Explct and Implct Temperature Constrants n Energy Harvestng Communcatons Abdulrahman Baknna, Omur Ozel 2, and Sennur Ulukus Department of Electrcal and Computer Engneerng, Unversty of Maryland, College Park, MD 2 Department of Electrcal and Computer Engneerng, Carnege Mellon Unversty, Pttsburgh, PA Abstract We consder an energy harvestng communcaton system where the temperature dynamcs s governed by the transmsson power polcy. Dfferent from the prevous work, we consder a dscrete tme system where transmsson power s kept constant n each slot. We consder two models that capture dfferent effects of temperature. In the frst model, the temperature s constraned to be below a crtcal temperature at all tme nstants; we con ths model as explct temperature constrant model. We nvestgate throughput optmal power allocaton for multple energy arrvals under general, as well as temperature and energy lmted regmes. In the second model, we consder the effect of the temperature on the channel qualty; we con ths model as mplct temperature constrant model. As the dynamc range of the temperature changes sgnfcantly, the change n the thermal nose becomes non-neglgble, affectng the sgnal to nterference plus nose rato SINR. In effect, transmtted sgnals contrbute as nterference for all subsequent slots. In ths case, we nvestgate throughput optmal power allocaton under general, as well as low and hgh SINR regmes. I. INTRODUCTION We study two dfferent effects of temperature change on the optmal power allocaton, and hence, on the system performance of a sngle-user energy harvestng system. We consder explct and mplct temperature constraned systems. In the explct temperature constraned model, a peak temperature constrant prevents the system from overheatng. In the mplct temperature constraned model, the effect of the temperature on the channel qualty controls the system temperature. The schedulng-theoretc approach for energy harvestng communcatons was studed n varous settngs, see [] [5]. Prevous works consdered sngle-user channel [] [4], broadcast channel [5], multple access channel [6], [7], nterference channel [8], two-hop channel [9], [], two-way channel [], [2], and damond channel [3]. Temperature effects are studed n [4], [5]. In [4], a peak temperature constrant s consdered and the optmal contnuous power allocaton s studed. Although extensve nsghts and propertes of the optmal polcy were derved, only the sngle energy arrval case was fully solved. The fact that the power can take any contnuous functon makes the problem challengng, as the problem then becomes a functonal optmzaton problem and the optmal such functon should be obtaned. In ths paper, we consder the slotted verson of the temperature evoluton model consdered n [4], [5]. Ths work was supported by NSF Grants CNS 3-4733, CCF 4-22, CCF 4-2229, and CNS 5-2668. envronment temperature T e data queue E Tx Fg.. System model: the system heats up due to data transmsson. In the frst model we consder here, whch we con as the explct temperature constraned model, we consder an explct peak temperature constrant. Increasng the transmsson power ncreases the throughput and the temperature. Hgher temperature levels mean smaller admssble transmsson power levels for future slots. Ths model s the slotted verson of [4]. We study the optmal power allocaton for the multple energy arrval case, for whch we develop a generalzed waterfllng algorthm. Then, we study suffcent condtons under whch the system operates n the lmtng cases of non-bndng temperature or energy constrants. When the temperature constrant s not bndng, the problem reduces to the sngle-user energy harvestng channel studed n []. When the energy constrant s not bndng, the system effectvely becomes a temperature constraned system wth all the needed energy arrvng before the communcaton begns. In ths case, we show that the optmal powers are bounded and non-ncreasng, and the temperature of the system s non-decreasng. In the second model we consder here, whch we con as the mplct temperature constraned model, the temperature s not explctly constraned, however, the temperature affects the thermal nose whch affects the channel qualty. Ths arses when the dynamc range of the temperature s large, and s smlar to that presented n [6] but n a schedulng-theoretc settng. In ths case, the transmt powers used n prevous channel uses affect the thermal nose and therefore the channel qualty n future channel uses, and hence, the channel becomes a use dependent or acton dependent channel, see [7] [9]. Specfcally, each slot sees the prevous transmssons as nterference through a certan temperature flter. Ths flter arses naturally from dscretzng tme nto tme slots. For the general sgnal to nterference plus nose rato SINR, the problem s non-convex and s a sgnomal problem for whch we acheve a local optmal soluton usng the sngle N + Rx
condensaton method proposed n [2]. We then propose a heurstc algorthm whch mproves upon the local optmal soluton and may acheve the global optmal soluton. Then, we study the extreme settngs of low and hgh SINR regmes. We show that for low SINR regme t s optmal to transmt zero powers for all the slots except for the last, n whch all the harvested energy s transmtted. For the hgh SINR regme, the optmal soluton can be found usng geometrc programmng. II. SYSTEM MODEL We consder an energy harvestng communcaton system n whch the transmtter harvests energy Ẽ n the th slot, see Fg.. We consder the same temperature model consdered at [4], [5]. In ths model, the temperature, Tt, s defned by the followng dfferental equaton, dtt = apt btt T e dt where T e s the envronment temperature, Tt s the temperature at tme t, and a,b are non-negatve constants. Wth the ntal temperature T = T e, the soluton of s: t Tt = e bt e bτ apτdτ +T e 2 In what follows we assume that the duraton of each slot s equal to, whch can take any postve value. Let us defne T T as the temperature level by the end of the th slot, P as the power level used n the th slot. Usng 2, T can be expressed as: T = αt + +γ 3 where α = e b, = a b [ α] and γ = T e[ α]. The effect of n 3 appears through the constantsα,,γ. As the slot duraton ncreases, the values of, γ ncrease whle the value of α decreases; as the slot duraton ncreases, the temperature at the end of the slot becomes more dependent on the power transmtted wthn ths slot and less dependent on the ntal temperature at the begnnng of the slot. We now elmnate the prevous temperature readngs n T makng the temperature a functon of the powers only. We can do ths by recursvely substtutng by T n T to have T k = α k +T e 4 Ths formula shows that the temperature at the end of each slot depends on the power transmtted n ths slot and all prevous slots through an exponentally decayng temperature flter. We note that ths s the same formula that was developed n [6] n whch the slot duraton was assumed to be unty. III. EXPLICIT PEAK TEMPERATURE CONSTRAINT We now consder the model n whch we have an energy harvestng transmtter wth a peak temperature constrant. The nose varance s the same throughout the communcaton sesson and s set to σ 2. We consder a slotted system wth a constant power per slot. It follows from [4, equaton 47], that the temperature s monotone wthn the slot duraton. Hence, for the peak temperature constraned case, t suffces to constran the temperature only at the end of each slot; we begn the communcaton wth the system havng temperature T e. In ths case, the problem can be wrtten as, max {} T k T c, 2 log + P σ 2 Ẽ, k 5 where n the objectve functon and the energy constrant s the slot duraton. In what follows, wthout loss of generalty, we drop snce t s just a constant multpled n the objectve functon and we defne E = Ẽ. We rewrte problem 5 makng use of 4 as, max {} 2 log + P σ 2 α k T c T e, E, k 6 In the last slot, ether the temperature or the energy constrant has to be satsfed wth equalty. Otherwse, we can ncrease one of the powers untl one of the constrants s met wth equalty and ths strctly ncreases the objectve functon. Ths problem s a convex problem n the powers, whch can be solved optmally usng the KKTs and the Lagrangan s: L = log + P σ 2 + k α k=λ k T c T e + k E 7 k=µ Dfferentatng wth respect to and equatng to zero we get, = α D k= λ kα k + D k= µ σ 2 8 k In the optmal soluton, f nether constrant was tght n slot < D, then the power n slot + s strctly less than the power n slot. Ths follows from complementary slackness snce f at slot, f both constrants were not tght then we have λ = µ = whch, usng 8, mples that > +. To fnd the optmal soluton, we begn wth an ntal feasble power allocaton. If for ths power allocaton there exst non-negatve Lagrange multplers whch satsfy 8 and the complementary slackness condtons, then ths s the optmal power allocaton. Otherwse, these power allocatons should be modfed by pourng water away from the slots wth negatve Lagrange multplers to the slots wth hgher Lagrange multplers untl non-negatve Lagrange multplers are found and the complementary slackness condtons are satsfed. Snce ths problem s a convex optmzaton problem, any soluton for the KKTs acheve the global maxmum.
A. Energy Lmted Case In ths subsecton, we study a suffcent condton under whch the system becomes energy lmted. For all slots j n whch the followng s satsfed j E T c T e the temperature constrant cannot be tght. In partcular, when t s satsfed for k = D, then the temperature constrant can be completely removed from the system. To prove ths, we assume for the sake of contradcton that we have at slot j j E Tc Te whle the temperature constrant s tght, whch mples: T c T e = j α j < j 9 j E whch contradcts the assumpton j E Tc Te. The strct nequalty follows snce α <. The structure of the optmal soluton for ths case s studed n []. B. Temperature Lmted Case In ths part, we frst study a suffcent condton for problem 6 to be temperature lmted. The energy constrant s never tght f the followng condton s satsfed: T c T e k < E, k {,...,D} k For the temperature lmted case, an upper bound on the transmsson powers s equal to Tc Te. Hence, s suffcent to satsfy k < k E. In what follows, we study the structure of the optmal polcy for the temperature lmted case. In the last slot, the temperature constrant s satsfed wth equalty. The optmal powers are monotoncally decreasng n tme. The proof follows by contradcton. Assume for some ndex j that we have Pj < P j+. We now form another polcy, denoted as { }, whch has = P for all slots j,j +, whle we change the powers of slots j,j + to be P j = Pj + δ and P j+ = Pj+ δ for small enough δ. Ths δ always exsts as Pj < Pj+ mples that j k= αj k Pk < Tc Te. Snce the objectve functon s strctly concave, ths new polcy yelds a strctly hgher objectve functon, whch contradcts the optmalty of Pj < P j+. Now t remans to check that wth ths new polcy, the temperature constrant s stll feasble for any slot k j + whch follows from:, j,j+ < =, j,j+ α k P +α k j Pj +α k j Pj+ α k P +αk j P j +αk j P j+ 2 α k P < T c T e 3 Moreover, the optmal temperature levels are non-decreasng n tme. To prove ths, t suffces to show that: k+ α k P α k+ P, k = {,...,D } 4 We rewrte 4 as follows, α α k P Pk+, k = {,...,D } 5 Snce, we know that the last slot has to be satsfed wth equalty then we know D αd P = Tc Te. Hence, for the constrant at k = D we have: D α D P T c T e = α D P 6 whch can be wrtten as follows, D α α D P PD 7 whch proves 5 for k = D. Now assume for the sake of contradcton that 5 s false for k = D 2,.e.: D 2 PD < α Substtutng ths n 7, we get: α D 2 P 8 P D = αp D + αp D 9 D 2 < α α α D 2 P + αpd 2 D = α α D P PD 2 But snce we know that n the optmal polcy the powers are non-decreasng, ths s a contradcton and 5 holds for k = D 2. The same argument follows for any k < D 2. In optmal soluton, f the constrant s satsfed wth equalty for two consecutve slots then the power n the second slot must be equal to α Tc Te. To obtan ths, the two consecutve constrants whch are satsfed wth equalty are solved smultaneously for the power n the second slot. Hence, when the temperature hts the crtcal temperature, the optmal transmsson power n all the subsequent slots becomes constant and equal to α Tc Te. Ths follows snce the temperature s ncreasng, thus whenever the constrant becomes tght, t remans tght for all subsequent slots. We now conclude that the transmsson power s bounded as α T c T e T c T e, = {,...,D} 22 The lower bound follows from the dscusson above whle the upper bound follows from the feasblty of the constrants. We then proceed to fnd the optmal power allocatons. Snce the problem s convex, a necessary and suffcent condton s
to fnd a soluton satsfyng the KKTs. The optmal power s gven by settng {µ } = n 8. To ths end we present an approach to obtan the optmal powers. We use lne search to search over the tme slot at whch the temperature constrant becomes tght, whch we denote as. Then slots = { +,...,D} have power allocaton equal to α Tc Te, whle the power allocatons for slots = {,..., } are strctly decreasng and strctly hgher than α Tc Te. Hence, we ntalze = D and search for a soluton for the powers satsfyng the KKTs. If we obtan a soluton then we stop and ths s the optmal soluton. Otherwse, we decrease by one and repeat the search. IV. IMPLICIT TEMPERATURE CONSTRAINT We now consder the case when the dynamc range of the temperature ncreases. In ths case, we need to consder the change n the thermal nose of the system due to temperature changes. The thermal nose s lnearly proportonal to the temperature [2, Chapter ]. The objectve functon s: 2 log + ct +σ 2 23 where c s the proportonalty constant between the thermal nose and the temperature. In ths settng, the nose varance n each slot s determned by the value of the temperature at the begnnng of the slot. The maxmum temperature the system can reach s equal to T max D E + T e. Ths occurs when the transmtter transmts all ts energy arrvals n the last slot. The value of T max s useful n determnng the maxmum possble temperature on the system. As we show n the low SINR case later, the optmal power allocaton results n system temperature equal to T max. Usng 4 n 23, the problem can now be wrtten n terms of only transmsson powers as follows: max {} 2 log + c k= α k P k +T e +σ 2 E, P k, k {,...,D} 24 The problem n ths form hghlghts the effect of prevous transmssons on subsequent slots. The transmsson power at tme appears as an nterferng term at slot ndces greater than wth an exponentally decayng weght due to the flterng n the temperature. Ths problem s non-convex and determnng the global optmal soluton s generally a dffcult task. Next, we adapt the sgnomal programmng based teratve algorthm n [2] for the energy harvestng case. Ths algorthm provably converges to a local optmum pont. The problem n 24 can be wrtten n the followng equvalent sgnomal mnmzaton problem mn {} D c k= α k P k +T e +σ 2 c k= α k P k +T e +σ 2 + E, P k, k {,...,D} 25 The objectve functon n 25 s a sgnomal functon whch s a rato between two posynomals. Note also that the energy harvestng constrants n 25 are posynomals n p. In each teraton we approxmate the objectve by a posynomal. We do ths by approxmatng the posynomal n the denomnator by a monomal. Approprate choce of an approxmaton whch satsfes the condtons n [22] guarantees convergence to a local optmal soluton. Let us denote the posynomal n the th denomnator evaluated usng a power vector P by u P,.e., we have + u P vkp = c α k P k + +ct e +σ 2 26 k= k= where for k = {,..., } we have v k P = cα k P k, v P = and v + P = ct e +σ 2. Usng the arthmetc-geometrc mean nequalty we approxmate each posynomal by a monomal as follows: cα k θ k P θ k cte +σ 2 θ + u P k= θ k θ θ + 27 where + k= θ k = for all = {,...,D}. We now solve the prevous problem teratvely where we ntalze the power allocaton to any feasble power allocaton P. Then, we approxmate the posynomals u P usng the arthmetc-geometrc mean nequalty shown above. In each teraton j where the power allocaton s P j we choose θk as a functon of the posynomals and power allocaton as follows: θ k Pj = v k Pj u P j 28 whch satsfes + k= θ k Pj =. Ths choce of θ k Pj guarantees that the teratons converge to a KKT pont of the orgnal problem [22]. In partcular, for each teraton ths s a geometrc program and as requred by [22], ths can be transformed nto a convex problem; see also [2]. The above teratve approach converges to a local optmal soluton. Achevng the global optmal soluton s of exponental complexty. Alternatvely, to get to the optmal soluton, an approach ntroduced n [23] can be used. Ths approach solves the followng problem teratvely: mn {},t t OP t, t t α, E, P k 29 where OP s the objectve functon of 25 and α s chosen to be a number whch s slghtly more than and t can be ntalzed to be the soluton of problem 25 and then updated as the optmal solutons resultng from 29.
A. Low SINR Case The low SINR case occurs when the ncomng energes are very small wth respect to the nose varance. In ths case, an approxmaton to the logarthm functon n the objectve functon s the lnear functon. Hence, the objectve functon can be rewrtten as, whch s smlar to [24, equaton 4]: c 3 k= α k P k +T e +σ 2 In ths case, the optmal allocaton s zero for all tme slots, except n slot D n whch t s equal to P D = D E. Ths follows snce 3 can be upper bounded by D ct e +σ 2 E ct e +σ 2 3 and t s acheved by the specfed allocaton. Hence, a suffcent condton to have a low SINR regme s D E ct e +σ 2. Note also that the non-causal energy arrval knowledge s not necessary here as all energy arrvals are used only at the very last slot. The temperature at the end of the communcaton sesson s equal to T max = D E +T e. B. Hgh SINR Case When the values of c and σ are small, SINR s hgh and we approxmate the objectve functon by gnorng nsde the logarthm. In ths case, the problem n 24 has the Lagrangan: L = 2 log c + k k=µ k= α k P k +T e +σ 2 E 32 Takng the dervatve wth respect to gves, L = + j=+ and then equatng to zero gves: j=+ cα j c + j k= αj k P k +T e +σ 2 cα j c = j k= αj k P k +T e +σ 2 k= µ k 33 µ k 34 k= Although ths problem s non-convex, t s a geometrc program and we show next that any local optmal soluton for ths problem s also a global optmal soluton. To show ths we propose the followng equvalent problem: mn {x } 2 log c k= α k e x k +T e +σ 2 e x e x E, x R k {,...,D} 35 Ths equvalent problem s obtaned by substtutng = e x. Problem n 35 s a convex optmzaton problem snce the objectve s a convex functon n the form of a log-sumexponent and the constrant set s a convex set [25]. Hence, the KKTs are necessary and suffcent for global optmalty. We next show that any soluton for the KKTs of the orgnal non-convex problem also yelds a global optmal soluton. To prove ths, we frst wrte the Lagrangan of problem 35 as: L = 2 log c k= α k e x k +T e +σ 2 e x + ν k e x E 36 k= Takng the dervatve wth respect to x gves, L x = + j=+ cα j e x c j k= αj k e x k +Te +σ 2 D +e x ν k 37 k= whch provdes the followng necessary condton: e x + j=+ cα j c + j k= αj k e x k +Te +σ 2 ν k = k= 38 Usng the transformaton = logx and settng ν = µ, we observe that any soluton of 34 satsfes 38. Also, complementary slackness correspondng to 32 s satsfed f and only f t s satsfed by those for 35. Snce 35 s convex, any soluton satsfyng the KKTs s global optmal and through the transformaton = logx,µ = ν s also global optmal n the orgnal problem. The equvalent problem n 35 can be solved usng any convex optmzaton toolbox. V. NUMERICAL RESULTS We frst consder the explct peak temperature constraned model. As shown n Fg. 2, n general the power allocaton does not possess any monotoncty. The optmal power allocaton s close to the mnmum of the power allocaton of the energy and temperature lmted cases. We study the temperature lmted case n Fg. 3. When temperature s strctly ncreasng power s strctly decreasng. When the temperature reaches the crtcal temperature, the power remans constant. We then study the mplct temperature constraned model. For the general SINR case, we ntalze the sgnomal programmng problem usng a feasble power allocaton of = mn E n all slots. For the case shown n Fg. 4, the objectve functon takes the value.895 at the the global optmal and our experment verfes that the sngle condensaton method also yelds.895. In general, we observe that the sngle condensaton gves solutons close the global optmal. We then present the hgh SINR case n Fg. 5. In all cases we studed,
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