Renewables powered cellular networks: Energy field modeling and network coverage

Transcription

1 Title Renewales powered cellular networks: Energy field modeling and network coverage Authors Huang, K; Kountouris, M; Li, VOK Citation IEEE Transactions on Wireless Communications, 4, v 4 n 8, p Issued Date 4 URL Rights IEEE Transactions on Wireless Communications Copyright IEEE; 4 IEEE Personal use of this material is permitted Permission from IEEE must e otained for all other uses, in any current or future media, including reprinting/repulishing this material for advertising or promotional purposes, creating new collective works, for resale or redistriution to servers or lists, or reuse of any copyrighted component of this work in other works; This work is licensed under a Creative Commons Attriution-NonCommercial-NoDerivatives 4 International License

2 Renewale Powered Cellular Networks: Energy Field Modeling and Network Coverage Kaiin Huang, Marios Kountouris and Victor O K Li arxiv:4474v3 [csit] 8 Mar 5 Astract Powering radio access networks using renewales, such as wind and solar power, promises dramatic reduction in the network operation cost and the network caron footprints However, the spatial variation of the energy field can lead to fluctuations in power supplied to the network and therey affects its coverage This warrants research on quantifying the aforementioned negative effect and designing countermeasure techniques, motivating the current work First, a novel energy field model is presented, in which fixed maximum energy intensity γ occurs at Poisson distriuted locations, called energy centers The intensities fall off from the centers following an exponential decay function of squared distance and the energy intensity at an aritrary location is given y the decayed intensity from the nearest energy center The product etween the energy center density and the exponential rate of the decay function, denoted as ψ, is shown to determine the energy field distriution Next, the paper considers a cellular downlink network powered y harvesting energy from the energy field and analyzes its network coverage For the case of harvesters deployed at the same sites as ase stations BSs, as γ increases, the moile outage proaility is shown to scale as cγ πψ +p, where p is the outage proaility corresponding to a flat energy field and c is a constant Susequently, a simple scheme is proposed for counteracting the energy randomness y spatial averaging Specifically, distriuted harvesters are deployed in clusters and the generated energy from the same cluster is aggregated and then redistriuted to BSs As the cluster size increases, the power supplied to each BS is shown to converge to a constant proportional to the numer of harvesters per BS Several additional issues are investigated in this paper, including regulation of the power transmission loss in energy aggregation and extensions of the energy field model Index Terms Cellular networks, renewale energy sources, energy harvesting, stochastic processes I INTRODUCTION The exponential growth of moile data traffic causes the energy consumption of radio access networks, such as cellular and WiFi networks, to increase rapidly This not only places heavy urdens on oth the electric grid and the environment, ut also leads to huge network operation cost A promising solution for energy conservation is to power the networks using alternative energy sources, which will e a feature of future green telecommunication networks [] However, the spatial randomness of renewale energy can severely degrade the performance of large-scale networks, hence it is a fundamental issue to address in network design Considering a cellular network with renewale powered ase stations BSs, K Huang and V O K Li are with the Dept of Electrical and Electronic Engineering at The University of Hong Kong, Hong Kong huangk@eeehkuhk, vli@eeehkuhk M Kountouris is with the Mathematical and Algorithmic Sciences La, Huawei France R&D, Paris, France marioskountouris@huaweicom Part of this work has een presented at IEEE Intl Conf on Comm Systems ICCS in Nov 4 this paper addresses the aforementioned issue y proposing a novel model of the energy field and quantifying the relation etween its parameters and network coverage Furthermore, the proposed technique of energy aggregation is shown to effectively counteract energy spatial randomness Studying large-scale energy harvesting networks provides useful insight to network planning and architecture design This has motivated researchers to investigate the effects of oth the spatial and temporal randomness of renewales on the coverage of different wireless networks spread over the horizontal plane [] [5] Poisson point processes PPPs are used to model transmitters of a moile ad hoc network MANET in [] and BSs of a heterogeneous cellular network [3] Energy arrival processes at different transmitters are usually modeled as independent and identically distriuted iid stochastic processes, reducing the effect of energy temporal randomness to independent on/off proailities of transmitters Therey, under an outage constraint, the conditions on the network parameters, such as transmission power and node density of the MANET [] and densities of different tiers of BSs [3], can e analyzed for a given distriution of energy arrival processes The assumption of spatially independent energy distriutions is reasonale for specific types of renewales that can power small devices, such as kinetic energy and electromagnetic EM radiation, ut does not hold for primary sources, namely wind and solar power To some extent, energy spatial correlation is accounted for in [4], [5] ut limited to EM radiation It is proposed in [4] that nodes in a cognitive-radio network opportunistically harvest energy from radiations from a primary network, esides intelligent sharing of its spectrum The idea of deploying dedicated stations for supplying power wirelessly to energy harvesting moiles in a cellular network is explored in [5] In [4], [5], radiations y transmitters with reliale power supply form an EM energy field and its spatial correlation is determined y the EM wave propagation that does not apply to other types of renewales such as wind and solar power It is worth mentioning that the analysis and design of large-scale wireless networks using stochastic geometry and geometric random graphs [6] [8] has een a key research area in wireless networking in the past decade Similar mathematical tools have also een widely used in the area of geostatistics concerning spatial statistics of natural resources including renewales, where research focuses on topics such as model fitting, estimation, and prediction of energy fields [9], [] The two areas naturally merge in the new area of large-scale wireless networks with energy harvesting It is in this largely uncharted area that the current work makes some

3 radio link radio link power-transmission line ase stations on-site harvester aggregator ase station distriuted harvesters a Base station powered y an on-site harvester Base stations powered y distriuted harvesters Fig Base stations are powered y energy harvesting using either a on-site harvesters or distriuted harvesters initial contriutions Intermittence of renewales introduces stochastic constraints on availale transmission power for a wireless device This calls for revamping classic information and communication theories to account for such constraints and it has recently attracted extensive research efforts [] [7] As shown in [] from an information-theoretic perspective, it is possile to avoid capacity loss for an AWGN channel due to energy harvesting provided that a attery with infinite capacity is deployed to counteract the energy randomness From the communication-theoretic perspective, optimal power control algorithms are proposed in [], [3] for single-user systems with energy harvesting, which adapt to the energy arrival profile and channel state so as to maximize the system throughput These approaches have een extended to design more complex energy harvesting systems, such as interference channels [4] and relay channels [5], to design medium access protocols [6], and to account for practical factors such as non-ideal atteries [7] The prior work assumes fast varying renewale energy sources such as kinetic activities and EM radiation for which adaptive transmission proves an effective way for coping with energy temporal randomness In contrast, alternative sources targeted in this paper, namely wind and solar radiation, may remain static for minutes to hours, which are orders of magnitude longer than the typical packet length of milliseconds In other words, these sources are characterized y a high level of spatial randomness ut very gradual temporal variations Thus, this paper focuses on counteracting energy spatial randomness instead of adaptive transmission that is ineffective for this purpose In practice, spatial renewale energy maps are created y interpolating data collected from sparse measurement stations separated y distances typically of tens to hundreds of kilometers [8] [] The measurement data is too coarse for constructing the energy field targeting next-generation small cell networks with cell radiuses as small as tens of meters Furthermore, the field depends not only on atmospherical conditions eg, cloud formation and moility ut also on the harvester/bs deployment environment such as locations and heights of uildings, trees and cell sites Mapping the energy field requires complex datasets that are difficult to otain Even if such datasets are availale, the resultant energy-field model may not allow tractale network performance analysis This motivates the current approach of modeling energy fields using spatial random processes derived from PPPs Such processes with a well developed theory are widely agreed to e suitale models for natural phenomena and resources [9], [] as well as random wireless networks [6] and thus provide a desirale tradeoff etween tractaility and practicality Note that stochastic-geometry network models in the literature were developed to overcome the same difficulty of lacking real data [6] In the proposed model, the random locations of fixed maximum energy intensity denoted as γ, called energy centers, are modeled as a PPP with density λ e, corresponding to sites on top of tall uildings with exposure to direct sunshine or strong winds From an energy center, the energy intensity decays exponentially with the squared distance normalized y a constant ν, called the shape parameter, specifying the area of influence y the said center It is worth mentioning that the decay function is popularly used in solar-field mapping [] and atmospheric mapping [3] The energy intensity at an aritrary location is then given y the decayed intensity with respect to the nearest energy center The characteristic parameter ψ of the energy field, defined as ψ = λ e ν, is shown later to determine its distriution It is worth mentioning that general insights otained in this work using the aove model, such as that increasing energy spatial correlation reduces outage proaility, also hold for other models so long as they

4 3 oserve the asic principle of renewale energy field: energy spatial correlation etween two locations increases as their separation distance decreases and vice versa [9] In the paper, BSs of the cellular network are assumed to e deployed on a hexagonal lattice while moiles are distriuted as a PPP The network is assumed to operate in the noise-limited regime where interference is suppressed using techniques such as orthogonal multiple access or multi-cell cooperation The regime is most interesting from the energy harvesting perspective since network performance is sensitive to variations of transmission powers or equivalently, harvested energy For instance, it has een shown that in the interference limited regime, the network coverage is independent of BS transmission power since it varnishes in the expression for the signal-to-interference-and-noise ratio with noise removed [4] A moile is said to e under network coverage if an outage constraint is satisfied and the outage proaility is the performance metric Each BS allocates transmission power simultaneously to moiles, either y equal division of the availale power, called channel-independent transmission, or y channel inversion, called channel inversion transmission As illustrated in Fig, each BS is powered y either an onsite harvester or a remote energy aggregator that collects energy generated y a cluster of neary distriuted harvesters over transmission lines Aggregators and distriuted harvesters are deployed on hexagonal lattices with densities λ a and λ h, respectively Connecting harvesters to their nearest aggregators form harvester clusters Note that energy aggregation is ased on the same principle of tackling energy randomness y energy sharing as other existing techniques designed for renewale powered cellular systems see eg, [5], [6] Prior work focuses on algorithmic design while the current analysis targets network performance The main contriution of this work is the estalishment of a new approach for designing large-scale renewale powered wireless networks that models the energy field using stochastic geometry and applies such a model to network design and performance analysis The analysis ased on the approach allows the interplay of different mathematical tools such as stochastic geometry theory, proailistic inequalities and large deviation theory, leading to the following new findings Consider the on-site harvester case If the characteristic parameter ψ is small ψ /π and the maximum energy densityγ is large γ, the outage proaility monotonically decreases with increasing ψ and γ in the form of cγ πψ + p with p eing the proaility corresponding to a flat energy field and c a constant The result holds for oth channel inversion and channelindependent transmissions Despite this similarity, the former outperforms the latter y adapting transmission power to moiles channels Next, consider distriuted harvesters and define the size of a harvester cluster as λ h /λ a As the cluster size increases, energy aggregation is shown to counteract the spatial randomness of the energy field and therey stailize the power supply for BSs Specifically, the power it distriutes to each BS converges to a constant proportional to the numer of harvesters per BS In other words, the energy field ecomes a reliale power supply for the network However, an insufficiently high voltage used y harvesters for power transmission to aggregators can incur significant energy loss It is found that the loss can e regulated y increasing the voltage inversely with the aggregator density to the power of 4 Last, key results are extended to two variations of the energy field model characterized y a shot noise process and a power-law energy decay function, respectively The remainder of the paper is organized as follows The mathematical models and metrics are descried in Section II The energy field model is proposed and its properties characterized in Section III The network coverage is analyzed for the cases of on-site and distriuted harvesters in Sections IV and V, respectively The extensions of the energy field model are studied in Section VI Simulation results are presented in Section VII followed y concluding remarks in Section VIII II MODELS AND METRICS The spatial models for the energy field, energy harvesters, and cellular network are descried in the susections The notations are summarized in Tale I A Energy Harvester Model Aggregation loss arises in the scenario of distriuted harvesters, corresponding to the power loss due to transmissions from harvesters to aggregators over cales see Fig It is impractical to assume high voltage transmission from distriuted harvesters having small form factors and thus aggregation loss can e significant, which is analyzed in the sequel On the other hand, each aggregator supplies power to λ /λ a BSs also over cales, assuming that λ /λ a is an integer for simplicity Aggregators are assumed to e much larger than harvesters and thus can afford having relatively large transformers Thus, high voltages can e reasonaly assumed for power distriution such that the power loss is negligile Consequently, the specific graph of connections etween aggregators and BSs has no effect on the analysis except for the numer of BSs each aggregator supports The energy field is represented y Ψ, which is determined y the function gx mapping a location X R to energy intensity The discussion of the stochastic geometry model of the energy field is postponed to Section III, whereas its relation with energy harvesting is descried elow Let gx,t represent the time-varying version of gx with t denoting time Harvesters are assumed to e homogeneous and time is partitioned into slots of unit duration Let η, denote a constant comining factors such as the harvester physical configuration and conversion efficiency, referred to as the harvester aperture y analogy with the antenna aperture Then the amount of energy harvested at X in then-th slot isηg n X or equivalently η n+ n gx, tdt B Cellular Network Model As illustrated in Fig, the traditional model of a cellular network is adopted, in which the BSs are deployed on a hexagonal lattice with density λ, denoted as Φ, and consequently

5 4 Symol Φ e,λ e Φ h, λ h Φ, λ Φ a, λ a TABLE I SUMMARY OF NOTATIONS Meaning Process of energy centers and its density Set of harvesters on a hexagonal lattice and its density Set of BSs on a hexagonal lattice and its density Set of aggregators on a hexagonal lattice and its density fd Energy decay function of distance d gx Energy intensity at location X γ ψ η B, P C, K R,n, H,n p out, ǫ P V τ Maximum energy intensity of the energy field Characteristic parameter of the energy field Harvester aperture Typical BS and its transmission power Cell served y B and the numer of moiles in the cell Propagation distance and channel coefficient for the n-th moile in C Outage proaility and its constraint Reduction of P due to aggregation loss Harvester voltage for power transmission Fixed multiplier of P in, representing regulated aggregation loss + Collocated BS and harvester a On-site harvesters co-located with BSs + BS Harvester Aggregator the plane is partitioned into hexagonal cells with areas of/λ Let B, P and K denote the typical BS, its transmission power and the numer of simultaneous moiles the BS serves, respectively Since adaptive transmission is effective for counteracting energy spatial randomness as mentioned earlier,p is assumed to e fixed and equal to the power supplied to the BS so as to keep the exposition concise In addition, circuit power consumption of each BS is assumed to e negligile compared with P Moiles are assumed to e distriuted as a PPP with density λ u and all moiles are scheduled for simultaneous transmissions Then K is a Poisson random variale with mean λ u /λ A signal transmitted y a BS at X R with power P is received at a moile at Y R with power given as PH XY X Y α, where X Y is the Euclidean distance etween X and Y, α > the path-loss exponent and the random variale H XY, called a channel coefficient, models fading or shadowing The channel coefficients are assumed to e iid Assuming unit noise variance, the received power also gives the received SNR In the noise-limited network, the condition for reliale data decoding at a moile is specified y the outage constraint that the received SNR exceeds a given threshold except for a small proaility ǫ Define an outage event as one that the receive SNR of a typical active moile is elow The outage proaility is the network performance metric and defined for the two transmission strategies as follows First, consider channelindependent transmission, where a BS equally allocates supplied power for transmission to moiles Let R and H The analysis can e modified to account for fixed BS circuit power, denoted as ϕ, y replacing P with P ϕ in the definitions of outage proaility in and in the sequel However, the notation and derivation steps are made tedious while the analytical methods and key insights remain unchanged Distriuted harvesters Fig Geometric patterns of hexagonal cells, BSs, harvesters and aggregators plotted/marked using solid lines, crosses, dots and circles, respectively Power transmission lines from harvesters to aggregators and those from aggregators to BSs are plotted using dashed and solid lines, respectively denote the propagation distance and the channel coefficient of a typical user in the typical cell, respectively Then, the outage proaility, denoted as p id out, can e written as p id P H R α out = Pr < K where given equal power allocation, the transmission power for each of the K users in the typical cell is P /K Alternatively, the outage proaility can e defined with respect to a set of users in the same cell as the proaility that the received SNR for least one user is elow the threshold Following similar steps as in the susequent analysis, one expect the resultant outage proailities to have similar expressions as those in Propositions -4 that arise mainly from the distriution of the energy field instead of the the specific definition of outage

6 5 proaility Next, consider channel inversion transmission Transmission power required for each moile is computed y channel inversion in an attempt to ensure that the receive SNR is equal to Specifically, the n-th moile in the typical cell is under coverage if the allocated transmission power is no smaller than R,n α /H,n, where R,n and H,n represent the corresponding propagation distance and channel coefficient, respectively It is difficult to write down the outage proaility for a typical active moile, denoted asp iv out, since it depends on the channels of other active moiles sharing the BS However, it can e ounded using the union ound as p iv out P Pr K < R α,n H n=,n The upper ound in is the proaility of the event that at least one user is in outage, which is the union of the outage events of individual users III ENERGY FIELD MODEL AND ITS PROPERTIES A Energy Field Model The energy field Ψ refers to the set of energy intensities at different locations in the horizontal plane with the maximum denoted as γ > The PPP modeling for the energy centers and its density are denoted as Φ e R and λ e, respectively The energy intensity function gx is defined for a given location X R as follows: gx = γ max Y Φ e f X Y 3 where the energy decay function f is fd = e d /ν, d > 4 Note that gx is a Boolean random function [7] The positive parameter ν in 4 controls the shape of f, thus called the shape parameter, and therey determines the area of influence of an energy center Then the characteristic parameter ψ is defined as ψ = νλ e A large value of ψ corresponds to an energy field with gradual spatial variation and a small value indicates that the field has a lot of shadows, where energy intensities are much lower than the peak As oserved from the plots in Fig 3, increasing λ e or decreasing ν introduces more ripples in the energy field; the field is almost flat for a large characteristic parameter eg, ψ = or λ e = and ν = An alternative model of the energy field can result from replacing the max operator in the energy intensity function in 3 with a summation The two models are shown in Section VI-A to have similar stochastic properties in the operational regime of interest Furthermore, the energy field ased on a different energy decay function is considered in Section VI-B and its effect on the network coverage is analyzed B Energy Field Properties First, the distriution function of the energy intensity can e easily otained y relating it to a Boolean model To this end, define rx as the distance from an energy center to a location with the decayed energy intensity x Then rx can e otained from the equation frx = x with f in 4 as follows: x rx = f = νln γ γ x 5 Moreover, let BX,r denote a disk in R centered at X and with a radius r The region of the energy field where energy intensities exceed a threshold x corresponds to a Boolean model X Φ e BX,rx Then for given X and x [,γ], the distriution function of gx can e written in terms of the model and otained as PrgX x = Pr X / BX,rx X Φ e = e πλer x πψ x = 6 γ where the last equality is otained y sustituting 5 and using the definition of ψ Next, the mean and variance of the energy intensity function can e directly derived using the distriution function in 6 as follows: E[gX] = πψγ +πψ, 7 E[g X] = πψγ +πψ, vargx = πψγ +πψ+πψ As ψ, E[gX] is seen to converge to γ while vargx diminishes inversely with ψ On the other hand, as ψ, oth quantities ecome proportional to ψ Last, the joint distriution of the energy intensities gx and gx at two different locations X and X is derived They are independent if X X rx +rx : PrgX x,gx x = PrgX x If X X < rx +rx, PrgX x πψ x x = γ 8 PrgX x,gx x πνλe x x 9 = γ e λe BX,rx BX,rx where the operator applied on a set gives its measure area For ease of notation, rewrite rx and rx as r and r and define d = X X The area of the overlapping region

7 6 8 8 Energy Intensity 6 4 Energy Intensity a λ e =,ν = λ e =,ν = 8 8 Energy Intensity 6 4 Energy Intensity c λ e =,ν = d λ e =,ν = Fig 3 Energy field for different cominations of the energy center density λ e and shape parameter ν and different realizations of the energy-center process etween BX,r and BX,r is known to e given as r BX,r BX,r = r cos +d r + dr r +r +d r cos r d r dr +d r d For the special case of x = x = x thus r = r = r, PrgX x,gx x = where d/r and πνλe d/r x γ y = π cos y y 4 y 4π Comparing 9 and, one can see that reducing the distance etween two locations increases the joint cumulative distriution function of the corresponding energy intensities as they ecome more correlated IV NETWORK COVERAGE WITH ON-SITE HARVESTERS A Network Coverage with Channel-Independent Transmission To facilitate analysis, the outage proaility is decomposed as p id out = p +p a with p and p a defined as P H R α p a = Pr < K H K Rα Pr K H, p = Pr Rα K H Rα > The component p a is the proaility that the energy intensity at the typical BS site is so low as to cause an outage event at the typical moile even though the maximum intensity is sufficiently high for avoiding such an event The other component p represents the proaility that the maximum energy intensity at the typical BS is insufficiently high for avoiding an outage event at the typical moile, which occurs due to the comined effect of the numer of simultaneous users K eing too large, the channel gain eing too small, and the propagation distance R eing too long In other

8 7 words, p a and p are the components of outage proaility arising from the energy field spatial variation and the limit on the maximum harvested power, respectively The outage proaility is characterized y analyzing p a and p separately To this end, it is useful to define a random variale R as R = λ R that gives the propagation distance of a moile uniformly distriuted in a cell of unit area Note that a hexagon of unit area can e outer ounded y a disk with the area of π 3 Let D denote the distance from the center of the disk to 3 a random point uniformly distriuted in the disk Then Pr R x PrD x = 3 3 x, x The proaility p a is shown in the following lemma to e an exponential function of the energy field characteristic parameter ψ, when ψ is sufficiently small The proof of the lemma is provided in Appendix A Lemma The proaility p a can e upper ounded as c E [ H ] minπψ, λu p a 4 +απψ where c = 3 3 α γηλ +α One can oserve from that p is the tail proaility of the product of multiple random variales and thus it is difficult to derive a closed-form expression for the proaility Following typical approaches, we apply a proailistic inequality, namely Markov s inequality, to otain an upper ound on p and large deviation theory to characterize its asymptotic scaling The following lemma is otained ased on Markov s inequality with the proof in Appendix B Lemma The proaility p in can e upper ounded as p c 3λ u E [ H ] where the constant c 3 = +α λ + α 3 3 α, 5 Given p id out = p +p a, comining the results in Lemmas and leads to the following first main result as follows Proposition Consider the scenario where BSs adopt channel-independent transmission and are powered y on-site harvesters The outage proaility can e ounded as p id out c E [ H ] minπψ, λu + c 3λ u E [ H ] +απψ λ +α γη λ +α Consider the scenario where the maximum harvested power γη is large and the characteristic parameter ψ is small Then the outage proaility decreases monotonically with these parameters following the scaling law of p id out γη πψ 6 which is verified y simulation in the sequel Next, p is analyzed using large deviation theory To this end, it is necessary to consider a specific type of distriution for the channel coefficient H Let F denote the complementary cumulative distriution function CCDF of the random variale RV H in Assume that H is a regularly varying random variale defined y the following condition: lim t Fζt Ft = ζ ω 7 where ω > is the exponent of the distriution and ζ > One example is that H has the chi-squared distriution that is typical for wireless channels: PrH < t = Γω t x ω e x dx, t 8 where ω > is a positive integer and the distriution parameter and Γ denotes the gamma function Then H is a regularly varying RV with the exponent ω Based on the assumption and applying Breiman s Theorem see Lemma 7 in Appendix C, an asymptotic ound on p is otained as shown elow Lemma 3 Suppose that the inverse of a channel coefficient is a regularly varying RV As, the proaility p is ounded as α λ p E[K ω ]Pr 3 3 H, where E[K] ω is given as ω m E[K ω λu m m ] = m k k ω 9 m k m= λ k= In particular, if a channel coefficient is a chi-squared RV with parameter ω, p E[K] ω λ 3 3 α ω +O ω, Remark: It is possile to derive asymptotic upper ounds on p for other types of channel coefficient distriution For instance, [8, Theorem ] can e applied to otain a ound for the su-exponential distriution defined y the condition F Fx lim x = where * denotes convolution Such Fx results will change the second ut not the first term of the upper ound on p id out in Proposition presented shortly Comining the results in Lemmas and 3 leads to the second main result as follows: Proposition Consider the scenario where BSs adopt channel-independent transmission and are powered y on-site harvesters Suppose that the inverse of a channel coefficient is Two asymptotic relation operators, and, are defined as follows Two functions hx and qx are asymptotically equivalent, denoted as hx hx hx gx, if lim x = The case of limx gx qx is represented y hx qx

9 8 a regularly varying RV, the outage proaility can e ounded as as follows: p id λ + α minπψ,+ γη out +απψ c E [ H ] λu α ω E[K ω ] λ 3 Pr 3 H < In particular, if channel coefficients follow iid chi-squared distriutions, as, p id λ + α minπψ, γη out + +απψ c Γω λ u E[K ω ] Γω + λ 3 3 α ω +O ω Comparing the results in Propositions and, oth upper ounds on p id out have identical first term, which is dominant when is large The second term in, proportional to γη ω, decays faster than the counterpart in Proposition, proportional to γη, when ω > This is due to a tighter ound on p derived using large deviation theory with respect to that otained using Markov s inequality B Network Coverage with Channel Inversion Transmission Following the same method as in the preceding susection, the outage proaility in can e decomposed as p iv out p c +p d where K p c = Pr P < p d = Pr Pr K n= K n= R α,n H n=,n R α,n H,n R,n α > H,n K n= R α,n H,n, Their key difference from those of their counterpartsp and p a in the preceding section is that the transmission power of a typical BS for the current case is a compound Poisson random variale However, the expected transmission powers for oth cases are identical due to the following equality: E [ K R α,n H n=,n ] = E[K ]E [ R,n α ] [ E H,n] Given this equality, the ounds onp a in andp in can e shown to hold for p c in and p d in, respectively, yielding the following proposition Proposition 3 Consider the scenario of on-site harvesters The upper ound on the outage proaility for the case of channel-independent transmission as given in Proposition also holds for the case of channel inversion transmission Despite the identical upper ounds, the outage proaility for channel inversion transmission is lower than that for channel-independent transmission The performance gain is due to adapting transmission-power allocation to multiuser channel states to minimize the numer of moiles in outage Next, p c is analyzed using large deviation theory To this end, some useful definitions and results are introduced as follows The distriution of a RV X has a light tail if E [ e δx] < for some δ > and a heavy tail if E [ e δx] = for all δ > The following result is from [9, Theorem ] Lemma 4 Let {X m } e a set of iid random variales having a heavy-tailed distriution and N have a light-tailed distriution independent of those of {X m } Then N Pr X m > t E[N]PrX > t as t m= To apply the result, the upper ound on p d in is simplified using the inequality R,n λ /3 3 as K p d Pr H,n > λ /3 3 α 3 n= One can see that the upper ound is the tail proaility of a compound Poisson RV, K n= H,n Note that the Poisson distriution of K has a light tail since it decays faster than the exponential rate as oserved from the following inequality see eg, [3, Theorem 54]: PrK x e λm/λ eλ m /λ x x x, if x > λ m /λ Then the lemma elow follows from 3 and Lemma 4, which is the channel inversion counterpart of Lemma 3 Lemma 5 Suppose that the inverse of a channel coefficient has a heavy-tailed distriution As, the proaility p d is ounded as α p d λ λ u 3 3 Pr H, 4 λ In particular, if a channel coefficient is a chi-squared RV with parameter ω, p d λ u λ λ 3 3 α ω +O ω, As mentioned, the ound on p a in holds for p c in Comining this result, the ounds on p d in Lemma 5 and the inequality p iv out p c +p d gives the following result Proposition 4 Consider the scenario where BSs adopt channel inversion transmission and are powered y on-site harvesters Suppose that the inverse of a channel coefficient is a heavy-tailed RV, the outage proaility can e ounded as follows: as, p iv out +απψ λ u λ Pr λ +α minπψ,+ γη c E [ H ] λu α λ H < 3 3

10 9 In particular, if channel coefficients follow iid chi-squared distriutions, as, minπψ, p iv out λ +α γη + +απψ c Γω λ u λ u Γω +λ λ 3 3 α ω +O ω Comparing the results in Propositions and 4, the outage proaility for channel inversion transmission follows the same scaling laws as those for channel-independent transmission except for some difference in linear factors As a result, p iv out is smaller than p id out, as also oserved in the simulation results in the sequel V NETWORK COVERAGE WITH DISTRIBUTED A Network Coverage HARVESTERS In this section, it is shown that aggregating energy harvested y many distriuted harvesters stailizes the power supplied to the BSs y the law of large numers It is assumed that the energy aggregation loss is regulated such that the scaling factor of BS transmission powers due to such loss is smaller than a constant τ, The design requirement under this constraint is analyzed in the next susection The harvester lattice partitions the plane into small hexagonal regions, each with area of /λ h Whether each region contains an energy center can e indicated y a set of independent Bernoulli random variales {Q n } with proailities exp λ e /λ h and [ exp λ e /λ h ] for the values of and, respectively Despite the independence of the numers of energy center in different regions, it is important to note that the energy intensities at harvesters are correlated [see their joint distriution in 8 and 9] The energy intensity at each harvester is at least exp 3 if the corresponding small region contains an 3νλ h energy center or otherwise takes on some positive value Based on the aove points, the transmission power of the typical BS is lower ounded as P τγηλ a λ M Q n e n= 3 3νλ h = τγηλ h λ λ am λ h M M n= Q n e 3 3νλ h 5 wherem represents the numer of harvesters connected to the typical aggregator Consider the scenario of sparse aggregators with each connected to many harvesters, corresponding to λ a As a result, M and λ a M /λ h Then it follows from 5 that lim P λ a τγηλ h λ a λ lim M M Q n e M n= 3 3νλ h 6 The law of large numers gives the following lemma Lemma 6 With the harvester density λ h fixed, as the aggregator density λ a, the transmission power for the typical BS converges as lim P λ a τγηλ h e λe λ h e 3 3νλ h, as 7 λ a λ In other words, P is lower ounded y a constant and thus its randomness due to energy spatial variation diminishes If the energy centers are dense λ e λ h and the shape parameter ν is large νλ h, the transmission power approaches its upper ound γηλ h /λ The power stailization y the spatial averaging of the energy field removes one random variale from the outage proaility Consider the case of channel-independent transmission and the corresponding outage proaility in Using Lemma 6 and applying Markov s inequality, the outage proaility for small aggregator density can e ounded as lim λ a pid out λ a λ E[H ]E[K]E[R α ] 8 τγηλ h e λe λ h e 3 3νλ h Sustituting E[K] = λ u /λ and the result in 48 yields the result in Proposition 5 as given elow The result is proved to also hold for channel-inversion transmission using and following a similar procedure Proposition 5 With the harvester density λ h fixed, as the aggregator density λ a, the outage proailities for oth the channel-independent and channel inversion transmissions can e ounded as lim λ a pid outλ a, lim λ a piv outλ a The results in Proposition 5 suggest that c 3 E [ H ] λ u τγηλ α λ h e λe λ h e 3 3νλ h lim λ pid outλ a, lim a λ piv outλ a a γη λ u λ α λ h /λ λ where the factors represent in order the inverses of the maximum power generated y a single harvester, the numer of harvesters per BS, the expected numer of active moiles per cell, and the expected propagation loss B Energy Aggregation Loss It is well known that the power line loss, P, for transmitting the power of P to a receiver is given as P = βp d V 9 where V is the voltage and β is a constant depending on the power line characteristics, such as resistivity and cross-section area In other words, the total transmission power isp + P Consider an aritrary harvester in a typical cluster Let V, g, and d denote the transmission voltage and energy intensity at this harvester and the corresponding distance for power transmission to the connected aggregator, respectively Then the power harvested at and transmitted y the harvester is ηg By the definition of transmission efficiencyτ, the transmission

11 loss, denoted as P, should e no larger than the fraction τ of transmission power: P τηg 3 Since the power effectively transmitted from the harvester to the aggregator is τηg, it follows from 9 that P = βτηg V d 3 Comining 3 and 3 leads to a requirement on the voltage βηg d V τ τ 3 3 3λ a since The distance d can e ounded as d the harvester lies in a hexagon centered at the connected aggregator and with an area of /λ a Using this ound as well as g γ, a sufficient condition for meeting the requirement in 3 is otained as shown in the following proposition Proposition 6 Consider distriuted harvesters and aggregators with fixed densities A sufficient condition on the harvester voltage for achieving a given transfer efficiency τ, is V = τ β τ 3 3 λ 4 a 33 Given fixed τ, one can oserve from Proposition 6 that as the size of harvester clusters increases y letting λ a, it is sufficient for the harvester voltage to scale asλa /4 multiplied y a constant, resulting in reliale power supply to all BSs It is also interesting to consider the case with fixed V for which there exists a conflict etween suppressing energy spatial randomness y energy aggregation and reducing the resultant power-transmission loss Specifically, ased on 33, decreasing the aggregator density λ a reduces the transmission efficiency τ, which in turn increases the outage-proaility upper ound in Proposition 5 This makes it necessary to optimize λ a for minimizing the outage proaility Solving the prolem is non-trivial and outside the scope of this paper ut warrants further investigation VI EXTENSIONS AND DISCUSSION A Shot-Noise Based Energy Field Model In this susection, the energy field model proposed in Section III is compared with an alternative model descried as follows By replacing the max operator in the energy intensity function in 3 with a summation, the result, denoted as gx, represents spatial interpolation of the energy centers: gx = γ Y Φ e f X Y 34 One drawack of the resultant alternative model is that the maximum energy intensity of the field can no longer e retained as γ as it ecomes a random variale with unounded support, which is impractical As another drawack, the random function in 34 is known as a shot-noise process and its distriution function has no simple form [3] The expectation of the energy density in the alternative model in 34 can e otained using Campell s theorem []: E[ gx] = γλ e e r ν πrdr = πγψ 35 Hence, the expectation ratio E[ gx]/e[gx] = + πψ As ψ is typically much smaller than, the ratio is close to one Using the result and the technique in [3] for ounding the tail proaility of a shot-noise process, it can e otained that Pr gx > x = PrgX > x+prgx x Pr f X Y > x γ Y Φ BX,f x/γ PrgX > x+prgx x [ ] E Y Φ BX,f x/γ f X Y x/γ = PrgX > x+prgx x πλ e ν The inequality can e rewritten as Pr gx > x PrgX > x πλ e ν PrgX x πλ e νǫ 36 where the constantǫ, is related to the maximum outage proaility of the renewale powered network and is close to zero in the operating regime of interest It follows that the tail proailities for oth energy field models are similar B Energy Field Model with Power Law Energy Decay and Its Effect on Network Coverage In the preceding sections, the outage proaility is analyzed ased on the energy field model characterized y the exponential energy decay function of distance in 4 In this section, the analysis is extended to the alternative power law function f defined as f d = + d 37 ν where f d d if d is large Note that f d as d and f d as d, having the desirale properties for an energy decay function It is assumed that the BSs adopt channel-independent transmission and that the channel coefficients follow a chi-squared distriution with parameter ω A similar procedure can e followed to otain results for other cases, however the key insights are identical to those from the current analysis Therefore, the details are omitted for revity Recall that the outage proaility can e decomposed as p id out = p a + p with p and p a defined in and From their definitions, the change on the energy field model y modifying the energy decay function affects onlyp a ut not p The resultantp a is analyzed as follows By a slight ause of notation, let the distance function r in 5 and the energy intensity function g in 3 also denote their counterparts corresponding to f in 37 Then rx = f x/γ = νγ/x 38

12 This results in the following distriution of the energy intensity gx for an aritrary location X R : PrgX x = e πλer x = e πψγ x Then from, p a can e upper ounded as gxh R α p a Pr K η [ = e πψ E exp πψγηh ] K R α ω = e πψ E[K ω πψ ]E[Rαω ] ω αω = e πψ πψ 3 E[K ω ] 39 3λ where E[K ω ] is given in 9 Comining the last inequality with Lemmas and 3 gives the main result of this section Proposition 7 Consider the scenario where BSs adopt channel-independent transmission and are powered y on-site harvesters Suppose that the channel coefficients follow iid chi-squared distriutions and that the energy density decay follows the function f in 37, the outage proaility satisfies ω αω p id out e πψ πψ 3 E[K ω ]+ c 3λ u E [ H ] 3λ λ +α Moreover, as, [ ω ] p id out c 5 e πψ + ω +O ω πψ Γω + where the constant ωα c 5 = E[K] ω ω λ 3 3 Recall that the energy field influences the outage proaility via its spatial variation and maximum harvested power, determined y the parameters ψ and, respectively The results show that the component of the outage proaility due to energy randomness scales as ψ ω if ψ is small Moreover, as increases, the proaility decreases following the power law ω as oserved from Proposition 7 The scaling law is more gradual due to a loose ound resulting from the use of Markov inequality in the derivation Comparing the results in Proposition 7 with those in Proposition corresponding to the exponential energy decay function, the outage proaility for the current case is less sensitive to the changes on ψ ut more sensitive to those on The reason is that the power law energy decay function results in less spatial fluctuation in the energy field with respect to the doule-exponential function VII SIMULATION RESULTS Unless otherwise specified, the simulation settings are as follows The BSs and moiles have densities of λ = 78/km and λ m = 78/km, respectively, corresponding to an average cell radius of 5 meter and users per cell For Outage Proaility Chan Indep TX Chan Indep TX Upper Bound Chan Invers TX Chan Invers TX Upper Bound 5 5 Characteristic Parameter of Energy Field Fig 4 Outage proaility versus the characteristic parameter of the energy field for the scenario of on-site harvesters propagation, the reference path loss is 7 db measured for a propagation distance of m [33], the pathloss exponent α = 4, and the noise power is 9 dbm The product of the harvester aperture and the maximum energy intensity, γη, gives the maximum power a harvester can generate, which is fixed as γη = kw for an on-site harvester and W for a distriuted one For the case of distriuted harvesters, the harvester density is 56/km that is twice of the BS density The SNR threshold is 8 corresponding to a spectrum efficiency of aout 3 it/s/hz The fading coefficients are iid and distriuted as max CN,,, where the CN, random variales model Rician fading and the truncation at accounts for the avoidance of deep fading y scheduling Consider the scenario of on-site harvesters The curves of outage proaility versus the characteristic parameter ψ are plotted in Fig 4 for oth the channel-independent and channel inversion transmissions The upper ounds on the outage proailities as given in Propositions and 3 are also shown The outage proailities are oserved to decay exponentially with increasingψ as predicted y analysis Channel inversion transmission perform etter than the other transmission scheme The proaility ound for channel inversion transmission is not as tight as that for channel-independent transmission as the former is ased on the union ound [see ] As the product γη is large, significant power can e harvested even at locations far from energy centers Consequently, the outage proaility is close to zero even for a small characteristic parameter eg, Next, consider the scenario of distriuted harvesters where the characteristic parameter is fixed as 5 In Fig 5, the outage proaility is plotted against the numer of harvesters connected to a single aggregator for energy aggregation that is approximately equal to λ h /λ a For comparison, the figure also shows the asymptotic upper ound on outage proailities as given in Proposition 5 as well as those generated y simulation and replacing transmission powers with the asymptotic

13 Asymp Upp Bnd Prop 5 Chan Invers TX Chan Indep TX It follows that πψ [ p a E[K πψ γη ]E H πψ ] [ E R απψ ] 4 Outage Proaility Asymp Upp Bnd Lem 6, Chan Invers TX Asymp Upp Bnd Lem 6, Chan Indep TX Numer of Harvesters for Energy Aggregation Fig 5 Outage proaility versus the numer of harvesters for energy aggregation for the scenario of distriuted harvesters The characteristic parameter is fixed as ψ = 5 lower ound in Lemma 6 Aggregation loss is omitted assuming sufficiently high transmission voltage It is oserved that energy aggregation dramatically reduces outage proailities, indicating energy randomness as the main reason for outage events Most of the aggregation gain can e achieved with less than 5 harvesters per aggregator For a large numer of harvesters, the limits of outage proaility depend only slightly on the randomness of the energy field and is mostly affected y channel fading as well as moile random locations for the case of channel inversion transmission Last, the asymptotic upper ound from Proposition 5 is oserved to e loose due to the use of Markov s inequality ut the other asymptotic ounds ased on Lemma 6 are tight VIII CONCLUSION In this paper, a novel spatial model of renewale energy has een presented and applied to quantify the effect of energy spatial randomness on the coverage of cellular networks powered y energy harvesting Moreover, the proposed technique of energy aggregation has demonstrated that new network architectures can e designed to cope with energy spatial variations This work provides a useful analytical framework for designing large-scale energy harvesting networks Moreover, it opens an interesting research direction of redesigning communication techniques eg, multi-cell cooperation and resource allocation, as a means to achieve network reliaility in the presence of energy randomness A Proof of Lemma APPENDIX By sustituting P = ηgb and 6 into, πψ [ K p a = E H πψ ] γη Rα K H Rα Pr K H Rα γ By sustituting R into 4, p a γηλ α πψ [ E[K πψ ]E H πψ ] E [ Rαπψ ] 4 It follows from the inequality R D and the distriution of D in 3 that [ Rαπψ ] E cπψ 4 +απψ where c is defined in the lemma statement By comining the inequality with 4, p a c +απψ γηλ α πψ E[K πψ ]E [ H πψ ] 43 If πψ, the upper ound on the outage proaility can e reduced using Jensen s inequality as p a +απψ c E[K ]E [ H γηλ α ] πψ 44 If πψ >, since larger ψ leads to more harvested energy and thus smaller outage proaility, the inequality in 43 holds y setting πψ = : p a c E[K ]E [ H ] 45 +απψγηλ α Comining 44 and 45 and sustituting E[K ] = λ u /λ gives the desired result B Proof of Lemma Using Markov s inequality, p defined in can e ounded as p E[ K H ] Rα = E[K ]E [ H ] E[R α ] 46 = λ ue [ H ] E[R α ] 47 λ where the equality in 46 holds since K, H and R are independent and 47 follows y sustituting E[K ] = λ u /λ Using the definition of R and the inequality in 3, it is otained that α E[R] α λ Sustituting 48 into 47 gives the desired result