Cooperative Dynamic Game-Based Optimal Power Control in Wireless Sensor Network Powered by RF Energy

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1 sensors Artcle Cooperatve Dynamc Game-Based Optmal Power Control n Wreless Sensor etwork Powered by RF Energy Manx Wang 1, Hatao Xu 2, * ID and Xanwe Zhou 2 1 State Key Laboratory of Complex Electromagnetc Envronment Effects on Electroncs and Informaton System, Luoyang , Chna; cemee@vp.163.com 2 School of Computer and Communcaton Engneerng, Unversty of Scence and Technology Bejng, Bejng , Chna; xwzhoul@sna.com * Correspondence: xuhatao@ustb.edu.cn; Tel.: Receved: 1 June 2018; Accepted: 16 July 2018; Publshed: 23 July 2018 Abstract: Ths paper focuses on optmal power control n wreless sensor networks powered by RF energy, under the smultaneous wreless nformaton and power transfer SWIFT protocol, where the nformaton and power can be transmtted at the same tme. We am to maxmze the utlty for each sensor through the optmal power control, consderng the nfluences of both the SIR and the harvested energy. The utlty maxmzaton problem s formulated as a cooperatve dynamc game of a gven tme duraton. All the sensors cooperate together to control ther transmsson power to maxmze the utlty and agree to act cooperatvely so that a team optmum can be acheved. As a result, a feedback ash equlbrum soluton for each sensor s gven based on the dynamc programmng theory. Smulaton results verfy the effectveness of the proposed approach, by comparng the grand coalton solutons wth the non-cooperatve solutons. Keywords: cooperatve dynamc game; power control; wreless sensor network; RF energy transfer 1. Introducton As an mportant component of wreless networks, wreless sensor networks have drawn lots of academc and ndustral research nterest for a long tme, as wreless sensor networks wth sensng, computaton, and communcaton capabltes can work autonomously [1]. More and more sensor nodes are arranged to consttute the wreless sensor networks to realze the concept of the Internet of Thngs IoT [2]. In a tradtonal wreless sensor network, the wreless sensors should constantly transmt the collected nformaton to the access pont. Meanwhle, they ether have lmted battery energy, or are powered by grd energy sources [3,4]. For the wreless sensors wth lmted battery power, ther workng tme s restrcted by ther lmted energy. For the wreless sensors powered by grd energy sources, they wll be restrcted to a fxed area. Rado frequency RF-based wreless energy transfer can be ntroduced to solve the lmted energy and non-statonary power supply problems [5]. Through RF-based wreless energy transfer, the wreless sensors can replensh ther energy from varous energy sources [6]. Applyng the wreless energy transfer nto wreless sensor networks, can enhance the lfe cycle of the sensor nodes, and mprove the network performance [7]. In wreless sensor networks, even when the wreless sensors are powered by the RF energy, the energy consumpton problem s stll a severe problem [8], because of the ncreasng demand for computng and communcaton tasks [9]. Therefore, how to control the energy consumpton n wreless sensor networks, especally how to control the sensors power level for nformaton transmsson, s stll Sensors 2018, 18, 2393; do: /s

2 Sensors 2018, 18, of 10 a contnung scentfc problem that needs to be solved [10]. Lots of works have been done n the power control problems n the wreless sensor networks powered by RF energy [11 14]. In [11], a wreless powered sensor network wth mnmal power requrements s desgned. The energy consumpton of the crcut and nformaton processng are consdered n the proposed model. The effect of the number of sensors are also taken nto account. In [12], a proper MAC protocol s desgned to solve the problem of the tradeoff between the RF energy transfer and data communcaton. Meanwhle, a correspondng Markov chan m and steady-state probabltes are derved to fulfll the performance analyss. In [13], a fuzzy power allocaton and rate adaptaton model s proposed. Throughput and energy performance are analyzed for the proposed scheme. Kwan et al. [14] desgned an optmal protocol for data transmsson consderng RF energy harvestng. A mult-source selecton and tmng allocaton algorthm s proposed and a closed-form mathematcal soluton s gven. evertheless, none of the above works to acheve optmal power control n wreless sensor networks powered by RF energy consder the dynamc characterstcs of the battery energy, and do not consder the optmzaton n gven a tme perod. In ths paper, we try to use dynamc game theory by consderng the dynamc characterstcs of the battery and try to solve the optmal power control problem n a tme perod. The dynamc game theory, also called dfferental game theory, was frstly proposed by Isaace [15], s one of the most practcal and complex branches of game theory and can be used to solve a class of resource allocaton problems, under whch the evoluton of the state s descrbed by a dfferental equaton and the players act throughout a tme nterval [16]. A non-cooperatve dynamc game-based power control model has been proposed n [17], to solve the power control problem n wreless powered sensor network usng feedback control. In ths paper, based on the cooperatve dynamc game, we pay attenton to the power control problem n wreless sensor networks whch are powered by RF energy, to maxmze the utlty based on the SIR requrements and the energy state. We am at fndng an optmal strategy for the sensors mappng the power level to the SIR requrements and the energy state. The game solutons are gotten n the condton of grand coalton and non-cooperatve feedback ash equlbrum. The Shapely theorem s used to acheve farly allocaton. The man contrbutons of ths work are summarzed as follows: Frstly, we formulate the system model of the wreless sensor network powered by RF energy, whch conssts of one access pont and sensor nodes, where the sensor nodes can harvest energy and transmt nformaton smultaneously. Secondly, a dynamc game model s proposed to formulate the power control problem n the proposed network. The energy varatons are consdered as the system state, and the objectve functon s composed by the SIR and energy requrements. Fnally, two knds of analyses are gven, whch are the grand coalton solutons and non-cooperatve solutons for the sensors. The remander of the paper s organzed as follows: Secton 2 ntroduces the system model of wreless sensor networks powered RF energy and the power control problem n a dynamc game. Secton 3 provdes the grand coalton solutons and the feedback ash equlbrum solutons for each wreless sensor. umercal smulatons are gven n Secton 4. Fnally, we conclude the work n Secton System Model and Problem Formulaton 2.1. System Model Consder a wreless sensor network powered by RF energy wth one access pont AP and sensors, where the sensors are equpped wth rechargeable batteres and can obtan energy from the AP based on RF energy transfer, as shown n Fgure 1. Located at an approprate place, the AP has abltes to transfer energy to all sensors, and can work as a data gatherng pont to collect and transmt nformaton for all sensors. The AP s connected to a constant power supply, and the broadcast energy

3 Sensors 2018, 18, of 10 Sensors 2018, 18, x FOR PEER REVIEW 3 of 11 over RF sgnals s assumed to be fxed on a stable level for all sensors. In ths paper, we assume that the AP can be serve as a snk node [4] for nformaton transmsson, whch operates on 2.4 GHz. Then all on 2.4 GHz. Then all the sensors can transmt nformaton to the AP drectly. For wreless energy the sensors can transmt nformaton to the AP drectly. For wreless energy transfer, the AP can utlze transfer, the AP can utlze the spectrum at 350 MHz to 3 GHz to carry RF energy to the sensor nodes the spectrum at 350 MHz to 3 GHz to carry RF energy to the sensor nodes [18]. As the sensors are [18]. As the sensors are equpped wth lmted rechargeable batteres, they need to harvest energy equpped wth lmted rechargeable batteres, they need to harvest energy from the AP and use the from the AP and use the harvested energy to transmt nformaton. The smultaneous wreless harvested energy to transmt nformaton. The smultaneous wreless nformaton and power transfer nformaton and power transfer SWIFT s appled. Both AP and sensors are equpped wth two SWIFT s appled. Both AP and sensors are equpped wth two antennas, wreless energy transfer antennas, for wreless energy transfer WET and wreless nformaton transmsson WIT WET and wreless nformaton transmsson WIT ndvdually. Meanwhle, we assume that the ndvdually. Meanwhle, we assume that the wreless energy transfer and the nformaton wreless transmsson energy operate transfer over and orthogonal the nformaton frequency transmsson bands wth operate dentcal over bandwdth, orthogonal and frequency thus the sensor bands wth nodes dentcal can harvest bandwdth, energy and and thus the transmt sensor nformaton nodes can harvest at the energy same and tme. transmt Wreless nformaton energy at and the same nformaton tme. Wreless transmsson energy are and operated nformaton at the transmsson same frequency, are operated based on at the same harvest then transmt frequency, based on the protocol harvest-then-transmt [19], as as shown n n Fgure protocol To [19], smplfy as shown the analyss, n Fgurethe 2. tme To smplfy duratons the for analyss, energy transfer the tme duratons and and nformaton energy transmsson transfer and are nformaton assumed to transmsson be the same n are ths assumed paper. to be the same n ths paper. Fgure 1. Wreless sensor network powered by RF energy. Fgure 1. Wreless sensor network powered by RF energy. Phase for Energy Transfer Phase for Energy Transfer Phase for Informaton Phase for Informaton Transmsson Transmsson 1/2 1/2 1/2 1/2 Fgure 2. Tme swtchng TS protocol. Fgure 2. Tme swtchng TS protocol. As the sensors have lmted energy, so t s essental to control the nformaton transmsson As the sensors have lmted energy, so t s essental to control the nformaton transmsson power As the for sensors all the sensors, have lmted even they energy, are so powered t s essental by the to RF control energy. the Then nformaton the target transmsson of ths paper power s to power for all the sensors, even they are powered by the RF energy. Then the target of ths paper s to for get allthe theoptmal sensors, uplnk evennformaton they are powered transmsson by thepower RF energy. n the Then wreless the sensor target of networks. ths paper We smodel to get get the optmal uplnk nformaton transmsson power n the wreless sensor networks. We model the the optmal power uplnk control nformaton problem as transmsson cooperatve power dynamc n the game, wreless where sensor all sensors networks. try We to cooperate model the the power control problem as a cooperatve dynamc game, where all sensors try to cooperate power together. control Meanwhle, problem we as awll cooperatve consder dynamc the requrements game, where of sgnal to nterference plus nose all sensors try cooperate together. rato together. Meanwhle, we wll consder the requrements of sgnal to nterference plus nose rato Meanwhle, SIR and we resdual wll consder energy after the requrements nformaton transmsson of sgnal-to-nterference-plus-nose for model constructon. rato SIR and SIR and resdual energy after nformaton transmsson for model constructon. resdual energy after nformaton transmsson for model constructon Energy State Energy Energy State State Durng the process of downlnk WET, the sensors wll harvest energy from the AP, and prepare Durng the process of downlnk WET, the sensors wll harvest energy from the AP, and prepare enough Durng energy the process for the uplnk of downlnk nformaton WET, transmsson. the sensors wll The harvest amount energy of harvested from the energy AP, and at sensor prepare enough energy for the uplnk nformaton transmsson. The amount of harvested energy at sensor enough s denoted energy by for q and thecan uplnk be expressed nformaton as follows: transmsson. The amount of harvested energy at sensor s s denoted by q and can be expressed as follows: denoted by q and can be expressed as follows: q Gq, 1 q Gq, 1 where q s the transferred energy from the q = AP. ηg As q, the transferred energy from the AP s 1 a where q s the transferred energy from the AP. As the transferred energy from the AP s a broadcastng energy, t s assumed to be the same for all sensors. η s the energy converson effcency. where broadcastng energy, t s assumed to be the same for all sensors. η s energy converson effcency. Let η q= s 1 the for smplfcaton. transferred energy G denotes from the the AP. channel As the power transferred gan between energythe fromap the and APsensor s a broadcastng. As nose energy, Let η = 1 for smplfcaton. G denotes the channel power gan between the AP and. As nose can be t gnored s assumed for energy to be transfer, the samewe forassume all sensors. there are η sno the harvested energy converson energy from effcency. nose. Let η = 1 can be gnored for energy transfer, we assume there are no harvested energy from nose.

4 Sensors 2018, 18, of 10 for smplfcaton. G denotes the channel power gan between the AP and sensor. As nose can be gnored for energy transfer, we assume there are no harvested energy from nose. Sensors equpped wth rechargeable batteres can use the harvested energy for nformaton transmsson. In ths paper, we assume all the sensors can transmt nformaton and harvest energy at the same tme. Let x denote the batteres energy of all the sensors, whch can be consdered as the state of the system. Assumng that the batteres energy beng decreased by the uplnk power consumed by nformaton transmsson and beng ncreased by the harvested energy from AP n a lnear relatonshp. Let p denote the nformaton transmsson power of sensor, then the evaluaton of x can be expressed by the followng dfferental equaton: dx/dt = q p + δx = ηg q p + δx, 2 where δ s a tme-varyng parameter of energy, and can be expressed as δ = ηq/e, wth E s the maxmum battery capacty of the sensors. Durng the process of uplnk WIT, the sensors wll control ther nformaton transmsson power based on the SIR requrements. Because the wreless sensors can share the same spectrum for the uplnk WIT and the downlnk WET, the nterference to sensor should manly come from the WET of AP. Assumng n 0 s the power spectral densty of the addtve whte Gaussan nose, then the SIR for sensor can be expressed as: γ = p n 0 + q = 1 p α. 3 In 3, as q s a constant power for all sensors, we can re-wrte the above formula wth α and α = n 0 + q. Then we have p = α γ, and Equaton 2 can be reformulated as follows: 2.3. Problem Formulaton dx/dt = ηg q p + δx = ηg q α γ + δx. 4 Based on 3, we can see that the SIR s n drect proporton to the uplnk nformaton transmsson power level. For each sensor, t expects to ncrease the uplnk WIT power to acheve hgher SIR, whch means the sensors can earn more proft for hgher SIR when ncreasng the uplnk WIT power level. Assumng there s a SIR threshold for each sensor and s denoted by γ, then the proft for havng a hgher SIR can be expressed as: Pr SIR = 1 2 γ γ 2. 5 Besdes the hgher SIR proft, proft of battery energy s also consdered n our model. We defne the proft of battery energy s a lnear form of the battery energy and can be expressed as: Pr battery = π x, 6 where all sensors contrbutons for battery proft are denote by the contrbutons parameter π. To maxmze the SIR and the fnal energy among all sensors, the utlty of each sensor s defned as the combnaton of achevable SIR and energy level, whch s gven: s.t. 4. maxux, t = 0 Pr ds = 0 Pr SIR + Pr battery = β 1α p γ + π x ds ds = 0 12 γ γ 2 + π x ds, 7

5 Sensors 2018, 18, of 10 ow, we formulate the optmal power control for all sensors as a cooperatve dynamc game, as follows: Players: All wreless sensors. Strategy space: All wreless sensors can cooperatvely choose ther nformaton transmt power to maxmze the utlty gven n 7. State: The battery energy state s denoted by vector x, where the state s controlled by the dynamc constrant n Equaton 4. Objectve functon: All of the wreless sensors act to maxmze ther utlty. 3. Solutons and Analyss In ths secton, we wll analyse the solutons to the game problem gven n 7 based on the dynamc optmzaton programmng technque, whch was ntroduced by Bellman. We try to get the feedback ash equlbrum solutons for all the sensors. We consder the case when all the sensors cooperate together to control ther transmsson power to maxmze the proft, and agree to act so that a team optmum could be acheved. In the cooperatve dynamc game, the group ratonalty and ndvdual ratonalty should be satsfed at any nstant of nterval tme. Lemma 1. For the optmzaton Equatons 7, an n-tuple of strateges p t, x, f or } consttutes a feedback ash equlbrum soluton f there exsts a functonal V t, x, defned on the tme nterval [0, T] and satsfyng the followng relatons for each [20,21]: V t, x == p α γ + π x ds p α γ + π x ds. 8 Then, we wll gve the process for obtanng the cooperatve solutons as follows Computaton of Optmal Cost of Grand Coalton For each sensor, ts target s to maxmze the proft gven n 7. In order to get the optmal soluton to the game 7, frstly we should defne the value functon based on the dynamc optmzaton programmng. The value functon W, x, t must satsfy the Bellman equaton: rw, x, t = max γ [ ] 1 2 γ γ 2 + π x + W x, x, t Performng the ndcated mnmzaton n 9 yelds: [ α γ + δx + ηg q ]}. 9 γ = γ + α W x, x, t p = α γ + α 2 W x, x, t. 10 Substtutng γ upon nto 9 and solvng, we can yeld the value functon as follows: π W, x, t = rr δ α γ 1 α 2 π 2 β r δ + rx + ηg q. 11 Let π = π, then we have:

6 Sensors 2018, 18, of 10 W, x, t = π rr δ α γ 1 α 2 π + rx + 2 r δ ηg q }. 12 Based on 12, the optmal SIR and transmt power of sensor can be gven by: γ = γ + α π r δ p = α γ + α 2 π r δ and we can get the optmal trajectory of battery energy as follows: x t = expδtx0 + 1 δ. 13 α γ } 1 expδt. 14 Based on the above equatons, we have obtaned the optmal SIR and transmsson power for each sensor and the maxmzed utlty n grand coalton. The battery energy of all the sensors n grand coalton condton, whch are the state of the wreless powered sensor networks, can also be obtaned based on 14. From 14, we can fnd that the optmal trajectory of the battery s a functon of the optmal SIR for each sensor, wth an ntal energy level x0. It can be seen that the optmal varaton of the energy s an exponental functon, whch fts the physcal meanng of the battery. Through 14, we can obtan the optmal varaton of the energy state n the proposed wreless sensor networks, under grand coalton condton Computaton of Feedback ash Equlbrum To solve the feedback ash Equlbrum for the game 7, the followng Bellman equaton should be satsfed: [ ]} rv 1 x = max γ 2 γ γ 2 + π x + Vxx α γ αj γ j + δx + ηg q, for. 15 j=1,j = Smlar to Secton 3.2, we can get the ndcated mnmzaton of 15 as follows: γ = γ + α V xx p = α γ + α 2 V xx. 16 Substtutng 16 nto the Bellman Equaton 15 and solvng, we can yeld the followng results: V xx = [ V π x = α rr δ j γ j α2 j π j 2β j=1 j r δ π r δ, 17 j=1,j = α 2 j and the feedback ash equlbrum level can then be obtaned as: 2β j ] π j r δ + rx + ηg q }. 18 γ = γ + α π r δ p = α γ + α 2 π r δ. 19 The dfference between ash equlbrum obtaned n 19 and those obtaned for the grand coalton n 13 s that player takes nto account the sum of all coalton members and not only hs own one.

7 Sensors 2018, 18, of Computaton of Optmal Cost for Intermedate Coaltons The value functon WK, x, t for the players n coalton K K < must satsfy the followng Bellman equaton: rwk, x, t = max [ γ, +W x K, x, t α γ [ 12 γ γ 2 + π x] j K Performng the ndcated mnmzaton to 20 yelds: α j γ j + δx + ηg q ]} 20 γ K = γ + α W x K, x, t p = α γ + α 2 W x K, x, t, 21 W x K, x, t = Substtutng 21 and 22 nto the Equaton 20 and solvng yeld: π WK, x, t = rr δ α γ α 2 2 π r δ. 22 π r δ α j γ j + α jπ j + rx + β j K j r δ 3.4. Defnton of the Characterstc Functon and Computaton of the Shapley Value The values of the characterstc functon are gven by: v}; x, t = V π x = rr δ j=1 α j γ j α2 j 2β j vk; x, t = WK, x, t = α j γ j + j K π j r δ π rr δ α 2 j 2β j=1,j = j α jπ j + rx + β j r δ ηg q α γ α 2 π j + rx + r δ π 2 r δ ηg q. 23 ηg q, 24 }. 25 In order to be convenent for computng the Shapley value and clarfyng our model, we suppose = 3, then we have: φ v x, t = π rr δ j=1,j = π rr δ h K,h = 3.5. Computaton of IDP Functons α j γ j α2 j β j α h γ h α h 2 2β h π j r δ + π r δ π π r δ j K x + ηg q α j γ j + α } jπ j β j r δ. 26 In [22], the authors defned the Imputaton Dstrbuton Procedure IDP beng Bt = B 1 t, B 2 t,..., B t}, and for the tme constant B t, t can be calculated as follows: B t = φ v xt, t d dt φv xt, t In 27, we can fnd that the IDP functon s a functon of the Shapley values. Combnng the Shapley values obtaned n the Secton 3.4, we can get the fnal allocaton for each sensor. 27

8 3.5. Computaton of IDP Functons In [22], the authors defned the Imputaton Dstrbuton Procedure IDP beng B t B1 t, B2 t,..., B t, and for the tme constant B t, t can be calculated as follows: d Sensors 2018, 18, 2393 B v, v t xt t xt, t of 10 dt In 27, we can fnd that the IDP functon s a functon of the Shapley values. Combnng the 4. umercal Shapley values Results obtaned n the Secton 3.4, we can get the fnal allocaton for each sensor. In ths secton, we wll smulate the method proposed n Secton 3. Based on [10], assumng there 4. umercal Results are three wreless nodes powered by one access pont. Each sensor needs to control the nformaton In ths transmsson secton, we wll power smulate to maxmze the method the proposed network n Secton proft. 3. The Based grand on [10], coalton assumng and there feedback are three wreless nodes powered by one access pont. Each sensor needs to control the nformaton ash equlbrum solutons ntroduced n Secton 3 are smulated to get dfferent results under transmsson power to maxmze the network proft. The grand coalton and feedback ash equlbrum dfferent stuatons. solutons ntroduced n Secton 3 are smulated to get dfferent results under dfferent stuatons. Fgure 3 shows the optmal power level of each sensor for nformaton transmsson. In Fgure 3a, Fgure 3 shows the optmal power level of each sensor for nformaton transmsson. In Fgure the power 3a, the for power energy for energy transfer transfer s set s to set be to 3 Watt, be 3 Watt, where where t s t set s to set be to 6 be Watt 6 Watt n Fgure n Fgure 3b. 3b. It It can can be be seen that seen the sensors that the can sensors havecan more have energy more energy for nformaton nformaton transmsson transmsson when when they they can harvest can harvest more more energy formenergy the RFform energy. the RF Two energy. kndstwo of solutons knds of solutons are obtaned are obtaned for all sensors, for all sensors, whch whch are grand are grand coalton solutons coalton andsolutons non-cooperatve and non cooperatve solutons respectvely. solutons respectvely. The power level The power for nformaton level nformaton transmsson s hgher transmsson n grand coalton s hgher n than grand thecoalton non-cooperatve than the non cooperatve solutons. Thsolutons. represents Ths that represents the power that s the more effcently power used s more n effcently grand coalton. used n Ingrand other coalton. words, the In other grand words, coalton the grand can nspre coalton thecan sensors nspre workng the effcently sensors amng workng ateffcently maxmzeamng the proft. at maxmze the proft. a q 3 watt b q 6 watt Fgure Fgure 3. Optmal 3. Optmal power power level level for for nformaton transmsson transmsson of of each each sensor. sensor. Fgure 4 shows the maxmzed network proft of the wreless sensor networks. The concluson s Fgure that the 4 network shows the proft maxmzed s ncreased network wth the proft tme of varaton. the wreless Meanwhle, sensor the networks. network proft The concluson s hgher s that n thegrand network coalton proft that s ncreased that non cooperatve wth the tmecondton. varaton. In Meanwhle, the condton theof network grand coalton, proft sbased hgher n grand on coalton the smulatons that that gven n n non-cooperatve Fgure 3, the sensors condton. wll have Inmore the condton power for of nformaton grand coalton, transmsson, on the smulatons then they have gven more nwllng Fgure to 3, cooperatve the sensors together wll have to maxmze more power the network for nformaton proft. Fgure transmsson, 5 shows then they maxmzed have more proft wllng of each to cooperatve sensor under together the grand to maxmze coalton condton the network and the proft. non cooperatve Fgure 5 shows the maxmzed condton respectvely. proft of each sensor under the grand coalton condton and the non-cooperatve condton Sensors respectvely. 2018, 18, x FOR PEER REVIEW 9 of 11 a q 3 watt b q 6 watt Fgure 4. etwork proft. Fgure 4. etwork proft.

9 a q 3 watt b q 6 watt Sensors 2018, 18, 2393 a q 3 watt b q 6 watt 9 of 10 Fgure 4. etwork proft. Fgure 4. etwork proft. a q 3 watt b q 6 watt a q 3 watt b q 6 watt Fgure 5. Proft of each sensor. Fgure 5. Proft of each sensor. Fgure 6 shows the energy varaton of the wreless sensor networks. Wth the energy transfer, Fgure the energy Fgure 6 shows of 6 the shows the wreless the energy sensor varaton network s of ncreased the wreless Wth the energy transfer, wth sensor the tme networks. varaton. Wth In the guarantee energy transfer, of the the qualty the energy energy of of servces, of the the wreless each sensor sensor wll network try to s reserve s ncreased more wth energy the tme to maxmze varaton. the In In network the the guarantee utlty. of of the the qualty qualty of of servces, each each sensor wll try to to reserve more energy to maxmze the network utlty. a q 3 watt b q 6 watt a q 3 watt b q 6 watt Fgure 6. Battery energy varaton. Fgure Battery energy varaton. 5. Conclusons 5. Conclusons 5. Conclusons In ths paper, we have proposed a cooperatve dynamc game based model that maxmzes the network In ths utlty paper, consderng we have the proposed SIR requrements a cooperatve and dynamc energy game based varatons, acheved model that by maxmzes cooperatvely the In ths paper, we have proposed a cooperatve dynamc game-based model that maxmzes the optmal network utlty allocaton consderng of the nformaton the SIR requrements transmsson and power. energy varatons, In the proposed acheved game by cooperatvely model, the network utlty consderng the SIR requrements and energy varatons, acheved by cooperatvely optmal allocaton of the nformaton transmsson power. In the proposed game model, the optmal allocaton of the nformaton transmsson power. In the proposed game model, the researched wreless sensor networks are powered by the RF energy sources. The energy varatons are consdered as the system state of the wreless sensor networks, and the sensors can control ther nformaton transmsson power based on the grand coalton solutons and the non-cooperatve ash equlbrum. Based on the smulaton results, t can be seen that our proposed model can acheve optmal power control. Author Contrbutons: H.X. conceved the man dea and the dynamc game theory model; all authors contrbuted to data analyss, smulatons and the wrtng of ths paper. Fundng: Ths work s supported by the atonal Scence and Technology Key Projects o Acknowledgments: The authors would lke to thank the edtor and the anonymous revewers for ther valuable comments and suggestons that mproved the qualty of ths paper. Conflcts of Interest: The authors declare no conflcts of nterest. The foundng sponsors had no role n the desgn of the study; n the collecton, analyses, or nterpretaton of data; n the wrtng of the manuscrpt; or n the decson to publsh the results.

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