Energy-Efficient Power Allocation for M2M Communications with Energy Harvesting Transmitter

Transcription

1 nergy-ffcent ower Allocaton for M2M Communcatons wth nergy Harvestng Transmtter Derrck Wng Kwan Ng and Robert Schober Char of Dgtal Communcatons, The Unverstät rlangen-nürnberg Abstract In ths paper, power allocaton for energy-effcent pont-to-pont machne-to-machne (M2M) communcaton systems wth multple energy harvestng sources s studed. Under a determnstc system settng, we formulate the power allocaton problem as a non-convex optmzaton problem over a fnte horzon takng nto account the crcut energy consumpton, fnte battery storage capactes, and a mnmum requred data rate. The consdered non-convex optmzaton problem s transformed nto a convex optmzaton problem by explotng the propertes of fractonal programmng whch results n an effcent optmal off-lne teratve power allocaton algorthm. In each teraton, the transformed problem s solved by usng dual decomposton and a recursve power allocaton soluton s obtaned for maxmzaton of the energy effcency of data transmsson (bt/joule delvered to the recever). I. INTRODUCTION Recently, a large amount of work has been devoted to machne-to-machne (M2M) communcaton due to ts wde spread applcatons n e-health care, smart cty, and remote montorng, etc. In practce, M2M sensor type devces are usually small and nexpensve whch puts strngent constrants (.e., bandwdth and energy consumpton) on the system desgn [1]. On the other hand, green communcaton has receved much attenton n recent years drven by envronmental concerns [2], [3]. As a result, M2M communcaton systems are not only envsoned to be energy-effcent, but also to be selfsustanable. In the lterature, a tremendous number of green technologes/methods have been proposed for maxmzng the energy effcency (bt-per-joule) of wreless communcaton systems [3]-[7]. Among these technologes, energy harvestng s partcularly appealng and sutable for M2M communcaton snce each M2M devce can harvest energy from natural renewable energy sources such as solar, wnd, and vbraton, etc, thereby reducng substantally the operatng cost of the servce provders. The ntroducton of energy harvestng capabltes nto M2M systems poses many nterestng new challenges for the transmsson desgn due to the tme varyng avalablty of renewable energy sources. In [4] and [5], optmal packet schedulng and power allocaton algorthms were proposed for energy harvestng systems to mnmze the transmsson completon tme, respectvely. However, these works assumed an nfnte battery capacty n the energy harvester and the obtaned results may not be applcable to the case of fnte battery storage. Besdes, they dd not take nto account the crcut energy consumpton and the maxmum energy effcency of these systems s stll unknown even for the case of pontto-pont communcaton. In [6] and [7], the authors proposed an optmal power control tme sequence for maxmzng the throughput by a deadlne wth a sngle energy harvester for dfferent channel scenaros. Yet, the ntermttent nature of energy harvestng of a sngle energy source wll cause the power avalablty at the M2M devce to be hghly random. In other words, such sngle energy harvester desgn may not be able to guarantee the demandng qualty of servce requrements of M2M applcatons such as a mnmum data rate requrement. In ths paper, we address the above ssues. We study the structure of the optmal off lne power allocaton soluton where we assume that non-causal nformaton of channel state nformaton (CSI) and energy arrvals s avalable at the transmtter. The derved off lne soluton consttutes a performance upper bound whch sheds some lght on the desgn of effcent on lne solutons n future research. We formulate the power allocaton problem for energy-effcent M2M communcaton wth multple energy harvesters as an optmzaton problem. By usng nonlnear fractonal programmng, the consdered non-convex optmzaton problem n fractonal form s transformed nto an equvalent optmzaton problem n subtractve form wth a tractable soluton, whch can be found wth an teratve algorthm. In each teraton, dual decomposton s used and a recursve closed-form power allocaton soluton s computed for maxmzaton of the system energy effcency. II. SYSTM MODL We consder a sngle lnk contnuous tme narrowband M2M communcaton system. The transmsson tme s T seconds. We assume that the transmtter adapts the power allocaton L tmes for a gven value of T. Note that the optmal value of L and the tme nstant of each adapton operaton wll be found n the next secton. The data symbol receved at the user at tme nstant t, t T, s gven by y(t) = (t)g(t)h(t)x(t)+z(t), (1) where (t) and x(t) are the transmtted power and the transmtted symbol at tme t, respectvely. h(t) and g(t) are the small scale fadng coeffcent and the path loss between transmtter and recever at tme t, respectvely. z(t) s the addtve whte Gaussan nose (AWGN) at tme t wth zero

2 Transmtter Recever,, 1,1 1,2,,,1,2 Transmtter sgnal processng core nergy harvestng battery1 max1 nergy harvestng battery max ower amplfer Users data nergy flow Jont energy supply Channel gan nergy consumed by the A stemng from energy harvester n (1) = [1] l1,1,2,3,4,5 ε [7] l ε 7 ε [11] l11 poch 1 poch 7 poch 11 n (4) = n (2) =,1 n (6) = n (7) =,2 T,, N,1 N,2 nergy harvestng batteryn maxn Fg. 2. An llustraton of epoches and the meanng of n ( ) for dfferent events and dfferent arrval tme for energy harvester. Fadng changes and energes are harvested at tme nstants denoted by and, respectvely. Fg. 1. A transmtter wth multple energy harvestng sources. mean and varancen W, wheren s the nose power spectral densty and W s the sgnal bandwdth. A. Tme Varyng Fadng Model and nergy Supply Model We adopt a tme varyng system model smlar to the one n [6]. At the transmtter, there are N dfferent types of energy harvesters for supplyng the energy requred for transmsson by the power amplfer (A), cf. Fgure 1. For nstance, the M2M sensor can extract wnd energy or solar energy wth the help of a wnd harvester and a solar energy harvester, respectvely. As a result, the nstantaneous total rado frequency (RF) transmt power at the A n tme nstant t can be wrtten as (t) = (t), t T, (2) where (t) s the nstantaneous power transmtted by the power amplfer, whch s fueled by energy harvester, {1,...,N}. We assume that the changes n the channel gans and energy arrvals n energy harvester are stochastc processes n tme whch can be modeled as osson countng processes wth rates λ F and λ, respectvely [6], [7]. Therefore, changes n the channel gans and the energy arrvals, respectvely, occur n a countable number of tme nstants, whch are ndexed as t F 1,tF 2... and t 1,t 2..., respectvely. The nter-occurrence tmes t F a t F a 1,a {1,2,...}, and t b t b 1,b {1,2,...}, are exponentally dstrbuted wth means 1/λ F and 1/λ, respectvely. Note that we set t = t F = for convenence. A block fadng tme varyng communcaton channel model s consdered. In other words, the fadng level n < t t F 1 s constant but changes to an ndependent value n the next tme nterval t F 1 < t t F 2, and so on. Smlarly,,a unts of energy arrve (to be harvested) at tme t a n energy harvester, cf. Fgure 2. The ncomng energes are collected by the N energy harvesters and buffered n the battery before they are used n data transmsson. On the other hand, we assume that unts of energy arrve/(are avalable) n the battery of energy harvester at t = and the maxmum amount of energy storage n the battery s denoted by max. In the followng, we refer to a change n the channel gans or n the energy level n any one of the batteres as an event and to the tme nterval between two consecutve events as an epoch. Specfcally, epoch a s defned as the tme nterval [t a 1,t a ), where t a 1 and t a are the tmes at whch successve events happen, cf. Fgure 2. B. hyscal Constrants on the nergy Harvesters There are two nherent constrants on each energy harvester: t b δ b 1 C1: ε (u)du,j, b {1,2,...}, (3) d (t) C2: j=,j t j= ε (u)du max, t T,,(4) where,j s the amount of energy harvested by energy harvester at energy arrval j at tme t j. δ s an nfntesmal postve constant for modelng purpose. d (t) = arg max a {t a : t a t} and ε 1 s a constant whch accounts for the neffcency of the A. For nstance, f ε = 5, 5 Watts of power are consumed n the A for every 1 Watts of power radated n the RF and the power effcency s 1 ε = 1 5 = 2%. Constrant C1 mples that n every tme nstant, f the transmtter draws energy from energy harvester to cover the energy requred at the A, t s constraned to use at most the amount of stored energy currently avalable n energy harvester (causalty), although there wll be possbly more energy arrvals n the future. Constrant C2 states that the energy level n energy harvester never exceeds max to prevent the occurrence of an energy overflow n the battery. III. OWR OTIMIZATION ROBLM FORMULATION In the followng, we desgn the power allocaton algorthm based on an nformaton theoretc approach whch nherently assumes that the data buffer at the transmtter s always full 1. 1 In practce, f there s no data n the buffer, the transmtter can smply shut off the A and store all the harvested energy f possble.

3 A. Instantaneous Channel Capacty In ths subsecton, we defne the adopted system performance measure. Gven perfect CSI at the recever, the channel capacty between the transmtter and recever over a transmsson perod of T second(s) wth bandwdth W s gven by T ) C() = W log 2 (1+(t)Γ(t) dt and Γ(t) = g(t) h(t) 2 N W, (5) where Γ(t) s the receved channel gan-to-nose rato (CNR) at the recever at tme t and = { (t),, t T} s the power allocaton polcy. On the other hand, we take nto account the total energy consumpton of the system by ncludng t n the optmzaton objectve functon. For ths purpose, we model the weghted energy dsspaton n the system as the sum of two terms whch can be expressed as U T () = T ε φ (t)dt+ C T, (6) where φ > s a non-negatve constant weght mposed on the use of energy harvester and φ φ k, k. In partcular, the value of φ can be nterpreted as the cost or preference n usng energy harvester. For nstance, f energy harvester explots solar energy, the transmtter may prefer to use battery on sunny days for transmsson by settng φ. On the other hand, C n (6) s the constant requred sgnal processng power 2 at each tme nstant whch ncludes the power dsspatons n the mxer, transmt flter, frequency syntheszer, and dgtal-to-analog converter (DAC), etc. Hence, the weghted energy effcency of the consdered system over a tme perod of T seconds s defned as the total average number of receved bts/joule C() U eff () = U T (). (7) B. Optmzaton roblem Formulaton The optmal power allocaton polcy,, can be obtaned by solvng max U eff () (8) s.t. C1, C2, C3: C() R mn, C4: (t) max, t T, C5: (t),, t T, where C3 specfes the mnmum system data rate requrement R mn. C3 can also be nterpreted as a delay constrant for data transmsson snce at least R mn amount of data has to be transmtted by the end of tme T. In partcle, such constrant 2 We assume that there s a constant energy supply from a non-renewable energy source (e.g. from power grd) for supplyng the energy requred n sgnal processng. Note that we can ncorporate the constant energy supply nto (6) by treatng t as the N-th energy harvester whch has the hghest value of weght φ N. s needed for real tme M2M communcaton servces such as vehcle and asset trackng. Note that although varable R mn n C3 s not an optmzaton varable n ths paper, a balance between energy effcency and system capacty can be struck by varyng R mn. C4 s a constrant on the maxmum transmt power of the transmtter. For nstance, f Zgbee s used for M2M communcaton, the maxmum transmt power s max = 1 W n the US. C5 s the non-negatve constrant on the power allocaton varables. IV. SOLUTION OF TH OTIMIZATION ROBLM The optmzaton problem n (8) s non-convex due to the fractonal form of the objectve functon. We note that there s no standard approach for solvng non-convex optmzaton problems. In order to derve an effcent power allocaton algorthm for the consdered problem, we ntroduce the followng transformaton. A. Transformaton of the Objectve Functon The objectve functon n (8) can be classfed as nonlnear fractonal program [8] and has some nterestng propertes that wll be ntroduced n the followng. Wthout loss of generalty, we defne the maxmum energy effcencyq of the consdered system as q = C( ) U T ( ) = max C() U T (). (9) Then, we can establsh the followng theorem. Theorem 1: The maxmum energy effcency q s acheved f and only f the optmal power allocaton polcy satsfes the followng condton: max C() q U T () (1) = C( ) q U T ( ) =, for C() and U T () >. roof: We can follow a smlar approach as n [9] to prove Theorem 1. The detaled proof s omtted here because of space constrants. By Theorem 1, for any optmzaton problem wth an objectve functon n fractonal form, there exsts an equvalent 3 objectve functon n subtractve form, e.g. C() q U T () n the consdered case. As a result, we can focus on ths equvalent objectve functon n the rest of the paper. B. Iteratve Algorthm for nergy ffcency Maxmzaton In ths secton, we adopt an teratve algorthm (known as the Dnkelbach method) for solvng (8) wth an equvalent objectve functon. The proposed algorthm s summarzed n Table I and the convergence to the optmal energy effcency s guaranteed f we are able to solve the nner problem (11) n each teraton. roof: lease refer to [9] for a proof of convergence. 3 Here, equvalent means that both problem formulatons lead to the same optmal power allocaton polcy.

4 TABL I ITRATIV OWR ALLOCATION ALGORITHM. Algorthm 1 Iteratve ower Allocaton Algorthm 1: Intalze the maxmum number of teratons L max and the maxmum tolerance ɛ 2: Set maxmum energy effcency q = and teraton ndex n = 3: repeat {Man Loop} 4: Solve the nner loop problem n (11) for a gven q and obtan power allocaton polcy { } 5: f C( ) qu T ( ) < ɛ then 6: Convergence = true 7: return { } = { } and q = C( ) U T( ) 8: else 9: Set q = C( ) U T( ) and n = n+1 1: Convergence = false 11: end f 12: untl Convergence = true or n = L max As shown n Table I, n each teraton of the man loop, we solve the followng optmzaton problem for a gven parameter q: max C() qu T () s.t. C1, C2, C3, C4, C5. (11) Soluton of the Man Loop roblem: Although the objectve functon s now n a subtractve form whch s easer to handle, there s stll an obstacle n solvng the above problem. The optmal power allocaton polcy s expected to be tme varyng n the consdered duraton of T seconds. However, t s unclear how often the transmtter should update the power allocaton polcy whch s a hurdle for desgnng a practcal power allocaton algorthm. In order to strke a balance between soluton tractablty and computatonal complexty, we ntroduce the followng lemma whch provdes valuable nsght nto the tme varyng dynamc of the optmal power allocaton polcy. Lemma 1: The optmal power allocaton polcy maxmzng the system energy effcency does not change wthn an epoch. roof: lease refer to the Appendx for a proof of Lemma 1. As revealed by Lemma 1, the optmal power allocaton polcy must be kept constant n each epoch for maxmzng the system energy effcency. As a result, we can dscretze the ntegrals and contnuous varables nvolved n (11). In other words, the number of constrants n (11) reduce to countable quanttes. Wthout loss of generalty, we assume that the channel states change M tmes and energy arrves K tmes n the N energy sources n the duraton of [,T]. Specfcally, we have L = M+K epoch(s) for the consdered duraton of T seconds whch ncludes the epoch caused by at t = for all energy harvesters. Besdes, tme nstant T s treated as an addtonal fadng epoch wth zero channel gan to termnate the process. We defne the length of each epoch as l j = t j t j 1 where epoch j {1,2,...,L} s defned as the tme nterval [t j 1,t j ), cf. Fgure 2. Note that t s defned as t =. For the sake of notatonal smplcty and clarty, we replace the contnuous tme varables wth the correspondng dscrete tme varables,.e., (t) [j], (t) [j], and Γ(t) Γ[j]. Then, the weghted average system throughput and the total weghted energy consumpton can be re-wrtten as C() = l j C[j] and U T () = l j C + l j ε [j]φ, (12) ( respectvely, where C[j] = W log 2 1+( ) N [j])γ[j] s the channel capacty between the transmtter and the recever n epoch l. As a result, the optmzaton problem n (11) s transformed nto the followng convex optmzaton problem: C1: C2: C3: max C() qu T () e e l j ε [j] n [j], e, r r 1 n [j] l j C[j] R mn, εl j [j] max, r, C4: l e [e] l e max, e, C5: [e],,e, (13) where e {1,2,...,L} and r {2,...,L + 1}. In (13), n [j] s defned as the energy whch arrves n epoch j n battery. Hence, n [j] =,a for some a f event j s an energy arrval and n [j] = f event j s a channel gan change, cf. Fgure 2. Now, the transformed problem s jontly concave wth respect to all optmzaton varables 4, and under some mld condtons [1], solvng the dual problem s equvalent to solvng the prmal problem. C. Dual roblem Formulaton In ths subsecton, we solve the power allocaton and schedulng optmzaton problem by solvng ts dual. For ths purpose, we frst need the Lagrangan functon of the prmal problem whch can be wrtten as L(γ,β,ρ,µ,) = l j (1+ρ)C[j] ρr mn ( j γ,j l m ε [m] ( L q l j C + j ) l j ε [j]φ ) n [m] 4 We can follow a smlar approach as n Appendx A to prove the convexty of the above problem for the dscrete tme model.

5 L+1 ( j β,j j=2 j 1 n [m] εl m [m] max ) ( N µ j l j [j] l j max ), (14) where γ s the Lagrange multpler vector assocated wth the causalty constrant C1 n drawng energy from each energy harvester wth elements γ,j, {1,...,N},j {1,...,L}. β s the Lagrange multpler vector correspondng to the maxmum energy level constrant C2 n the battery of the energy harvester wth elements β,j where β,1 =,. ρ s the Lagrange multpler correspondng to the mnmum data rate requrement R mn n C5. µ s the Lagrange multpler vector for constrant C4 on the maxmum power wth elements µ j. Note that the boundary constrants C5 are absorbed nto the Karush-Kuhn-Tucker (KKT) condtons when dervng the optmal soluton n Secton IV-D. Thus, the dual problem s gven by mn max L(γ,β,ρ,µ,). (15) γ,β,ρ,µ D. Dual Decomposton and Sub-roblem Soluton By Lagrange dual decomposton, the dual problem s decomposed nto two parts (nested loops): the frst part (nner loop) s known as sub-problem; the second part (outer loop) s the master problem [9]. Then, the dual problem can be solved teratvely, where n each teraton the transmtter solves the sub-problem (nner loop) by usng KKT condtons for a fxed set of Lagrange multplers, and the master problem (outer loop) s solved usng gradent method. Let [j] denotes the optmal power allocaton soluton of the subproblem for energy harvester n epochj. Wthout loss of generalty, we assume φ 1 < φ 2 <... < φ N for the sake of notatonal smplcty. Usng standard optmzaton technques and the KKT condtons, the optmal power allocatons for the N energy sources n epoch j are gven by the followng recursve equaton: [ ] + 1[j]= W(1+ρ) (ln(2)a 1 [j]) 1 and (16) Γ[j] [ ] + +1 [j]= W(1+ρ) (ln(2)a +1 [j]) 1 Γ[j] d [j],(17) where A [j]= γ,e ε e=j d=1 β,e+1 ε+qφ ε+µ j. (18) e=j The power allocaton solutons n (16) and (17) can be nterpreted as a form of water-fllng. In partcular, varable ρ forces the transmtter to assgn more power for transmsson f the data rate requrement R mn becomes strngent. Interestngly, the optmal values of [j],, have a undrectonal dependence wth each other accordng to the weghts φ,.e., the power drawn from an energy source wth a hgher weght depends on the power drawn from the energy sources wth lesser weghts, but not vce versa. Specfcally, as can be seen n (17),1[j] decreases the water-level n calculatng the value of+1 [j]. In other words, 1 [j] reduces the amount of energy drawn from the less preferable energy sources (hgher values of φ ) for maxmzaton of energy effcency.. Soluton of the Master Dual roblem To solve the master mnmzaton problem n (15),.e., to fnd γ, β, ρ, and µ for a gven, the gradent method can be used snce the dual functon s dfferentable. The gradent update equatons are gven by: [ γ,j (ς +1)= γ,j (ς) ξ 1 (ς) ( j [ β,r (ς +1)= β,r (ς) ξ 2 (ς) ( max n [m] ε [m]l m )] +,,j, (19) r r +,,r,(2) [m]+ εl m [m])] n [ ( L )] +, ρ(ς +1)= ρ(ς) ξ 3 (ς) l j C[j] R mn (21) [ ( µ j (ς +1)= µ j (ς) ξ 4 (ς) max +, j, [j])] (22) where j {1,... L}, r {2,... L}, ndex ς s the teraton ndex, and ξ u (ς), u {1,...,4}, are postve step szes. Then, the updated Lagrange multplers n (19)-(22) are used for solvng the subproblem n (15) va updatng the power allocaton soluton accordng to (16)-(18). Snce the transformed problem n (13) s convex, the dualty gap between dual optmum and prmal optmum s zero and t s guaranteed that the teraton between the master problem and the subproblem converges to the optmal soluton of (11) n the man loop, f the chosen step szes satsfy the nfnte travel condtons [1]. V. RSULTS AND DISCUSSIONS In ths secton, we evaluate the system performance usng smulatons. We assume a transmsson duraton of T = 1 seconds, a carrer center frequency of 2.4 GHz, a sgnal bandwdth of W = 1 khz, a nose power of N W = 134 dbm, and the dstance between transmtter and recever s 5 meters. The small scale fadng coeffcents of the transmtter and recever are generated as Raylegh random varables wth unt varances. The statc crcut power consumpton s set to C = 23 dbm [11], the mnmum data rate requrement of the system s R mn = 2 kbts/s, and the maxmum transmt power s 1 W. The number of energy sources wll be specfed n each case study and each energy harvester has a maxmum energy storage of max = 1 J, and an ntal energy =.5 J n the battery. The amount of energy that can be harvested by each energy harvester n each energy epoch s assumed to be unformly dstrbuted n [,1] J [7]. The channel changes wth rate λ f = 2 ms. On the other hand, we assume a power effcency of 35% n the A,.e., ε = 1.35 =

6 1.2 x Multple energy harvesters dversty gan nergy arrval rates λ roposed algorthm roposed algorthm, 1 teratons, 4 energy harvesters roposed algorthm, 5 teratons, 4 energy harvesters roposed algorthm, 1 teratons, 3 energy harvesters roposed algorthm, 5 teratons, 3 energy harvesters roposed algorthm, 1 teratons, 2 energy harvesters roposed algorthm, 5 teratons, 2 energy harvesters Baselne, 1 teratons Baselne, 5 teratons Average system capacty (kbt/s) roposed algorthm roposed algorthm, 1 teratons, 4 energy harvesters roposed algorthm, 1 teratons, 3 energy harvesters roposed algorthm, 1 teratons, 2 energy harvesters Baselne Baselne 225 Baselne nergy arrval rates λ nergy arrval rates λ Fg. 3. nergy effcency (bt-per-joule) versus energy arrval rate, λ, for the proposed algorthm and the baselne wth dfferent numbers of energy harvesters. Note that f the resource allocator s unable to guarantee the mnmum data rate R mn n T, we set the energy effcency and the average system throughput for that channel realzaton to zero to account for the correspondng falure. The average system energy effcency s obtaned by countng the number of bts whch are successfully decoded by the recever over the total energy consumpton averaged over the mcroscopc fadng. Unless further specfed, n the followng results, the number of teratons refers to the number of teratons of Algorthm 1 n Table I. A. nergy ffcency versus nergy Arrval Rates Fgure 3 llustrates the average energy effcency versus the energy arrval rates, λ, for dfferent numbers of energy harvesters. We defne a vector φ = [φ 1...φ...φ N ]. For the case study of 1, 2, 3, and 4 energy harvester(s), the weght(s) of φ s/are set to φ 1 = [1], φ2 = [.5 1], φ3 = [.1.5 1], and φ 4 = [ ], respectvely, where s a small postve constant for studyng the effect of multple energy harvester dversty. For an energy harvester wth weght φ = 1, a tradtonal contnuous constant energy supply wth an nstantaneous power of 1 W s assumed. The case of φ 1 = [1] s treated as a baselne scheme for comparson. The energy harvesters wth weghts φ < 1, represent some forms of clean energy such as solar energy and wnd energy, etc. The number of teratons for the proposed teratve resource allocaton algorthm s 5 and 1. It can be observed that the performance dfference between 5 and 1 teratons s neglgble whch confrms the practcalty of the proposed algorthm. On the other hand, the growth of energy effcency has a dmnshng return for hgh energy arrval rates. Indeed, when the energy arrval rate ncreases from a small value, the transmtter has a hgher energy level n each battery for performng power allocaton and thus the system energy effcency s enhanced. However, when the arrval rates of energy become exceedngly large, the transmtter s forced to dscharge the batteres n order to prevent a battery overflow, Fg. 4. Average system capacty (kbt/s) versus energy arrval rate, λ, for the proposed algorthm and the baselne. cf. C2 n (4). As a result, the transmtter has to transmt an excess amount of energy for dschargng the batteres whch decreases the system energy effcency gan due to a hgher energy arrval rate. It can be observed that there s an energy effcency gan f we swtch the case from φ 2 to φ 3. Ths s because n φ 3, a more energy effcent source s avalable for transmsson compared to φ 2. Besdes, a form of multple energy harvester dversty can be observed n the energy effcency when we swtch from φ 3 to φ 4. Snce, the performance gan s comng from the transmtter n explotng energy from dfferent energy sources whch changes the ntermttent nature of energy avalablty compared to the case of sngle energy source. On the other hand, the proposed algorthm provdes a sgnfcant performance gan compared to the baselne scheme. Ths s because the baselne scheme can only drawn energy from a less energy-effcent source. B. Average System Capacty versus nergy Arrval Rates Fgure 4 shows the average system capacty versus the energy arrval rates, λ, for dfferent numbers of energy harvesters. We compare the system performance of the proposed algorthm agan wth the baselne scheme. The number of teratons n the proposed algorthm s set to 1. It can be observed that the average system capacty of the proposed algorthm ncreases wth the energy arrval rates. Ths s because more energy s avalable for data transmsson whch results n a capacty gan. We note that, as expected, the baselne scheme acheves a smaller average system capacty than the proposed algorthm snce the proposed algorthm s able to explot energy from dfferent energy sources n T seconds. VI. CONCLUSION In ths paper, we formulated the power allocaton algorthm desgn for a pont-to-pont M2M communcaton systems wth multple energy sources as a non-convex optmzaton

7 problem, n whch the crcut energy consumpton, the fnte battery storage capacty, and the system data rate requrement were taken nto consderaton. By explotng the propertes of nonlnear fractonal programmng, the consdered problem was transformed nto an equvalent convex optmzaton problem wth a tractable soluton. An effcent teratve offlne power allocaton algorthm wth recursve closed-form power allocaton was derved for maxmzaton of the energy effcency. Smulaton results dd not only show that the proposed algorthm converges to the optmal soluton wthn a small number of teratons, but unveled also the achevable maxmum energy effcency. Interestng topcs for future work nclude studyng the optmal on-lne soluton n mult-channel M2M systems. ANDIX - ROOF OF LMMA 1 The proof of Lemma 1 s dvded nto two parts. In the frst part, we prove the convexty of the optmzaton problem n (11). Then, n the second part, we prove a necessary condton for the optmal power allocaton polcy based on the result n part one. 1) roof of the Convexty of the Transformed roblem n (11): We frst consder the concavty of the objectve functon on a per subcarrer bass wth respect to all optmzaton varables. For the sake of notatonal smplcty, we defne the channel capacty between the transmtter and the recever at tme nstant t as C(t) = W log 2 (1+(t)Γ(t)), respectvely. Let the objectve functon n (11) at tme nstanttbef(t,) = C(t) q(ε N φ (t)+ C t). Then, we denote the Hessan matrx of functon f(t,) by H(f(t,)) and the egenvalues of H(f(t,)) by ϕ 1, ϕ 2,..., and ϕ N, respectvely. After some algebrac manpulaton, the egenvalues of H(f(t, )) are gven by ϕ 1 = ϕ 2 =... = ϕ N 1 =, (23) Γ 2 (t)n N ϕ N = (t) ln(2)( N. (24) 2 (t)γ(t)+1) Hence, H(f(t, )) s a negatve sem-defnte matrx snce ϕ. Therefore, f(t,) s jontly concave wth respect to (w.r.t.) optmzaton varables (t) at tme nstant t. Then, the ntegraton of f(t,) over t preserves the concavty of the objectve functon n (11) [1]. On the other hand, the constrants C1-C5 n (11) span a convex feasble set and thus the transformed problem s a concave optmzaton problem. 2) Optmalty of a Constant ower Allocaton olcy n ach poch: Wthout loss of generalty, we consder a tme nterval [t 1,t 2 ) of epoch 1 and a tme nstant τ 1, where t 1 τ 1 < t 2. Suppose an adaptve power allocaton polcy s adopted n t 1 τ 1 < t 2 such that two constant power allocaton polces, { 1 } and { 2 }, are appled n t 1 t < τ 1 and τ 1 t < t 2, respectvely. We assume that { 1 } and { 2 } are feasble solutons to (11) whle 1 2. Now, we defne a thrd power allocaton polcy { 3 } such that 3 = 1(τ 1 t 1)+ 2(t 2 τ 1) t 2 t 1. Note that arthmetc operatons between any two power allocaton polces are defned element-wse. Then, we apply power allocaton polcy 5 { 3 } to the entre epoch 1 and ntegrate f(t,) over tme nterval [t 1,t 2 ) whch yelds: t2 f(t, 3 )dt (a) t2 τ 1 t 1 f(t, 1 )+ t 2 τ 1 f(t, 2 )dt t 1 t 1 t 2 t 1 t 2 t 1 = (τ 1 t 1 )f(t, 1 )+(t 2 τ 1 )f(t, 2 ) = τ1 t 1 f(t, 1 )dt+ t2 τ 1 f(t, 2 )dt, (25) where (a) s due to the concavty of f(t,). In other words, for any adaptve power allocaton polcy wthn an epoch, there always exsts at least one constant power allocaton polcy whch outperforms the adaptve approach. As a result, the optmal power allocaton polcy s non-adaptve wthn each epoch. RFRNCS [1] Y. Zhang, R. Yu, S. Xe, W. Yao, Y. Xao, and M. Guzan, Home M2M Networks: Archtectures, Standards, and QoS Improvement, I Commun. Magazne, vol. 49, pp , Apr [2] Y. Chen, S. Zhang, S. Xu, and G. L, Fundamental Trade-offs on Green Wreless Networks, I Commun. Magazne, vol. 49, pp. 3 37, Jun [3] T. Chen, Y. Yang, H. Zhang, H. Km, and K. Horneman, Network nergy Savng Technologes for Green Wreless Access Networks, I Wreless Commun., vol. 18, pp. 3 38, Oct [4] J. Yang and S. Ulukus, Optmal acket Schedulng n an nergy Harvestng Communcaton System, I Trans. Commun., vol. 6, pp , Jan [5] J. Yang, O. Ozel, and S. Ulukus, Broadcastng wth an nergy Harvestng Rechargeable Transmtter, I Trans. Wreless Commun., vol. 11, pp , Feb [6] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, and A. Yener, Transmsson wth nergy Harvestng Nodes n Fadng Wreless Channels: Optmal olces, I J. Select. Areas Commun., vol. 29, pp , Sep [7] K. Tutuncuoglu and A. Yener, Optmum Transmsson olces for Battery Lmted nergy Harvestng Nodes, I Trans. Wreless Commun., vol. 11, pp , Mar [8] W. Dnkelbach, On Nonlnear Fractonal rogrammng, Management Scence, vol. 13, pp , Mar [Onlne]. Avalable: [9] D. W. K. Ng,. S. Lo, and R. Schober, nergy-ffcent Resource Allocaton for Secure OFDMA Systems, I Trans. Veh. Technol., preprnt, May 212. [1] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty ress, 24. [11] Q. Wang, M. Hempstead, and W. Yang, A Realstc ower Consumpton Model for Wreless Sensor Network Devces, n Thrd Annual I Commun. Socety Conf. on Sensor, Mesh and Ad Hoc Commun. and Networks, vol. 1, Sep. 26, pp ower allocaton polcy { 3 } s also a feasble soluton to (11) by the convexty of the feasble soluton set.