energy can be estimated by the619 empirical model proposed in [33]. The model provides a620 year-round analysis of solar radiations from weather sta-621 tions and relates power levels to a quadratic equation on622 the time t of the day,

E ¼ ða1ðtþ a2Þ2 þ a3Þð1� sÞ: (12)624624

625The shape of Eq. (12) is determined by parameters a1 � a3626that vary seasonally for different months. For example, for627the month of May, a1 ¼ �1:1, a2 ¼ �13:5 and a3 ¼ 43:5. t1628and t2 are the respective time of sunrise and sunset

629(t1 ¼ �ffiffiffiffiffiffiffiffi� a3

a1

q� a2; t2 ¼

ffiffiffiffiffiffiffiffi� a3

a1

q� a2). s is the percentage of

630cloud cover from weather reports. For T days, energy har-

vested by SNs is

Es ¼ sXTi¼1

Z t2

t1

�a1ðtþ a2Þ2 þ a3

�ð1� siÞdt: (13)

631The wireless energy replenished by MCs into the net-632work is governed by the battery charging rates Ch=Tr. Thus,633the amount of wireless energy replenished by m MCs in T634can be calculated by Ew ¼ ðmTChÞ=Tr, m 6¼ 0. Then network635energy balance is achieved when

Ec � Es þEw

Ec <

�2

3h3 � 1

2h2 � 1

6h

ðet þ erÞ þ h2ðet þ esÞ

�pr2r�sT

� sXTi¼1

Z t2

t1

�a1ðtþ a2Þ2 þ a3Þ

�ð1� siÞdtþmTCh

Tr:

(14)637637

638Since pðhrÞ2s L2, we haveffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2=ðsr2pÞp � h. By plugging it

639into Eq. (14) and taking approximation et � er, we obtain a640relationship between s andm

LTretr�

3ffiffiffipp

Chr

4L2ffiffiffisp � pr2

ffiffiffisp

�Xsþ1

2et þ es

L2r�Tr

Ch� m;

(15)642642

643whereX is

X ¼�

Tr

ChT

XTi¼1ð1� siÞ

��a13t3 þ a1a2t

2 þ ða1a22 þ a3Þt��

t¼t2

t¼t1:

(16)645645

646

6475.2 Analysis of Budget and System Cost648The relationship between s and m in Eq. (15) can be649explained graphically. Fig. 3a shows a group of energy bal-650ance curves when N ¼ 250 750. Any point on a curve651serves the same purpose for balancing network energy and652there is no preference between choosing SNs or MCs as653long as the balance holds. In fact, these curves can also be654interpreted as the indifference curves in microeconomics. An655indifference curve shows a collection of different goods

Fig. 3. Network plans under budget constraints. (a) Energy balancecurves for different network sizes. (b) Optimal choices under budgetconstraints.


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656 between which the consumer is indifferent and every point657 on the curve results in the same utility. For example, when658 N ¼ 500, point A requires 2 SNs and 8 MCs, which is equiv-659 alent to point B of 4 SNs and 6 MCs. Note that keeping add-660 ing SNs reduces MCs at a diminishing marginal rate. This is661 because that when the number of SNs is small, adding more662 SNs helps alleviate routing cost significantly (saving663 energy); however, this benefit gradually diminishes as more664 SNs are deployed.665 Based on the budget, we can find whether a network666 plan is feasible to maintain energy balance. Since s is deter-667 mined by the SPP algorithm, the corresponding number of668 MCs m can be found from the budget line m ¼ � Pm

Pssþ B

Ps669 (point ðs;mÞ). If this point is above a balance curve, it means670 that the corresponding network size can satisfy energy bal-671 ance. For example, point ð6; 3Þ on budget line 1:672 m ¼ � 1

3 sþ 5 is above the balance curve of N ¼ 250 which673 indicates that the selection of s ¼ 6, m ¼ 3 is feasible. The674 feasible region for budget line 1 is marked as the shaded675 area. Furthermore, to find the maximum network size a676 given budget can sustain, we gradually increase N until677 point ðs;mÞ is no longer above the balance curve. For point678 ð6; 5Þ on budget line 2, the maximum network size is679 N ¼ 500. As the network size increases, the budget should680 be increased as well, which is to shift the budget line681 upward in Fig. 3b. In this way, network administrators can682 quickly find an appropriate network plan, given the budget683 and available choices of SNs and MCs.

684 5.3 Adaptive Re-Selection of Cluster Heads685 An imperfection of solar energy is that sunlight is not686 always available. For example, during raining seasons, the687 network could experience consecutive cloudy or raining688 days and SNs are unable to harvest enough energy. To sus-689 tain network operation, our framework should adaptively690 switch cluster heads to WNs for aggregating sensed data. In691 this section, we first discuss how to maintain energy balance692 in the absence of solar energy. Then, we propose an algo-693 rithm to re-select cluster heads among WNs.

694 5.3.1 Maintaining Energy Balance

695 Since the number of MCs m is fixed for a network plan, we696 cannot expectmore energy income from the energy replenish-697 ing side. To this end, we should reduce energy consumptions698 to restore energy balance. That is, the energy gap during699 cloudy days should be filled by reducing consumption for at700 least the same amount. Intuitively, introducing more cluster701 heads can effectively reduce energy consumption because702 more aggregation points would shorten packet relay paths. In703 other words, this is equivalent to having smaller k-hop clus-704 ters (k < h). The following property validates our intuition.

705 Property 4. For a network originally clustered by s SNs with706 cluster size h > 2, in the case that solar energy is unavailable,707 we can always restore network energy balance by reducing clus-708 ter size.

709 Proof. See Section 10.3. tu710 An anomaly is h ¼ 1; 2. Because the original cluster sizes711 are already very small, we cannot reduce energy consump-712 tion further by having smaller clusters. In these cases, a noti-713 fication message should be sent to request for more MCs.714 However, one-hop mobile data gathering only occurs in the

715worst case. Normally, we have 1 < k < h so our objective716is to calculate how many WN cluster heads are needed717given k. From the previous section, the solar radiation718model indicates the energy harvested peaks when s � 0719(perfect weather condition). Let us denote the number of720new WN cluster heads by s0 (s0 > s). The maximum721amount of energy harvested is E�s when s ¼ 0 in the ideal722case. EcðhÞ is the energy consumption with h-hop clusters.723XcðkÞ is the energy consumption for each k-hop cluster724(plugging k into Eq. (11) and getting rid of s0). Since we725require E�s � EcðhÞ � s0XcðkÞ, a range for s0 is

L2

pðkrÞ2 < s0 � EcðhÞ � E�sXcðkÞ : (17)

727727

728By fixing k, any s0 satisfying Eq. (17) will guarantee energy729balance of the network. Next, we develop a distributed algo-730rithm to findWN cluster heads.

7315.3.2 Head Re-Selection Problem

732We now further explore the Head Re-selection Problem (HRP)733which finds k-hop clusters with s0 cluster heads satisfying734Eq. (17). On one hand, since WN cluster heads will be tra-735versed by MCs for data collection, the number of such nodes736should beminimized to saveMCs’moving cost. On the other737hand, for heads to cover all the nodes within k hops, s0

738should be sufficiently large; otherwise, clusters will exceed k739hops and more likely break the energy balance. Hence, our740objective is to select a minimum number of heads and ensure741that the shortest path from any node to its nearest head does742not exceed k hops. It is not difficult to see that HRP is themin-743imum k-hop dominating set problemwhich is NP-hard [34].744A distributed algorithm for this problem is proposed745in [34] for ad-hoc networks. The algorithm requires two746rounds of k-hop message flooding for all the nodes. Since747flooding is usually less preferred in energy constrained748WSNs, we will not adopt the algorithm in [34]. Instead, we749leverage the range in Eq. (17) as a basis for HRP. That is, as750s0 grows, hop distance from a node to its nearest head751should decrease. Thus, we can start from the lower bound752and increase s0 iteratively until all the nodes are covered in753k hops or the upper bound is reached. To find which WNs754should become cluster heads, we extend the furthest first tra-755versal algorithm proposed in [35]. The algorithm selects the756node with the maximum distance from the current node to757become the head in the next round. Unfortunately, the algo-758rithm cannot be applied directly to our problem because: 1)759it is centralized and not efficient to implement in distributed760WSNs; 2) it may lead to inefficient selections. A new head761might be chosen in the vicinity of an established one thereby762causing a large overlap between neighboring clusters. This763is not efficient and may also violate Eq. (17). Hence, we764leverage the principle of furthest first traversal and propose a765new distributed algorithm.766When gathered data at an MC indicates solar energy is767not sufficient to support SNs, the MC sends a head notifica-768tion message to any arbitrary WN whose battery has just769been replenished and sets a counter to 0. The message speci-770fies the cluster size k hops (k ¼ h� 1 initially and decreased771by 1 in each trial till k ¼ 1). Upon receiving the head notifi-772cation message, the WN declares itself as a new head and773builds a shortest path tree (e.g., using Bellman-Ford algo-774rithm [36]). Each node also maintains a routing entry to


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775 store minimum hop distance to a head. Those entries are776 updated when a new shortest path tree is formed. If a node777 j’s entry indicates the minimum hop distance to a head i is778 less than or equal to k, it sends a join message to i to “join”779 the cluster as a member. Otherwise, it sends a resume mes-780 sage to node i to let the head selection continue. Within a781 timeout period, if the head receives a resume message, it782 means that there still exist some node(s) uncovered and the783 selection process should continue.784 If a resume message is received, the head computes a785 shortest path tree using the Bellman-Ford algorithm. To786 avoid inefficient head selection, nodes should also report to787 the head whether they are cluster members or not. Then a788 new head notification message is generated and sent along789 the shortest path tree to the node with the maximum hop790 distance and enough battery energy which is not a cluster791 member yet. The counter is then increased by one. Other-792 wise, if no resume message is received during the timeout793 period, the head declares that clustering is successful by794 sending a complete message to all the heads. Upon receiving795 the complete message, heads report to the MC of cluster796 information. Note that if the counter exceeds the upper797 bound in Eq. (17), the current k is not feasible to maintain798 energy balance so it should be further decreased. In this799 case, the head should broadcast a message to restart the800 whole process and choose a smaller k. The psuedocode of801 this algorithm is given in Table 6. Based on Property 4, the802 distributed HRP algorithm can always find a set of cluster803 heads in OðhSÞ rounds and the worse case message over-804 head is OðhSN2Þ, where S is the upper bound in Eq. (17).

805 6 MOBILE CHARGER LAYER: JOINT WIRELESS

806 CHARGING AND MOBILE DATA GATHERING

807 In this section, we focus on optimizing trajectories of MCs.808 Proposed in [5], the instrumenting radio modules on MCs

809realize joint wireless charging and mobile data gathering on810a single MC. This design certainly reduces manufacturing811cost of MCs and system cost. However, because the effective812wireless charging range is very limited (0.5-1 m), the813method in [5] requires the MC to stop at the exact WN loca-814tion to perform simultaneous data gathering and recharge.815However, in our framework, since SNs are powered by816solar energy, it is only necessary for the MC to enter the817transmission range (“touch” the transmission boundaries)818to collect data from SNs. This creates opportunities to fur-819ther optimize MCs’ trajectories.

8206.1 Planning of Joint Data Gathering and821Recharging Tours

8226.1.1 Initial Center Tour

823We assume MC i has been assigned a touring sequence. The824sequence defines an ordered set of nodes that starts from825the base station b, traverses through WNs wi and SNs (clus-826ter heads) aj, wi 2 N ; aj 2 S, and finally returns to the base827station for uploading data and recharging MC’s own bat-828tery. Recharge scheduling algorithm proposed in [9] can be829used conveniently to take recharging and data gathering830requests together and calculate an initial touring sequence831for each MC. Normally, the cluster size is larger than one832hop (h > 1), and the transmission range around SNs form833disjoint disks with identical radius. Since the initial834sequence does not distinguish an SN from a WN and stops835at the center of SN’s transmission radius, we call it “Initial836Center Tour” and denote its length as Lc. For such a tour837with nWNs and s SNs, we have the following property.

838Property 5. For an optimal tour with length L�r , when L�r is839much larger than transmission range r, Lc is within840

�1þ 8

pþ �� 3:55þ � to the optimal L�r .

841Proof. See Section 10.4. tu

8426.1.2 Exhaustive Search

843Although the initial center tour guarantees a (3:55þ �)-factor844approximation, the solution can be further improved if the845MC can take shortcuts through the disks. This problem is846known as the Traveling Salesmen Problem with Neighborhoods847and no efficient solution exists [37], [38]. However, in our848problem, we can take advantage ofWNs in the sequence and849greatly reduce computation complexity. For all WNs in a850sequence (and the base station), we order them in pairs851ðb; w1Þ, ðw1; w2Þ, . . . , ðwn; bÞ. For each pair ðwi; wiþ1Þ, there852could be at most s SNs in between. Let us start the analysis853with s ¼ 1. Fig. 4 shows that there are two cases: 1) the path854connecting wi and wiþ1 directly cuts through the disk855(Fig. 4a). In this case, the MC does not need to change direc-856tions. It only stops for a period of data uploading time (sev-857eral minutes) in the disk. 2) the disk does not intersect with858the path so there should exist a point on the boundaries of859the disk that can minimize the path (minðaþ bÞ in Fig. 4b). A860naive approach is to divide the disk perimeter into l seg-861ments and find out which one yields the minimum distance.862The method is used in [19] to find optimal hitting points on863disk boundaries and its accuracy is proportional to l.864A closer look at Fig. 4b suggests a possible reduction of865search space. Let us define the angle between lines connecting866wiai andwiþ1ai as u. For a point t on the arc outside the sector,867there is an angle b > 0 between lines wiþ1t and wiþ1ai.868Clearly, cþ d > eþ f so any point t outside the sector gives

TABLE 6Distributed Head Re-Selection Algorithm for WN i, i 2 N

MC sends HeadMsg to a WN (with enough energy), counterc 0,sets HeadMsg.hop to kðk < hÞ. Set of cluster headsH ;.If Recv(HeadMsg.ID ¼ i AND c � EcðhÞ�E�s

XcðkÞ )

dij ¼ minj2N HopCountði; jÞ (Bellman-Ford-SPT(i)),H Hþ i.Send new routing msg regarding new head i to all the nodes.Set time-out period T waiting for resumemessages.If Recv(ResumeMsg.ID ¼ i) within Tu ¼ argmaxminj2N HopCountði; jÞ, c cþ 1.Send HeadMsg to u.ElseClustering is completed and broadcast complete msg.End IfElse If Recv(NewRoutingMsg.ID is i)ANDminj2HHopCountði; jÞ > k.Send ResumeMsg to the new head.Else If Recv(NewRoutingMsg.ID is i)ANDminj2HHopCountði; jÞ � k.Send JoinMsg to u ¼ argminj2H HopCountði; jÞ,Declare as cluster member of u (Bu Bu þ i;N N � i).

Else If Recv(HeadMsg.ID ¼ i AND c >Ec�E�sXcðkÞ )

k k� 1, broadcast a restartmessage.Else Forward message according to routing entries.End If


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869 an inferior solution compared to a point within the sector of u.870 Thus, we can narrow down the search space to the points on871 the arc within angle u between wiai and wiþ1ai, so examina-872 tion of only a fraction f ¼ ul

2p points is enough. Nevertheless,873 computation complexity of exhaustive search still grows874 exponentially when there are s SNs between wi and wiþ1875 (with complexityOðfsÞ), thus a fastermethod is needed.

876 6.1.3 Minimizing Sum of Squared Distance

877 Exhaustive search quickly turns out to be impractical in878 reality. If the problem can be solved analytically, computa-879 tional complexity can be greatly reduced. Since the expres-880 sions of distance involving square roots tend to yield881 intractable computations, we minimize the sum of squared882 distance instead. In fact, sum of squared distance has been883 used in many applications such as the well-known K-means884 algorithm [36]. The estimation error to the actual sum of dis-885 tance will be evaluated by simulations. Likewise, we start886 our analysis from s ¼ 1 and derive the following property.

887 Property 6. The point p on disk ai that minimizes d ¼888 ðjwipjÞ2 þ ðjwiþ1pjÞ2 is the intersection between line wmai and889 the disk, where wm is the mid-point between the coordinates of890 wi and wiþ1.

891 Proof. See Section 10.5. tu892 Next, we consider the case of s ¼ 2without a direct cut as893 illustrated in Fig. 4c. To use our method, computing each894 touching point on a disk needs two fixed points. For two895 disks, since the touching points can change simultaneously,896 minimization of sum of distance ðaþ bþ cÞ by considering897 multiple variables is very difficult analytically. Instead, we898 use the center of aiþ1 as the reference point and calculate pi899 on disk ai to minimize ðaþ dÞ first. Then, based on pi, we900 calculate piþ1 on aiþ1 tominðbþ cÞ. In this way, each compu-901 tation only involves one variable. The method can be easily902 extended to the case when there are s SNs between wi and903 wiþ1, so a total of OðsÞ computation is needed, which904 reduces the exponential-OðfsÞ exhaustive search algorithm905 to linear time. We summarize the route improvement algo-906 rithm in Table 7.

907 6.2 Optimizing Recharging Time908 We now optimize the allocation of MCs’ recharge time for909 different WNs to avoid battery depletion. After the touring910 sequences have been found, the MCs need to traverse

911through WNs and the neighborhoods of SNs for wireless912charging and data gathering. We assume the energy913recharged into sensors’ batteries is roughly proportional to914the recharging time. In fact, most batteries exhibit such915property as the longer they are being recharged, the more916energy will be stored (under the battery capacity). An ideal917situation is to replenish all the nodes to full capacity and918hopefully no sensor depletes its battery before the recharge919begins [8]. However, due to the nontrivial recharge time,920the MCs have to reside at the WN locations for some time921(e.g., 30-78 mins). This may easily cause energy depletion of922subsequent nodes in the sequence.

9236.2.1 Linear Program Formulation

924We take a new approach that gives theMCsmore flexibility so925they can rechargemore sensors in fixed time and sustain their926battery energy at working levels. That is, instead of fully927replenishing sensors’ batteries [5], [6], [7], [8], [9], [10], we928allow partial recharge. In case a node is bound to deplete its929energy, the MC can expedite all recharge schedules before930that node.931Let us denote the touring sequence found in Section 6.1.3932by h1; 2; . . . ; i; . . . ; ni. There are two types of sojourn time at933sensor nodes. For WN i, the MC stays for a period of ti for934recharge; for SN j, the MC spends a fixed time tj to collect935data packets. tj ¼ pj=B, where pj is the amount of data at936SN j’s buffer and b is the data rate. A special case is when937the MC’s trajectory directly cuts through SN’s transmission938range (Fig. 4a). If

pjb � dc

v , tj ¼ dcv where dc is the length of

939chord within SN’s transmission range. It means that if940the data transmission time is less than MC’s traveling941time inside SN’s transmission range, the MC can collect942all the data without stopping. We denote MC’s traveling943time between two consecutive nodes in the sequence by944ti;iþ1 ¼ di;iþ1=v. Each WN i has a residual lifetime Li. Our945objective is to maximize the sum of recharge time under a946pre-determined packet delay Td. At the same time, we947should also guarantee that all the recharge requests are met948before their lifetime expirations. It can be formulated as a949Linear Programming (LP) problem,

P4 : maxXi2N

ti: (18)

951951

Fig. 4. Analysis of shortest path through feasible regions around SNs.

TABLE 7Route Improvement Algorithm for MCs

Input: Sequence hb; w1; . . . ; ai; . . . ; wj; . . . ; wn; bi.Set of SNs between wj and wjþ1, Sj, S ¼

Sj2N Sj.

Output: Coordinates ðxi; yiÞMC should visit near ai.

While Sj 6¼ ;For ai 2 Sj, find coordinates of WNs wj, wjþ1 in sequence.If aiþ1 is also between wj, wjþ1:xjþ1 ¼ xaiþ1 , yjþ1 ¼ yaiþ1 .Else ðxjþ1; yjþ1Þ is set to wjþ1’s coordinates.End IfEstablish cartesian coordinate system originated at center of ai.

xi ¼ ðxj þ xjþ1Þr=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxj þ xjþ1Þ2 þ ðyj þ yjþ1Þ2

q,

yi ¼ ðyj þ yjþ1Þr=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxj þ xjþ1Þ2 þ ðyj þ yjþ1Þ2

q.

Sj Sj � ai.EndWhile


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952 Subject to

Xj�1i¼1ðti;iþ1 þ tiÞ þ

Xi < j;i2S

ti � Lj; 2 � j � n (19)954954

955

0 < ti � Tr; 1 � i � n (20)957957

958

Xn�1i¼1

ti;iþ1 þXni¼1

ti þXi2S

ti � Td: (21)960960

961

962 Constraint (19) in fact contains ðn� 1Þ constraints and each963 one ensures that for a sensor, the sumof all the time spent dur-964 ing recharging, data gathering and traveling for all previous965 nodes in the sequence does not exceed its lifetime. Constraint966 (20) states that the maximum recharge time is to replenish the967 battery to full capacity and Constraints (21) imposes a delay968 bound Td for the entire recharge sequence.

969 6.2.2 Recharge Time Assignment

970 Although the problem can be calculated by standard LP971 solvers (e.g., using the simplex method), their worst case972 performance may take exponential time [41]. Thus, in this973 section, we develop a new algorithm by exploiting the par-974 ticular structure of the problem. Since ti;iþ1 and ti can be975 calculated (constants) once the recharge sequence has been976 determined, we can simplify Constraints (19) and (21) as,

977Pj�1

i¼1 ti < L0j,Pn

i¼1 ti < T 0d, where

L0j ¼ Lj �Xj�1i¼1

ti;iþ1 �X

i < j;i2Sti; T

0d ¼ Td �

Xn�1i¼1

ti;iþ1 �Xi2S

ti: (22)979979

980

981 In other words, L0j and T 0d represent the maximum982 recharge time from node 1 to j� 1 and the entire sequence983 respectively. The level of partial recharge is assigned984 dynamically and there are generally two cases.985 Case I: 8j 2 N , L0j ðj� 1ÞTr and T 0d nTr. In this case,986 the MC can recharge all the nodes to full capacity, i.e.,987 t�i ¼ Tr; 8i 2 N and

Pi2N t�i ¼ nTr.

988 Case II: 9j 2 N , L0j < ðj� 1ÞTr or T0d < nTr. In this case,

989 recharge time prior to jmay not take Tr so new assignments990 of recharge time should be performed. In fact, the maximi-991 zation should take the value of a node’s lifetime when its992 lifetime constraint is tight. Based on this observation and993 the iterative structure of Constraint (19), we propose a time994 assignment algorithm. It assigns recharge time proportional995 to the energy demands. The algorithm starts from the node996 with the minimum lifetime jm ¼ argminðL0j; T 0dÞ; j 2 N .997 Then for node i in the sequence (1 � i < jm), an amount of998 ti ¼ L0jmdi=ð

Pjm�1j¼1 djÞ recharge time is assigned. di is the

999 energy demand from node i. If assigned ti > Tr, ti should1000 be bounded by Tr according to Constraint (20). The remain-1001 ing time ti � Tr can be evenly distributed among the other1002 nodes from iþ 1 to jm. For the node’s lifetime, we have the1003 following property.

1004 Property 7. For the time assignment of node i prior to jm in the1005 recharge sequence i < jm, the constraint

Pi�1k¼1 tk < L0i holds.

1006 Proof. The property can be proved since

Xi�1k¼1

tk ¼ L0jmXi�1k¼1

dk=Xjm�1k¼1

dk

!< L0jm < L0i: (23)

10081008

1009 tu

1010Based on Property 7, we can see that once jm has been1011selected and recharge time is assigned for all the nodes1012before jm, the lifetime constraints still hold for these nodes.1013Hence, we can proceed to find the next node with minimum1014lifetime from ðjm þ 1Þ to n and repeat the same procedure.1015The iteration continues until the recharge sequence is1016exhausted. In most cases, the time complexity of the algo-1017rithm is Oðn2Þ since it needs to iterate through OðnÞ nodes1018and each one requires OðnÞ time assignments. In the worst1019case, the algorithm needs Oðn3Þ since an extra OðnÞ assign-1020ments are needed once the calculated recharge time exceeds1021Tr. The algorithm is summarized in Table 8.

10226.3 An Example of Proposed Hybrid Framework1023Finally, we demonstrate a complete example of the pro-1024posed framework in Fig. 5. In Fig. 5a, 8 SNs are placed to1025organize 250 WNs into clusters. Their initial locations cal-1026culated by the distributed SPP algorithm are marked by1027triangles and improved by the intra-cluster Weiszfeld1028algorithm shown as the crosses. In case of shortage of sun-1029light, Fig. 5b shows the results from the HRP algorithm to1030re-allocate cluster heads to WNs. Here, the loss in energy1031harvested from the 8 SNs can be compensated by introduc-1032ing 13 WN cluster heads to reduce hop distance. For wire-1033less charging and data gathering, Fig. 5c shows an initial1034center tour that covers 9 energy requests and 7 data1035uploading sites (SNs). The route is improved by our algo-1036rithm in Fig. 5d with a saving of 17 percent moving energy1037on the MCs.

10387 PERFORMANCE EVALUATIONS

1039In this section, we evaluate the performance of the frame-1040work by a discrete-event simulator and compare it with a1041network solely relied on wireless energy [5], [9], [11]. In the1042simulation, all the cluster heads are replenished by the MCs1043in the wireless-powered framework. N ¼ 500 nodes are uni-1044formly randomly distributed over a square field of L ¼ 1501045m. Sensors have identical transmission range of r ¼ 12 m,1046and consume es ¼ 0:05 J for generating a sensing packet and1047et ¼ er ¼ 0:02 J for transmitting/receiving a packet [23].1048Each time slot is 1 min. We have assumed an event-driven1049data generation model. Events occur at an average rate every105020 s and a packet is generated for each event. Thus, the Pois-1051son traffic rate is 3 pkt/min. WNs have battery capacity1052Ch ¼ 780 mAh and require Tr ¼ 78 mins for recharge [24].1053SNs have battery with larger capacity 2150 mAh [42]. The

TABLE 8Recharge Time Assignment Algorithm

Initialize js 1, jm ¼ argminðL0; T 0ÞWhile jm 6¼ nInitialize time variables ’i 0 (js � i � jm � 1).For i from js to jm � 1ti ¼ minðTr; L

0jmdi=Pjm�1

j¼js dj þ ’iÞ.If ti ¼ Tr, for iþ 1 � k � jm � 1

’k ’k þ ðL0jmdi=Pjm�1

i¼js di � TrÞ=ðjm � i� 1ÞEndEnd Forjs jm, update L0 fL0jmþ1; . . . ; L0n; T 0gFind the next, jm ¼ argminðL0; T 0ÞEndWhile


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1054 MC has battery packs and weighs around 20 lbs. Based on1055 the calculations from [43], the MC consumes energy at a rate1056 of 5 J/mwhilemoving at v ¼ 1m/s.We use real meteorolog-1057 ical trace at Stony Brook, NY from [39], which has a complete1058 archive of weather conditions. The simulation time starts1059 fromDecember and lasts for 12 months. The results are aver-1060 aged over a number of 50 runs.

1061 7.1 Evaluation of Algorithms1062 First, we evaluate the performance of the proposed1063 algorithms.

1064 7.1.1 Extended Weiszfeld Algorithm

1065 We validate the convergence property of the extended1066 Weiszfeld algorithm for finding the SN locations in continu-1067 ous space. For clarity, we randomly pick 4 SNs and trace1068 the evolution of their total costs in Fig. 6a. We normalize the1069 total cost for better visualization. First, we observe that the1070 Weiszfeld algorithm can converge extremely fast (within 5-1071 6 iterations) and offer an additional 2-4 percent energy sav-1072 ing. Second, since the deploying cost is usually much less1073 than the routing cost in our problem, the total costs are1074 dominated by the changes in the routing costs. Therefore,1075 the optimal values reach the minimum when the algorithm1076 converges. Otherwise, the algorithm will use the minimum1077 total cost during the iteration process as the final solution1078 (explained in Section 4.2).

1079 7.1.2 Route Improvement Algorithm

1080 We evaluate the route improvement algorithm by compar-1081 ing it with the algorithms proposed in [5], [40]. The algo-1082 rithm in [40] continuously finds the closest hitting points on1083 the boundaries of the disks and we call it nearest insertion1084 algorithm. The tour passing through the disk centers is used1085 in [5] for joint wireless charging and data gathering and we1086 denote it by initial center tour. Fig. 6b compares MC’s

1087moving energy using the three methods. First, we can see1088that our algorithm provides an average of 25 percent energy1089saving compared to the initial center tour. In fact, more1090energy saving can be achieved with a larger transmission1091range since an MC only needs to visit the transmission1092boundaries for gathering data. Second, the results further1093indicate 5-7 percent improvements over the nearest inser-1094tion algorithm [40]. This is because that selecting the closest1095hitting point on a disk cannot guarantee that the sum of dis-1096tance to the neighboring nodes is minimal. In contrast, our1097algorithm finds a point on the disk that minimizes the sum1098of squared distance. To examine the gap between minimiz-1099ing the sum of squared distance and the actual distance, we1100conduct more evaluations in Fig. 6c by considering a joint1101route comprised of WNs and SNs. Since WNs outnumber1102SNs by a considerable amount, we maintain a 10 to 1 ratio1103betweenWNs and SNs. To provide a baseline, an exact solu-1104tion is found by exhaustive search using [19].1105Hence, we examine this approximation ratio by simula-1106tions in Fig. 6c, which shows “surprising” results that mini-1107mizing the sum of square distance has only 1 percent1108difference compared to the sum of actual distance in the joint1109data gathering and recharge problem. Recall that our route1110improvement algorithm (Table 7) has made an approxima-1111tion by taking the mid-point which minimizes the sum of1112square distance of route segments. Surprisingly, this algo-1113rithm only has an average of 1 percent difference to the solu-1114tion found by exhaustive search. We found it surprising1115because our linear-time algorithm achieves almost the same1116results as the exponential exhaustive searchmethod.1117In addition, for mixed WNs and SNs in a joint tour, our1118algorithm enjoys even more improvement. This is because1119that for the touring sequence that only consists of SNs, we1120have taken further approximation. As shown in Fig. 4c,1121when there are more than 2 consecutive SNs in the touring1122sequence, we take the center of the second circle aiþ1 as a

Fig. 5. A complete example of the framework. (a) Placement of SNs. (b) Restoring energy balance by WNs. (c) Initial center tour. (d) Optimized jointtour with MC’s stopping time.

Fig. 6. Evaluation of the proposed algorithms. (a) Convergence of Weiszfeld algorithm. (b) Improvements of tour for only data gathering sites (SNs).(c) Improvements of joint wireless charging and mobile data gathering tour. (d) Average approximation ratios of the recharge time assignmentalgorithm.


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roof1123 reference point while calculating the hitting point on the

1124 first circle ai whereas the optimal reference point should be1125 on the arc of the second circle. Thus, more consecutive SNs1126 may introduce some inaccuracies. However, for the joint1127 tour, we have at most 2-3 consecutive SNs in our test so the1128 performance is better than Fig. 6b.

1129 7.1.3 Recharge Time Assignment Algorithm

1130 We also evaluate the performance of the recharge time assign-1131 ment algorithm and compare with optimal solutions from1132 standard LP solver. Recharge sequences from 5 to 100 nodes1133 are evaluated with nodes’ battery energy and lifetime ran-1134 domly distributed. The results are averaged over 100 simula-1135 tion runs. Fig. 6d illustrates difference between our algorithm1136 and the optimal solution. We observe that our algorithm is1137 able to achieve results very close (within 5 percent) to the opti-1138 mal LP solver. As we increase the number of nodes in the1139 sequence, the approximation ratio does not degrade as indi-1140 cated by the flat trend line. In the simulation, we discover that1141 in most cases, our algorithm can find the optimal solution.1142 However, in some special cases, there are several consecutive1143 nodes with limited lifetimes that require partial recharge of1144 previous nodes. Our algorithm conducts one recharge time1145 assignment at a time until Constraints (19) and (21) for all the1146 nodes are satisfied. In contrast, the optimal LP solver consid-1147 ers all the nodes together and assigns recharge time in one1148 shot. For these cases, our approachmay not always yield opti-1149 mal results but remains very close to them.

1150 7.2 Nonfunctional Nodes1151 One of the key performance metrics for recharging is non-1152 functional nodes. Once a node’s battery is depleted, it stops1153 working and becomes nonfunctional until its battery is1154 replenished. To sustain perpetual operations, nodes should1155 be alive all the time; otherwise, they will degrade sensing1156 qualities and node communications. Fig. 7 compares the1157 percentage of nonfunctional nodes between hybrid (full or

1158partial recharge) and wireless-powered frameworks [5], [9],1159[11]. The SPP algorithm generates s ¼ 11 SNs. To compare1160the performance, we change the number of MCs m. Fig. 7a1161shows the results from the hybrid framework when1162m ¼ 2 4. We can see that 2 MCs can keep the percentage1163of nonfunctional nodes around 10 percent and 4 MCs can1164almost achieve perpetual operations. Partial recharge offers1165even better performance with less than 10 percent nonfunc-1166tional nodes whenm ¼ 2.1167In contrast, m ¼ 2 for wireless-powered network results1168in 30 percent nonfunctional nodes in Fig. 7b and an increase1169tom ¼ 6 still barely eliminates all battery depletions at equi-1170librium. These observations clearly demonstrate that the1171hybrid framework can improve network performance sig-1172nificantly. Since for a wireless-powered network, cluster1173heads consume energy much faster, MCs need to visit them1174more frequently, which reduces the chances for other nodes1175to get recharged. However, SNs are replenished by solar1176energy which has much higher power density so MCs have1177more leverage to take care of the rest of the network.

11787.3 System Cost1179The cost to maintain network operations is another impor-1180tant performance metric. In our framework, there are two1181types of costs. The first type comes from network mainte-1182nance which basically involves energy expenditures at the1183MCs while moving and recharging. Since the energy1184expense for recharging is necessary for sensor nodes, we1185focus on the energy cost while the MCs are moving. Figs. 8a1186and 8b trace the evolution of MCs’ moving energy cost1187when our goals are maintaining nonfunctional percentage1188under 15 and 1 percent, respectively. First, let us examine1189the cost brought by partial recharge. Although it is not obvi-1190ous on Fig. 8a, by taking their mean values, we are able to1191see that partial recharge results in slightly higher cost (3.571192KJ versus 3.49 KJ). This is because that partial recharge1193allows MCs to recharge more nodes in a fixed time period1194(MCs move more frequently). Second, we compare the mov-1195ing cost to wireless-powered networks. Since it requires1196more MCs to maintain energy balance, their moving costs1197inevitably increase which are almost doubled if evaluated1198by their mean values (mean 6.91 KJ versus 3.49 KJ of the1199hybrid network). Similar results are observed in Fig. 8b1200when the objective is to maintain nonfunctional percentage1201below 1 percent. The difference is that the gap between the1202two curves becomes even wider (mean value 36 KJ versus1203105 KJ), which indicates that the wireless-powered network1204almost needs 3 times the energy of the hybrid network.1205From these results, we can see that hybrid networks are1206more energy efficient for higher performance requirements.

Fig. 7. Number of nonfunctional nodes. (a) Hybrid framework. (b)Wireless-powered framework.

Fig. 8. Evaluation of system cost. (a) Evolution of MC’s moving cost when nonfunctional rate is less than 15 percent. (b) Evolution of MC’s movingcost when nonfunctional rate is less than 1 percent. (c) Energy efficiency of MCs. (d) Trade-offs between performance and system cost.


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roof1207 We have also evaluated energy efficiency for different

1208 numbers ofMCs in Fig. 8c. Energy efficiency is defined as the1209 ratio between energy replenished and the total energy con-1210 sumed on the MCs. The hybrid network enjoys the highest1211 overall energy efficiency as more than 90 percent energy can1212 be used for rechargewhile partial recharge has slightly lower1213 efficiency due to possibly more movements. The wireless-1214 powered network has the lowest energy efficiency since the1215 MCs have tomove evenmore often to recharge cluster heads.1216 The second type of cost is the fixed cost of SNs and MCs.1217 The theoretical results in Section 5 indicate that by having1218 more SNs, we are able to save the cost on MCs since they are1219 usually more expensive. To see the trade-offs between the1220 numbers of SNs and MCs, we show the impact on nonfunc-1221 tional percentage in Fig. 8d in which the X and Y axes repre-1222 sent the numbers of SNs and MCs and the Z axis represents1223 the percentage of nonfunctional nodes. We observe that1224 introducing SNs is more cost-effective to reduce nonfunc-1225 tional percentage compared to MCs. For example, when we1226 have 1 MC and 1 SN, having 1 more SN helps reduce non-1227 functional percentage from 23 to 13 percent while having 11228 moreMC still results in over 20 percent nonfunctional nodes.1229 With more SNs, even fewer MCs can achieve similar perfor-1230 mance. For example, 8 SNs and 1-2MCs result in similar per-1231 formance to 6-8 MCs and 1 SN. These results suggest that1232 hybrid networks are more cost-effective and energy efficient1233 thanwireless-powered networks.

1234 7.4 Harvested Energy and Message Overhead1235 To validate our algorithm design, we evaluate the evolution1236 of SN’s energy and networkmessage overhead. Fig. 9a traces1237 SNs’ energy with weather conditions represented by per-1238 centage of solar exposure (1� s) obtained in [39]. We focus1239 on two typical nodes with light and heavy data traffic. We1240 can see that through the month of December, energy storage1241 continuously declines due to weak solar strength in winter.1242 In addition, there are also several consecutive snowing days1243 so SNs are unable to harvest enough energy and re-selection1244 of cluster heads among WNs is needed. This gives SNs1245 opportunities to recover their energy (during 40-50 days).1246 For the remaining simulation, although a few consecutive1247 raining days are observed, the energy gaps are quickly filled.1248 This is because that solar radiation has strengthened since1249 spring and energy storage is sufficient to sustain network1250 operations. For example, from summer to fall, SNs canmain-1251 tain their battery energy over 80 percent due to favorable1252 weather conditions and strong solar radiation.1253 Fig. 9b demonstrates the energy consumed by exchang-1254 ing control messages such as SN clustering, re-selection of1255 WN cluster heads and energy information requests. For

1256each month, MCs initiate a new calculation of SN’s place-1257ment pattern to reflect the updated geographical solar radia-1258tions (e.g., Fig. 2). These message overhead is shown as the1259blue spikes at the beginning of each month. Note that in the1260simulation, when SNs’ energy is less than 30 percent due to1261insufficient solar energy, they request to re-cluster by WNs.1262The overhead is represented by those higher spikes (in pur-1263ple), which corresponds to the time when SNs’ energy1264drops in Fig. 9a. Our results indicate that re-clustering using1265WNs only occur during winter time though there are some1266consecutive raining periods in other seasons. From Fig. 9,1267we have validated that our algorithm can adapt to weather1268conditions effectively.

12697.5 Geographical Distributions of Service1270Interruption1271Finally, we examine geographical distributions of service1272interruptions. Our objective is to see how long nodes are in1273nonfunctional status and their geographical distributions.1274Since cluster heads are responsible for aggregating and1275uploading sensed data, their survivals are critical for the1276entire network. A breakdown may lead to severe packet1277loss, network interruption and extended data latency. For1278fair comparison, we use the results from Section 7.2 and set1279m ¼ 2 for the hybrid framework and m ¼ 4 for the wireless-1280powered framework so that both cases have a similar num-1281ber of nonfunctional nodes. From Fig. 10a, we observe that1282the distribution of nonfunctional nodes is quite even in the1283hybrid framework. In contrast, nodes around the cluster1284heads (including the heads as well) are more prone to1285deplete battery energy in the wireless-powered network (201286percent more nonfunctional time in Fig. 10b). On average, a1287node in the hybrid framework has only 8.5 percent time in1288nonfunctional status whereas it would experience 16 per-1289cent nonfunctional time in the wireless-powered network.1290The sharp contrast is because that for the wireless-powered1291network, MCs need to not only take care of cluster heads1292but also their surrounding areas. This may cause the MCs to1293move frequently between head locations and overwhelm1294their recharge capabilities. However, for the hybrid frame-1295work, MCs do not need to recharge SNs so the resources1296can be re-distributed among WNs to reduce their nonfunc-1297tional rates.

12988 DISCUSSIONS

1299We now discuss a few interesting issues worth future study.1300First, for tractability, our solution breaks the optimization1301problem into many sub-problems. Indeed, system cost can1302be improved in a cross-layer fashion. For example, the

Fig. 9. Evolution of harvested solar energy and message energy over-head. (a) Energy variations of SNs. (b) Message energy overhead.

Fig. 10. Geographical distributions of service interruption. (a) Hybridframeworkm ¼ 2. (b) Wireless-powered frameworkm ¼ 4.


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1303 planning of SNs could be associated with route manage-1304 ment for the MC to reduce system cost. Such cross-layer1305 optimization brings variables from different aspects to1306 determine the number, allocation, location of sensors1307 together with route planning of the MC. Second, whether1308 the MC should visit the exact SN locations can depend on1309 the SN’s energy. In case it has low energy income, the MC1310 should move close to avoid draining sensor’s battery. We1311 plan to study these interesting problems in the future.1312 Second, a fixed time period has been used to adjust SN’s1313 locations. A more efficient solution is to develop an adap-1314 tive scheme that can reflect solar variations with minimal1315 overhead. If micro-climate solar logs can be obtained, we1316 can predict spatial-temporal energy distributions based on1317 historical data. Then we can schedule re-location more effi-1318 ciently. However, these data are usually available at low1319 resolutions (up to 10 km) [44]. Thus, in our future work, we1320 will exploit the MC for gathering such solar data.

1321 9 CONCLUSION

1322 In this paper, we consider a hybrid framework that com-1323 bines the advantages of wireless charging and solar energy1324 harvesting technologies. We study a three-level network1325 consisting of SNs, WNs and MCs levels. First, we study1326 how to minimize the total cost of deploying a set of SNs.1327 The problem is formulated into a facility location problem1328 and a 1:61ð1þ �2Þ-factor distributed algorithm is proposed.1329 The solution is further improved by using intra-cluster1330 Weiszfeld algorithm in continuous space. Second, we exam-1331 ine the energy balance in the network and develop a distrib-1332 uted head re-selection algorithm to designate some WNs as1333 cluster heads when solar energy is not available during1334 raining/cloudy days. Third, we focus on how to optimize1335 the joint tour consisting of both wireless charging and data1336 gathering sites for the MCs. A linear-time algorithm is pro-1337 posed that can approach very closely to the exact solution1338 and reduce at least 5 percent MC’s moving energy com-1339 pared to previous solutions. We also propose to partially1340 refill sensors’ energy to further reduce battery depletion1341 and develop an efficient algorithm to solve the problem1342 with high accuracy. Finally, based on real weather data, we1343 demonstrate through simulations the effectiveness and effi-1344 ciency of the hybrid framework that can improve network1345 performance significantly.

1346 10 PROOFS OF PROPERTIES

1347 10.1 Proof of Property 11348 We sequentialize the distributed algorithm into execution1349 rounds. For the distributed algorithm, each round consists of1350 a number of message sending and receiving by respective1351 WNs and SNs. In each round, the total offers received1352 from all the nodes are

Pj2N aj ¼

Pj2N cij þ fi. For some

1353 nodes already connected, the new offers areP

j2N aj ¼1354

Pj2N ci0j � cij, which should be deducted from the total

1355 offers to reflect the adjusted value. We can see that this result1356 is exactly the term in (5). Since the offer value is increased at1357 a rate ð1þ �Þ, an SN that meets the lowest total offer will be1358 selected in the earliest time, which is equivalent to selecting1359 the least average cost in the centralized algorithm. Therefore,1360 we can see that the mechanism of the distributed algorithm1361 is analogous to the centralized algorithm in [28].

136210.2 Proof of Property 31363Our proof of Property 3 is based on [28]. First, denote opti-1364mal offers in the centralized algorithm [28] by aj and the1365distributed algorithm by a0j, j 2 N . A Factor Revealing LP is1366constructed by [28]. For SN i, k ¼ jBij, the optimal solution1367is to solve the following maximization problem

P3 : zk ¼ maxXkj¼1

aj Xk

j¼1cij þ fi

!(24)

13691369

1370Subject to

aj � ajþ1; 8j 2 f1; 2; . . . ; k� 1g (25)13721372

1373

Xkj¼1

maxðaj � cil; 0Þ � fi; 8l 2 f1; 2; . . . ; kg (26)

13751375

1376aj � al þ cij þ cil; 8j; l 2 f1; 2; . . . ; kg (27) 13781378

1379

aj; cij; cil; fi 0; 8j; l 2 f1; 2; . . . ; kg: (28)13811381

1382For the maximization problem to be bounded, aj should1383also be bounded. It implies that at least one of the con-1384straints of (26) and (27) is tight (i.e., changing from inequal-1385ity into equality). Case 1: Eq. (26) is tight; Case 2: Eq. (27) is1386tight, thus al is also bounded.1387For the distributed algorithm, we can formulate it into a1388similar Factor Revealing LP except that constraint (26)

1389becomesPk

j¼1 maxð a0j

1þ�� cil; 0Þ � fi and constraint (27)

1390becomesa0j

1þ� � a0l þ cij þ cil since increasing offers at the1391same pace in the centralized scheme would deploy SN i at1392most ð1þ �Þ time earlier compared to the distributed algo-1393rithm. For Case 1, since fi 0 and the constraint is tight,

1394aj � cil 0 anda0j

1þ�� cil 0 hold for the centralized and

1395distributed algorithms, respectively. Thus,a0j

aj� 1þ �. For

1396Case 2:a0j1þ� ¼ a0l þ cij þ cil and

Pkl¼1 maxð a

0l

1þ�� cil; 0Þ � fi is1397also tight to bound a0l. The latter suggests that the ratio in

1398Case 1a0l

al� 1þ � can be applied here

a0j1þ �

� alð1þ �Þ þ cij þ cil � ð1þ �Þðal þ cij þ cilÞ: (29)

14001400

1401Then by taking the ratio of Eqs. (29) to (27), we have

1402a0j

aj� ð1þ �Þ2 for Case 2. Since the approximation ratio to the

optimal solution a�j is proved by [28],aja�j� 1:61. Thus, our

1403algorithm has at mosta0j

a�j� 1:61ð1þ �Þ2 approximation to the

optimal solution.

140410.3 Proof of Property 41405By assumption, for SNs to successfully aggregate and trans-1406mit data, E�s sh2r2pr�ðet þ erÞT . Plugging this into1407Eq. (14), we have

mCh

Tr>h� 2

3h3 � 3

2h2 � 1

6h�ðet þ erÞ þ h2ðet þ esÞ

ipr2r�s:

(30)14091409

1410For h > 2, 23h3 � 3

2h2 � 1

6h > 0 so mChTr

> h2r2prðet þ esÞ�s >

1411Nðet þ esÞ�. Nðet þ esÞ� is exactly the energy consumed by

sensors to generate and transmit data in one-hop communi-

cation to the MCs and the inequality states that the


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recharging rates from MCs is enough to support one-hopcommunications. Thus, we have proved that for h > 2, in

the worst case, we can always use one-hop mobile data

gathering to restore energy balance.

1412 10.4 Proof of Property 51413 Our proof is based on [38]. Since SNs can be represented by1414 disjoint disks, the sum of feasible areas for s SNs is spr2. We1415 consider a larger disk of radius 2r so any point in disk of1416 radius r can be enclosed. The total area spr2 should be less1417 than the area swept by the disk of 2r,

spr2 þ npr2w � 4rL�r þ 4r2p: (31)14191419

1420 The wireless charging range rw is much smaller than r1421 (rw � r). If we enforce the MC to go through disk centers,1422 an extra distance less than 2r has to be made (entering and1423 leaving the center). Thus, Lc is bounded by

Lc � L�r þ 2rs � L�r þ 2r4rL�r þ 4r2p� npr2w

pr2

Lc

L�r� 1þ 8

p

þ 8r

L�r� 1þ 8

pþ �:

(32)

14251425

1426 In the first step, we use Eq. (31) for s. In the last step, we omit1427 the last term nðrwr Þ2, as rw

r � 0. Since r is much smaller than1428 L�r , we denote 8r

L�r� �where � is a small fraction close to 0.

1429 10.5 Proof of Property 61430 Although the property seems to be true by visual judgment1431 of Fig. 4b, a geometric proof is difficult. Thus, we calculate p1432 in terms of cartesian coordinates. Denote coordinates of1433 wi; wiþ1 and p as ðxi; yiÞ; ðxiþ1; yiþ1Þ; ðx; yÞ, respectively.1434 Assume the origin of coordinate system resides at the disk1435 center. The function of the disk is x2 þ y2 ¼ r2. We use the1436 Lagrangian multipliermethod to findminimal sum of squared1437 distance. After taking partial derivatives, the variables are

Lx

@x¼ 4x� 2ðxi þ xiþ1Þ þ 2x�;

Ly

@y¼ 4y� 2ðyi þ yiþ1Þ þ 2y�

L�

@�¼ x2 þ y2 � r2:

(33)

14391439

1440 After some calculations, the coordinates for p are

x ¼ ðxi þ xiþ1Þrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi þ xiþ1Þ2 þ ðyi þ yiþ1Þ2

q ;

y ¼ ðyi þ yiþ1Þrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi þ xiþ1Þ2 þ ðyi þ yiþ1Þ2

q :

(34)

14421442

1443 On the other hand, the coordinates of wm are (xiþxiþ1

2 ;yiþyiþ1

2 ).

1444 We plug the function of line wmai, y ¼ yiþyiþ1xiþxiþ1 x into the disk

1445 function of ai, and obtain two intersection points

x ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2

ðyiþyiþ1xiþxiþ1Þ2 þ 1

vuut ; y ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r2

ðxiþxiþ1yiþyiþ1Þ2 þ 1

vuut : (35)

1446 We can see one of the solutions in Eq. (35) is exactly1447 Eq. (34), so the property is proved.

1448ACKNOWLEDGMENTS

1449The work in this paper was supported in part by the grant1450from the US National Science Foundation under grant1451number ECCS-1307576.

1452REFERENCES

1453[1] Powermat Technology. [Online]. Available: http://powermat.com/1454[2] UBeam Technology. [Online]. Available: http://ubeam.com/1455[3] FCC Rules for ISM bands. [Online]. Available: http://www.afar.1456net/tutorials/fcc-rules1457[4] PowerCast Corp. [Online]. Available: http://www.powercastco.1458com1459[5] M. Zhao, J. Li, and Y. Yang, “A framework of joint mobile energy1460replenishment and data gathering in wireless rechargeable sensor1461networks,” IEEE Trans. Mobile Comput., vol. 13, no. 12, pp. 2689–14622705, Dec. 2014.1463[6] B. Tong, Z. Li, G. Wang, and W. Zhang, “How wireless power1464charging technology affects sensor network deployment and1465routing,” in Proc. IEEE 30th Int. Conf. Distrib. Comput. Syst., 2010,1466pp. 438–447.1467[7] Y. Peng, Z. Li, W. Zhang, and D. Qiao, “Prolonging sensor net-1468work lifetime through wireless charging,” in Proc. 31st IEEE Real-1469Time Syst. Symp., 2010, pp. 129–139.1470[8] C. Wang, J. Li, F. Ye, and Y. Yang, “NETWRAP: An NDN based1471real-time wireless recharging framework for wireless sensor1472networks,” IEEE Trans. Mobile Comput., vol. 13, no. 6, pp. 1283–14731297, Jun. 2014.1474[9] C. Wang, J. Li, F. Ye, and Y. Yang, “A mobile data gathering1475framework for wireless rechargeable sensor networks with vehi-1476cle movement costs and capacity constraints,” IEEE Trans. Com-1477put., vol. 65, no. 8, pp. 2411–2427, Aug. 2016.1478[10] Z. Li, Y. Peng, W. Zhang, and D. Qiao, “J-RoC: A joint routing and1479charging scheme to prolong sensor network lifetime,” in Proc.148019th IEEE Int. Conf. Netw. Protocols, 2011, pp. 373–382.1481[11] S. He, J. Chen, F. Jiang, D. Yau, G. Xing, and Y. Sun, “Energy pro-1482visioning in wireless rechargeable sensor networks,” IEEE Trans.1483Mobile Comput., vol. 12, no. 10, pp. 1931–1942, Oct. 2013.1484[12] H. Dai, Y. Liu, G. Chen, X. Wu, and T. He, “SCAPE: Safe charging1485with adjustable power,” in Proc. IEEE 34th Int. Conf. Distrib. Com-1486put. Syst., 2014, pp. 439–448.1487[13] S. Nikoletseas, R. Theofanis, and R. Christoforos, “Low radiation1488efficient wireless energy transfer in wireless distributed systems,”1489in Proc. IEEE 35th Int. Conf. Distrib. Comput. Syst., 2015, pp. 196–1490204.1491[14] A. Kansal, J Hsu, S. Zahedi, and M. Srivastava, “Power manage-1492ment in energy harvesting sensor networks,” ACM Trans. Embed-1493ded Comput. Syst., vol. 6, no. 4, 2007, Art. no. 32.1494[15] B. Gaudette, V. Hanumaiah, M. Krunz, and S. Vrudhula,1495“Maximum quality of cover with connectivity in solar powered1496active wireless sensor networks,” ACM Trans. Sensor Netw.,1497vol. 10, no. 4, pp. 1–59, 2014.1498[16] R. Margolies, et al., “Energy harvesting active networked tags1499(EnHANTs): Prototyping and experimentation,” ACM Trans. Sen-1500sor Netw., vol. 11, no. 4, pp. 1–62, Nov. 2015.1501[17] R. S. Liu, K. W. Fan, Z. Zheng, and P. Sinha, “Perpetual and fair1502data collection for environmental energy harvesting sensor1503networks,” IEEE/ACM Trans. Netw., vol. 19, no. 4, pp. 947–960,1504Aug. 2011.1505[18] M. Ma, Y. Yang, and M. Zhao, “Tour planning for mobile data-1506gathering mechanisms in wireless sensor networks,” IEEE Trans.1507Veh. Technol., vol. 62, no. 4, pp. 1472–1483, May 2013.1508[19] B. Yuan, M. Orlowska, and S. Sadiq, “On the optimal robot rout-1509ing problem in wireless sensor networks,”IEEE Trans. Knowl. Data1510Eng., vol. 19, no. 9, pp. 1252–1261, Sep. 2007.1511[20] L. He, J. Pan, and J. Xu, “A progressive approach to reducing data1512collection latency in wireless sensor networks with mobile ele-1513ments,” IEEE Trans. Mobile Comput., vol. 12, no. 7, pp. 1308–1320,1514Jul. 2013.1515[21] V. Raghunathan, A. Kansal, J. Hsu, J. Friedman, andM. Srivastava,1516“Design considerations for solar energy harvesting wireless1517embedded systems,” in Proc. 4th Int. Symp. Inf. Process. Sensor1518Netw., 2005, pp. 457–462.1519[22] V. Rai and R. N. Mahapartra, “Lifetime modeling of a sensor1520network,” in Proc. Des. Autom. Test Europe, 2005, pp. 202–203.


http://powermat.com/

http://ubeam.com/

http://www.afar.net/tutorials/fcc-rules

http://www.afar.net/tutorials/fcc-rules

http://www.powercastco.com

http://www.powercastco.com

IEEE P

roof

1521 [23] W. R. Heinzelman, A. Chandrakasan, and H. Balakrishnan,1522 “Energy-efficient communication protocol for wireless microsen-1523 sor networks,” in Proc. 33rd Annu. Hawaii Int. Conf. Syst. Sci., 2000,1524 Art. no. 10.1525 [24] Panasonic Ni-MH battery handbook. [Online]. Available: http://1526 www2.renovaar.ee/userfiles/Panasonic_Ni-MH_Handbook.pdf1527 [25] X. Wu, G. Chen, and S. Das, “Avoiding energy holes in wireless1528 sensor networks with nonuniform node distribution,” IEEE Trans.1529 Parallel Distrib. Syst., vol. 19, no. 5, pp. 710–720, May 2008.1530 [26] D. Shmoys, E. Tardos, and K. Aardal, “Approximation algorithms1531 for facility location problems,” in Proc. 29th Annu. ACM Symp. The-1532 ory Comput., 1997, pp. 265–274.1533 [27] K. Jain and V. Vazirani, “Approximation algorithms for metric1534 facility location and k-median problems using the primal-dual1535 schema and Lagrangian relaxation,” J. ACM, vol. 48, no. 2,1536 pp. 274–296, 2001.1537 [28] K. Jain, M. Mahdian, E. Markakis, A. Saberi, and V. Vazirani,1538 “Greedy facility location algorithms analyzed using dual fitting1539 with factor-revealing LP,” J. ACM, vol. 50, no. 6, pp. 795–824, 2003.1540 [29] S. Guha and S. Khuller, “Greedy strikes back: Improved facility1541 location algorithms,” J. Algorithms, vol. 31, no. 1, pp. 228–248,1542 1999.1543 [30] E. Weiszfeld and F. Plastria, “(Translated version) On the point for1544 which the sum of the distances to n given points is minimum,”1545 Ann. Operations Res., vol. 167, no. 1, pp. 7–41, 2009.1546 [31] H. Kuhn, “A note on Fermat’s problem,” Math. Program., vol. 4,1547 no. 1, pp. 98–107, 1973.1548 [32] J. Eyster, J.White, andW.Wierwille, “On solvingmultifacility loca-1549 tion problems using a hyperboloid approximation procedure,” IIE1550 Trans., vol. 5, no. 1, pp. 1–6, 1973.1551 [33] N. Sharma, J. Gummeson, and D. Irwin, “Cloudy computing:1552 Leveraging weather forecasts in energy harvesting sensor sys-1553 tems,” in Proc. 7th Annu. IEEE Commun. Soc. Conf. Sensor Mesh Ad1554 Hoc Commun. Netw., 2010, pp. 1–9.1555 [34] A. Amis, R. Prakash, T. Vuong, and D. Huynh, “Max-min d-cluster1556 formation in wireless ad hoc networks,” in Proc. IEEE INFOCOM,1557 2000, pp. 32–41.1558 [35] T. Gonzalez, “Clustering to minimize the maximum intercluster1559 distance,” Theoretical Comput. Sci., vol. 38, pp. 293–306, 1985.1560 [36] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduc-1561 tion to Algorithms. Cambridge, MA, USA: MIT Press, 2001.1562 [37] S. Shmuel and O. Schwartz, “On the complexity of approximating1563 TSP with neighborhoods and related problems,” Comput. Complex-1564 ity, vol. 14, pp. 281–307, 2006.1565 [38] A. Dumitrescu and J. Mitchell, “Approximation algorithms for1566 TSP with neighborhoods in the plane,” in Proc. 12th Anu. ACM-1567 SIAM Symp. Discrete Algorithms, 2001, pp. 38–46.1568 [39] Weather underground. [Online]. Available: www.wunderground.1569 com/history/1570 [40] K. Elbassioni, A. Fishkin, N. Mustafa, and R. Sitters,1571 “Approximation algorithms for euclidean group TSP,” in Proc.1572 Int. Colloq. Automata Languages Program., 2005, pp. 1115–1126.1573 [41] C. H. Papadimitriou and K. Steiglitz, Combinatorial Optimization:1574 Algorithms and Complexity. Mineola, NY, USA: Dover Publications,1575 1998.1576 [42] Lenmar AA battery. [Online]. Available: http://www2.lenmar.1577 com1578 [43] Battery Calculator. [Online]. Available: http://www.evsource.1579 com/battery_calculator.php1580 [44] Solar GIS. [Online]. Available: http://www.nrel.gov/gis/data_1581 solar.html

1582 Cong Wang receivedQ2 the BEng degree in infor-1583 mation engineering from the Chinese University1584 of Hong Kong, the MS degree from Columbia1585 University, and the PhD degree in electrical engi-1586 neering from Stony Brook University, in 2017. He1587 will join Old Dominion University as an assistant1588 professor in the ECE Department in fall 2017. His1589 research interests include wireless sensor net-1590 works and performance evaluation of network1591 protocols and algorithms.

1592Ji Li received the BS degree in electrical engi-1593neering from Harbin Engineering University,1594Harbin, China, and the MS degree from Zhejiang1595University, Hangzhou, China. He is currently1596working toward the PhD degree in electrical and1597computer engineering at Stony Brook University.1598His research interests include wireless sensor1599networks and embedded systems.

1600Yuanyuan Yang received the BEng and MS1601degrees in computer science and engineering1602from Tsinghua University and the MSE and PhD1603degrees in computer science from Johns Hopkins1604University. She is a SUNY distinguished profes-1605sor of computer engineering and computer sci-1606ence and the associate dean for Academic1607Affairs in the College of Engineering and Applied1608Sciences, Stony Brook University, New York. Her1609research interests include wireless networks,1610data center networks, and cloud computing. She1611has published more than 360 papers in major journals and refereed con-1612ference proceedings and holds seven US patents in these areas. She is1613currently the associate editor-in-chief of the IEEE Transactions on Cloud1614Computing and an associate editor of the ACM Computing Surveys. She1615has served as an associate editor-in-chief and associated editor of the1616IEEE Transactions on Computers and associate editor of the IEEE1617Transactions on Parallel and Distributed Systems. She has also served1618as a general chair, program chair, or vice chair for several major confer-1619ences and a program committee member for numerous conferences.1620She is a fellow of the IEEE.

1621Fan Ye received the BE and MS degrees from1622Tsinghua University and the PhD degree from1623the Computer Science Department, UCLA. He is1624an assistant professor in the ECE Department,1625Stony Brook University. He has published more1626than 60 peer reviewed papers that have received1627more than 8,000 citations according to Google1628Scholar. He has 21 granted/pending US and1629international patents/applications. He was the co-1630chair for the Mobile Computing Professional1631Interests Community at IBM Watson for two1632years. He received IBM Research Division Award, five Invention1633Achievement Plateau Awards, and the Best Paper Award for Interna-1634tional Conference on Parallel Computing 2008. His current research1635interests include mobile sensing platforms, systems and applications,1636Internet-of-Things, indoor location sensing, wireless, and sensor1637networks.

1638" For more information on this or any other computing topic,1639please visit our Digital Library at www.computer.org/publications/dlib.


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http://www.evsource.com/battery_calculator.php

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