Wireless Networks with Energy Harvesting and Power Transfer: Joint Power and Time Allocation

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by  Zoran Hadzi-Velkov

ACCEPTEDFORJOURNALPUBLICATION 1
Wireless Networks with Energy Harvesting and
Power Transfer: Joint Power and Time Allocation
Zoran Hadzi-Velkov, Ivana Nikoloska, George K. Karagiannidis, and Trung Q. Duong
Abstract—In this paper, we consider wireless powered com- WPCNs with separated frequency channels for energy broad-
munication networks which could operate perpetually, as the cast and information transmissions (IT), were studied in [8].
base station (BS) broadcasts energy to the multiple energy
Furthermore, [9] studies the WPCN with full-duplex nodes,
harvesting(EH)informationtransmitters.Theseemploy”harvest
6 where the BS is equipped with two antennas. The authors in
then transmit” mechanism, as they spend all of their energy
1 harvested during the previous BS energy broadcast to transmit [10] consider WPCNs, where the nodes choose between two
0 theinformationtowardstheBS.Assumingtimedivisionmultiple power levels, i.e., a constant desired power, or a lower power
2
access (TDMA), we propose a novel transmission scheme for when its EH battery has stored insufficient energy. Finally, in

r jointly optimal allocation of the BS broadcasting power and [11] a three-noderelayingsystem was considered,where both
a time sharing among the wireless nodes, which maximizes the
M overall network throughput, under the constraint of average source and relay harvest energy from the BS, by using WPT.
transmit power and maximum transmit power at the BS. The The above-referenced works analyzed WPCNs (except for

1 proposed scheme significantly outperforms ”state of the art” [10]) and proposed optimal (either time or power) allocation
schemesthat employ onlytheoptimal time allocation. If asingle schemes that aim at maximizing the achievable sum infor-
] EHtransmitterisconsidered,wegeneralizetheoptimalsolutions
T for the case of fixed circuit power consumption, which refers to mation rate (i.e., throughput in WPCNs). In this paper, we
I a much more practical scenario. considera similarWPCN modelasin[7] andproposeanovel
.
s optimal scheme, which jointly allocates the BS broadcasting
IndexTerms—Energyharvesting,wirelesspowertransfer,pro-
c power and the time sharing among the wireless nodes. Our
[ cessing cost.
proposed scheme significantly improves the overall WPCN
2 I. INTRODUCTION throughput, compared to [7], which assumes a constant BS
v Energy harvesting (EH) is considered a revolutionary tech- transmit power. For the point-to-point scenario, the proposed
9
nology for energy-constrained wireless networks, such as scheme is generalized to account for the fixed (circuit) power
8
8 sensor and ad-hoc, due to its capability to provide everlasting consumption by the EHNs, which, to the best of authors’
4 power supply [1]-[2]. Performance of these systems is maxi- knowledge, has not been considered in the literature so far.
0 mized by adapting the output powers of the EH transmitters
.
1 given the energy causality constraint [3]-[4]. However, EH
II. SYSTEMANDCHANNEL MODEL
0 from the environment (e.g., solar or wind) is an intermittent
6 and uncontrollable process. In order to maintain reliable EH-
1 As depicted in Fig. 1(a), we consider a WPCN in a fading
basedcommunication,dedicatedfar-fieldradiofrequency(RF)
:
v radiation is used as energy supply for EH transmitters, which environment,consistingofahalf-duplexBSandK half-duplex
Xi is knownaswireless powertransfer(WPT) [5]. TheWPT can EHNs. The BS broadcasts RF energy to the EHNs, whereas
the EHNs transmit information back to the BS. The EHNs
be realized as a simultaneous wireless information and power
r
are equipped with rechargeable EH batteries that harvest the
a transfer [6], or, alternatively, over a dedicated (either time
RF energy broadcasted from the BS. The IT from EHNs to
or frequency) channel for energy transfer. The latter option
BS (IT phase), and the WPT from BS to EHNs (EH phase)
gives rise to the so-called wireless powered communications
are realized as successive signal transmissions using TDMA
networks (WPCNs), which typically transmit using the time-
over a common channel, where each TDMA frame/epoch is
divisionmultipleaccess(TDMA)[7]-[11].AnexampleWPCN
may be a sensor network consisted of a base station (BS) and of duration T. As depicted in Fig. 1(b), each (TDMA) epoch
consistsofanEHphaseandanITphase,whereastheITphase
multiple EH sensors deployed in a hostile environment, such
thattheysendinformation(e.g.,telemetry)overtheuplinkand itself consists of K successive ITs from EHNs to the BS. We
assume perfect synchronization among all the nodes.
receive energy broadcasted by the BS over the downlink.
In [7], the authors determine the optimal TDMA scheme The fading between the the BS and EHN j (1 ≤ j ≤ K)
among the half-duplexnodes(either BS or EHNs), depending is a stationary and ergodic random process, and follows the
on the channel fading states. The optimal time-sharing in quasi-staticblockfadingmodel,i.e.,thechannelisconstantin
each slot but changes from one slot to the next. The duration
Z. Hadzi-Velkov and I. Nikoloska are with the Faculty of Electrical En- of one fading block is assumed equal to T, and one block
gineeringandInformationTechnologies, Ss.CyrilandMethodius University,
coincides with a single epoch. For convenience,the downlink
1000Skopje, Macedonia (email: {zoranhv,ivanan}@feit.ukim.edu.mk).
G.K.KaragiannidisiswiththeDepartmentofElectricalandComputerEn- (BS-EHNj) and the uplink (EHNj-BS) channels are assumed
gineering, Aristotle University ofThessaloniki, 54124Thessaloniki, Greece, reciprocal,andtheirfadingpowergainsinepochiaredenoted
(e-mail: [email protected]). by x′ . They are normalized by the additive white gaussian
T.Q.Duongis withtheSchool ofElectronics, Electrical Engineering and ji
Computer Science, Queen’s University Belfast, Belfast BT7 1NN, U.K. (e- noise (AWGN) power,yieldingxji =x′ji/N0 with an average
mail:[email protected]). value of Ω =E[x′ ]/N , where E[·] denotes expectation.
j ji 0
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The optimal BS transmit power, p∗, is determined by
i
P , b >λ
p∗ = max i (4)
i (0, bi ≤λ.
The optimal duration of the EH phase, τ∗, is found as the
0i
root of the following transcendental equation,
b P τ∗ b P
log 1+ i max 0i +λP = i max , (5)
1−τ∗ max 1−τ∗ +b P τ∗
(cid:18) 0i (cid:19) 0i i max 0i
Fig.1. (a)WirelesspowerednetworkwithK EHNsandacommonBS,(b) whereas the optimal IT duration of EHNs j is given by
TDMAepoch/frame structure a
τ∗ =(1−τ∗) ji, 1≤j ≤K. (6)
III. WPCN THROUGHPUT MAXIMIZATION ji 0i bi
Let us consider M → ∞ epochs. In epoch i, the BS The constant λ is found from (1/M) Mi=1p∗iτ0∗i =Pavg.
transmits with power p . The duration of the EH phase is, Proof: Please refer to Appendix A.
i
P
τ T,andthe ITdurationofthe EHNj isτ T (1≤j ≤K). Inpractice,constantλisestimatedbyaniterativealgorithm,
0i ji
Note that, 0 < τ < 1 and K τ = 1,∀i, where τ ,∀ij such as the bisection method [13]. Basically, the value of λ is
ji j=0 ji ji
are referred to as the time-sharing parameters. updated following some rule (e.g., subinterval bisection) until
P
Weproposeaschemeforjointlyoptimalallocationofp and theconstraintC1in(3)ismetwithsomepredefinedaccuracy.
i
τ so as to maximize the WPCN overall average throughput,
ji
subject to the average available BS transmit power, Pavg, and IV. POINT-TO-POINTEH SYSTEMWITH PROCESSING COST
maximum BS transmit power, P . The proposedscheme is
max In practical EH transmitters, besides their transmit power,
feasible if, in each epoch i, the BS has perfect knowledge of
an additionalpower is also consumedby its non-idealelectric
the instantaneous fading gains in all BS-EHN channels.
j circuitry (e.g., AC/DC converter, analog RF amplifier, and
During the EH phase of epoch i, the amount of harvested
processor), denoted as the processing energy cost [12]. In
power by EHN j is E = η x N p τ T, where η is the
ji j ji 0 i 0i j this section, we extend the system model with non-negligible
energy harvesting efficiency of EHN j. Note, it is safe to
processing cost to a point-to-pointsystem, consisting of a BS
assume that η =1, because η always appears together with
j j andasingleEHN(K =1)andproposeajointlyoptimalpower
x throughtheproductη x ,whichonlyaffectstheparameter
ji j ji andtimeallocation.Weconsiderthefollowingpracticalmodel
Ω ,buttheformoftheendresultsisunaffected.DuringtheIT
j for the total power consumption of the EHN:
phase,EHNj completelyspendsitsharvestedenergy,E ,for
ji
transmittinga complex-valuedGaussian codewordof duration P +p , P >0
S c S
p = (7)
τjiT, comprised of n→∞ symbols, with an output power t (0, PS ≤0,
E N p x τ
Pj(i)= ji = 0 i ji 0i, (1) where PS is the EHN’s transmit power, and pc is the EHN’s
τjiT τji processing energy cost. Note that the EHN can adapt its
and an information rate, transmit power in each epoch, whereas pc has a fixed value.
The amount of harvested power by the EHN is E =
i
a p τ
Rj(i)=τjilog(1+Pj(i)xji)=τjilog 1+ ji i 0i , N0pixiτ0iT, which is completely spent in the successive IT
(cid:18) τji (cid:19)(2) of duration(1−τ0i)T as Ei =(PS(i)+pc)(1−τ0i)T. Thus,
the EHN’s transmit power in epoch i is given by
where a =N x2 is an auxiliary fading coefficient1.
ji 0 ji
SinceM →∞,theaverageWPCNthroughoutiscalculated P (i)= N0pixiτ0i −p , (8)
as(1/M) Mi=1 Kj=1Rj(i),andcanbemaximizedaccording S 1−τ0i c
to the following optimization problem and the achievable information rate in this epoch is given by
P P
M K R (i)=(1−τ )log(1+P (i)x ).
1 τ S 0i S i
pi,τ0i,mτjia,x1≤j≤K M i=1j=1τjilog(cid:18)1+ajipi τ0jii(cid:19) thrGouivgehnpuPt,av(g1a/nMd)PmMax, Rwe(ai)im. Iant m(3a)x,imwiezinsgetthKe av=erag1e,
XX i=1 S
M
1 introduce (8), and restate the optimization problem in terms
s.t. C1: M piτ0i ≤Pavg of the optimization vPariables PS(i) and τ0i as follows:
i=1
X
C2:0≤p ≤P ,∀i M
i max 1
K max (1−τ0i)log(1+PS(i)xi)
C3: τji =1,∀i PS(i),τ0i,∀i M Xi=1
C4:τXj=ji0>0,∀ij. (3) s.t. C1′ : M1 M 1−xτ0i (PS(i)+pc)≤N0Pavg
i
i=1
Theorem 1: Letusdefineaji =N0x2ji andbi = Kj=1aji. C2′ : (1−Xτ0i)(PS(i)+pc)≤N0Pmaxτ0ixi,∀i
1If the channels are not reciprocal, aji =N0 xjiyji wherePxji and yji C3′ : PS(i)≥0,∀i
denotetheuplinkanddownlinkchannel gains. C4′ : 0<τ0i ≤1,∀i. (9)
ACCEPTEDFORJOURNALPUBLICATION 3
APPENDIX A
bol) K = 1 PROOF OF THEOREM1
m 6
sy K = 5 Pc = 0 The optimization problem (3) is non-convex because of
s per 5 PPmmaaxx == 53*0PWavg the products and ratios of the optimization variables pi and
put (bit 4 τcjhia.nTgheeorfefvoarrei,abwleesreeifo=rmpuilτa0tei,tahnedptrroanbslefomrmby(3i)ntirnotoduaccinognvthexe
h Benchmark curves
ug problem in terms of ei and τji, as
o
ble thr 3 Pc = 0.001mW max 1 M K τ log 1+a ei
hieva 2 ei,τ0iτji,1≤j≤K M Xi=1Xj=1 ji (cid:18) jiτji(cid:19)
c
a
mum 1 s.t. C¯1: 1 M ei ≤P¯
xi M
a i=1
M 0 X
1 2 3 4 5 6 7 8 9 10 C¯2:0≤ei ≤Pmaxτ0i,∀i
Average transmit power (W)
C3, and C4 as in (3). (12)
Fig.2. Maximaumachievable throughput vs.BSaverageoutputpower
Since τ , 1 ≤ j ≤ K, do not appear in C1 and C2 in (3),
ji
In (9), C′ and C′ correspondto C andC in (3), C andC (12) can be split into two optimization problems: The first
1 2 1 2 3 4
are compressed into the single constraint C′, whereas C′ is a optimizes τji, 1≤j ≤K, for given ei and τ0i as
4 3
natural constraint for EHNs transmit power.
K
a e
Theorem 2: The optimal BS transmit power, p∗, is deter- F(e ,τ )= max τ log 1+ ji i
i i 0i ji
mined by 1τ≤jij>≤K0 Xj=1 (cid:18) τji (cid:19)
K
p∗i =(P0,max, −othλe2prcw+iseq. (λp4c)2 +λ<xi < p1c (10) s.t. Xj=1τji =1−τ0i, (13)
and the second optimizes e and τ , ∀i given the optimized
i 0i
The optimal duration of the EH phase, τ∗, is given by the τ , 1≤j ≤K,
0i ji
root of the following transcendental equation,
M
1
N P x2τ∗ max F(ei,τ0i), (14)
log 1−x p + 0 max i 0i +N λP ei,τ0i M
i c 1−τ∗ 0 max Xi=1
(cid:18) 0i (cid:19) 1 M
N P x2 s.t. C¯1: e ≤P¯
= 0 max i . (11) M i
(1−τ0∗i)(1−xipc)+N0Pmaxx2i τ0∗i C¯2:0≤Xi=e1≤P τ ,∀i.
i max 0i
The constant λ is found from (1/M) M p∗τ∗ =P .
i=1 i 0i avg Since (13) is convex optimization problem, its Lagrangian is
Proof: Please refer to Appendix B.
P written as
Note, when setting p = 0, the optimal solutions (10) and
c
(11) reduce to (4) and (5), respectively, when K = 1. This K a e K
L = τ log 1+ ji i −µ τ +τ −1 ,
validatesourproposedoptimalsolutions,because(11)and(10) 1 ji ji 0i
τ  
are derived based upon an alternative approach to that of the Xj=1 (cid:18) ji (cid:19) Xj=1
previous section, cf. (9) vs. (3). The respective generalization  (15)
where µ > 0 is the Lagrange multiplier associated with the
for the case K ≥ 2, that involves the processing cost, yields
constraint in (13). We now differentiate (15) with respect to
non-convexoptimization problem.
τ and set the derivative to zero, i.e., ∂L /∂τ =0 for each
ji 1 ji
j, 1≤j ≤K, which yields the followingset of K equations,
V. NUMERICAL RESULTS
We illustrate our results for EH systems with K = 1 and µ=log 1+ ajiei − ajiτejii , 1≤j ≤K. (16)
K =5 EHNs in Rayleigh fading. The AWGN power is set to (cid:18) τji (cid:19) 1+ajiτejii
N0 = 10−12 W, and the deterministic path loss (PL) of the Givenµ,theseequationscanbesatisfiedforallj,1≤j ≤K,
BS-EHN channel is PL=60 dB. Thus, E[x′1i]=1/PL and if
Ω1 =1/(N0PL)=106. We consider M =105 epochs. a1i = a2i = a3i =......= aKi =C, (17)
Fig. 2 depicts the average throughput (in bits/symbol) vs. τ1i τ2i τ3i τKi
Pavg (inwatts).Twopairsofcurvesarepresentedfor:(1)non- whereC isafunctionofµ.Consideringtheconstraintin(13),
zeroprocessingenergycost,pc =10µW(denotedaspc >0), C is found as C = (1−τ0i)−1 Kk=1aki.
and(2)zeroprocessingenergycost,pc =0.The”benchmark” Thus,we obtainthe optimalτji is givenby(6). Introducing
P
curve refers to the case of a WPCN with fixed BS transmit (6) into (13), we obtain
power(p =P =const.,∀i)andanoptimaltimeallocation
i avg b e
according to [7, Eq. (10)]. Clearly, the joint time and power F(e ,τ )=(1−τ )log 1+ i i , (18)
i 0i 0i 1−τ
allocation significantly outperforms the benchmark scheme. (cid:18) 0i(cid:19)
ACCEPTEDFORJOURNALPUBLICATION 4
M
Now, introducing (18) into (14), we obtain a convex opti- 1 θ p
s.t. C¯1′ : α + i c ≤N P
mization problem, whose Lagrangian is given by M i x 0 avg
i=1(cid:18) i (cid:19)
X
L12 =M M1 XiM=1(1−τ0i)Mlog(cid:18)1+ 1Mb−ieτi0i(cid:19) CC¯¯32′′ :: ααii+≥θ0i,(cid:18)∀ipxci +N0Pmax(cid:19)≤N0Pmax,∀i
−λ M ei−Pavg!+ qiei− µi(ei−Pmaxτ0i). C¯4′ : 0≤θi <1,∀i. (23)
i=1 i=1 i=1
X X X (19)
The Lagrangian of (23) can be written as
where the non-negative Lagrange multipliers λ, q and µ are
i i
associated with the constraints C¯1, the left-hand side of C¯2 1 M α x2
and the right-hand side of C¯2, respectively. L = θilog 1+ i i
M θ
Bydifferentiating(19)withrespecttoτ0i andei,weobtain: Xi=1 (cid:18) i (cid:19)
M
∂L2 =−log 1+ biei + − λ 1 αi+θipc −N0Pavg
∂τ0i (cid:18) 1−τ0i(cid:19) M Xi=1 xi !
1−τb0iie+i biei +µiPmax =0 (20) − M qi αi+θi xpc +N0Pmax −N0Pmax
i=1 (cid:18) (cid:18) i (cid:19) (cid:19)
X
∂L b M M
2 = i −λ+q −µ =0, (21)
∂ei 1+bi1−eτi0i i i + i=1µiαi+i=1υiθi, (24)
AccordingtotheKarush-Juhn-Tucker(KKT)conditions,com- X X
plementary slackness should be satisfied, ∀i: q e = µ (e − where the non-negative Lagrangian multipliers λ, µi, qi and
i i i i υ are respectively associated with C1′, C2′, C3′, and C4′ in
P τ )=0, where q ≥0 and µ ≥0. Following a similar i
max 0i i i
(23). By differentiating(24) with respect to α and θ , we get
mathematical approach as in [12, Section III.A], we consider i i
the following 3 cases. ∂L x2
toCepaosech1i:,Iif.eτ.,0ip∗i==0,0.thSeinnceeie=i =0 0a,ndthenoslapcokwneerssiscoanlldoictaiotends ∂αi = 1+ αiθixi2i −λ+µi−qi =0, (25)
requireq >0andµ =0.From(21),we obtainthecondition
i i ∂L α x2 α x2
qi =λ−bi ≥0. Thus, this case occurs when bi ≤λ. ∂θ =log 1+ θi i − θ +iαix2
Case 2: Let us assume 0 < τ0i < 1 and ei = Pmaxτ0i. i (cid:18) i (cid:19) i i i
Thiscasecorrespondstop∗i =Pmax.Theslacknessconditions −λpc −qi pc +PmaxN0 +υi =0 (26)
requireqi =0andµi >0.From(21),weobtainthecondition: xi (cid:18)xi (cid:19)
µ = bi −λ>0. (22) According to KKT conditions, the complementary
i 1+ biP1m−aτx0iτ0i sqla[cαkn+essθ (cpon/dxiti+onNs Pshould)−bNe Psatisfi]e=d,υ∀θi:=µ0i,αwi her=e
Introducing (22) and e = P τ into (20), we obtain (5). i i i c i 0 max 0 max i i
i max 0i µ ≥0,q ≥0, and υ ≥0. We consider three cases:
i i i
Based upon (5) and (22), it can be shown that the sufficient
Case 1: If θ =0, then α =0 and no power is allocated to
condition for the occurrence of this case is given by b >λ. i i
i epoch i, i.e., p∗ =0.
Case 3: Let us assume 0 < τ < 1 and 0 < e < P τ . i
0i i max i Case 2: Let us assume 0 < θ < 1 and α = (1 −
Then,theslacknessconditionsrequireµ =q =0.From(21), i i
i i θ )N P −p θ /x . This case corresponds to p∗ = P .
we obtaintheequalitye =(1−τ )(1/λ−1/b ).Introducing i 0 max c i i i max
i 0i i The slackness conditions require q > 0 and µ = υ = 0.
thisequalityinto(20)leadstotheconditionλ/b +log(b /λ)= i i i
i i From (25), we obtain the condition
1, which is satisfied for b =λ. This case occurs only for one
i
xbspejeiecaiitrfihececrvoa0nlut(ienCuoaosfeubsi1,r)abnoudrtoiPtmsmoavcxacrui(aCrbraelsenesc.e2T)ph.ruosb,atbhielitoypitsimzaelrop∗isincacne qi = 1−xipc+ Pxm2iax(1−θiθi)N0x2i −λ>0. (27)
Introducing(27)andα =(1−θ )N P −p θ /x into(26),
i i 0 max c i i
APPENDIX B weobtain(11).Basedupon(11)and(27),itcanbeshownthat
PROOF OF THEOREM2 the sufficientconditionfortheoccurrenceofthiscase isgiven
The optimization problem (9) is non-convex, because the by −λpc/2+ (λpc/2)2+λ<xi <1/pc.
constraints C1′ and C2′ are not convex and the objective Case 3: Letpus assume 0 < θi < 1 and 0 < αi < (1−
functionis notconcave in τ0i and PS(i). Therefore,we refor- θi)N0Pmax−pcθi/xi. Then, the slackness conditions require
mulate this problem by introducing the change of variables, µi = qi = υi = 0. From (25), we obtain the equality αi =
θi = 1−τ0i and αi = (1−τ0i)PS(i)/xi, and transform (9) θi(1/λ−1/x2i). Introducing this equality into (26) leads to
into convex problem is terms of θi and αi, the equality condition λ(1 −pcxi)/x2i +log(x2i/λ) = 1. It
implies that this case occursonly for one specific value of x ,
max 1 M θ log 1+ αix2i but its occurrence probability is zero since xi is a continuouis
αi,θi,∀i M i=1 i (cid:18) θi (cid:19) random variable. Thus, the optimal p∗i can be either 0 (Case
X 1) or P (Case 2).
max
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