Renewable Powered Cellular Networks: Energy Field Modeling and Network Coverage

Renewable Powered Cellular Networks: Energy Field Modeling and Network Coverage

Kaibin Huang, Marios Kountouris and Victor O. K. Li K. Huang and V. O. K. Li are with the Dept. of Electrical and Electronic Engineering at The University of Hong Kong, Hong Kong. Email: huangkb@eee.hku.hk, vli@eee.hku.hk. M. Kountouris is with the Mathematical and Algorithmic Sciences Lab, Huawei France R&D, Paris, France. Email: marios.kountouris@huawei.com. Part of this work has been presented at IEEE Intl. Conf. on Comm. Systems (ICCS) in Nov. 2014.
Abstract

Powering radio access networks using renewables, such as wind and solar power, promises dramatic reduction in the network operation cost and the network carbon footprints. However, the spatial variation of the energy field can lead to fluctuations in power supplied to the network and thereby 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 occurs at Poisson distributed 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 arbitrary location is given by the decayed intensity from the nearest energy center. The product between the energy center density and the exponential rate of the decay function, denoted as , is shown to determine the energy field distribution. Next, the paper considers a cellular downlink network powered by harvesting energy from the energy field and analyzes its network coverage. For the case of harvesters deployed at the same sites as base stations (BSs), as increases, the mobile outage probability is shown to scale as , where is the outage probability corresponding to a flat energy field and is a constant. Subsequently, a simple scheme is proposed for counteracting the energy randomness by spatial averaging. Specifically, distributed harvesters are deployed in clusters and the generated energy from the same cluster is aggregated and then redistributed to BSs. As the cluster size increases, the power supplied to each BS is shown to converge to a constant proportional to the number 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.

{keywords}

Cellular networks, renewable energy sources, energy harvesting, stochastic processes.

I Introduction

The exponential growth of mobile data traffic causes the energy consumption of radio access networks, such as cellular and WiFi networks, to increase rapidly. This not only places heavy burdens on both the electric grid and the environment, but also leads to huge network operation cost. A promising solution for energy conservation is to power the networks using alternative energy sources, which will be a feature of future green telecommunication networks [1]. However, the spatial randomness of renewable 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 renewable powered base stations (BSs), this paper addresses the aforementioned issue by proposing a novel model of the energy field and quantifying the relation between 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 both the spatial and temporal randomness of renewables on the coverage of different wireless networks spread over the horizontal plane [2, 3, 4, 5]. Poisson point processes (PPPs) are used to model transmitters of a mobile ad hoc network (MANET) in [2] and BSs of a heterogeneous cellular network [3]. Energy arrival processes at different transmitters are usually modeled as independent and identically distributed (i.i.d.) stochastic processes, reducing the effect of energy temporal randomness to independent on/off probabilities of transmitters. Thereby, under an outage constraint, the conditions on the network parameters, such as transmission power and node density of the MANET [2] and densities of different tiers of BSs [3], can be analyzed for a given distribution of energy arrival processes. The assumption of spatially independent energy distributions is reasonable for specific types of renewables that can power small devices, such as kinetic energy and electromagnetic (EM) radiation, but does not hold for primary sources, namely wind and solar power. To some extent, energy spatial correlation is accounted for in [4, 5] but 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, besides intelligent sharing of its spectrum. The idea of deploying dedicated stations for supplying power wirelessly to energy harvesting mobiles in a cellular network is explored in [5]. In [4, 5], radiations by transmitters with reliable power supply form an EM energy field and its spatial correlation is determined by the EM wave propagation that does not apply to other types of renewables 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, 7, 8] has been a key research area in wireless networking in the past decade. Similar mathematical tools have also been widely used in the area of geostatistics concerning spatial statistics of natural resources including renewables, where research focuses on topics such as model fitting, estimation, and prediction of energy fields [9, 10]. 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 initial contributions.

Intermittence of renewables introduces stochastic constraints on available 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 [11, 12, 13, 14, 15, 16, 17]. As shown in [11] from an information-theoretic perspective, it is possible to avoid capacity loss for an AWGN channel due to energy harvesting provided that a battery with infinite capacity is deployed to counteract the energy randomness. From the communication-theoretic perspective, optimal power control algorithms are proposed in [12, 13] 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 been extended to design more complex energy harvesting systems, such as interference channels [14] and relay channels [15], to design medium access protocols [16], and to account for practical factors such as non-ideal batteries [17]. The prior work assumes fast varying renewable 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 by a high level of spatial randomness but 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 renewable energy maps are created by interpolating data collected from sparse measurement stations separated by distances typically of tens to hundreds of kilometers [18, 19, 20]. 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 (e.g., cloud formation and mobility) but also on the harvester/BS deployment environment such as locations and heights of buildings, trees and cell sites. Mapping the energy field requires complex datasets that are difficult to obtain. Even if such datasets are available, the resultant energy-field model may not allow tractable 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 be suitable models for natural phenomena and resources [21, 9] as well as random wireless networks [6] and thus provide a desirable tradeoff between tractability 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 ), called energy centers, are modeled as a PPP with density , corresponding to sites on top of tall buildings with exposure to direct sunshine or strong winds. From an energy center, the energy intensity decays exponentially with the squared distance normalized by a constant , called the shape parameter, specifying the area of influence by the said center. It is worth mentioning that the decay function is popularly used in solar-field mapping [22] and atmospheric mapping [23]. The energy intensity at an arbitrary location is then given by the decayed intensity with respect to the nearest energy center. The characteristic parameter of the energy field, defined as , is shown later to determine its distribution. It is worth mentioning that general insights obtained in this work using the above model, such as that increasing energy spatial correlation reduces outage probability, also hold for other models so long as they observe the basic principle of renewable energy field: energy spatial correlation between two locations increases as their separation distance decreases and vice versa [19].

In the paper, BSs of the cellular network are assumed to be deployed on a hexagonal lattice while mobiles are distributed 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 been 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 [24]. A mobile is said to be under (network) coverage if an outage constraint is satisfied and the outage probability is the performance metric. Each BS allocates transmission power simultaneously to mobiles, either by equal division of the available power, called channel-independent transmission, or by channel inversion, called channel inversion transmission. As illustrated in Fig. 1, each BS is powered by either an on-site harvester or a remote (energy) aggregator that collects energy generated by a cluster of nearby distributed harvesters over transmission lines. Aggregators and distributed harvesters are deployed on hexagonal lattices with densities and , respectively. Connecting harvesters to their nearest aggregators form harvester clusters. Note that energy aggregation is based on the same principle of tackling energy randomness by energy sharing as other existing techniques designed for renewable powered cellular systems (see e.g., [25, 26]). Prior work focuses on algorithmic design while the current analysis targets network performance.

The main contribution of this work is the establishment of a new approach for designing large-scale renewable powered wireless networks that models the energy field using stochastic geometry and applies such a model to network design and performance analysis. The analysis based on the approach allows the interplay of different mathematical tools such as stochastic geometry theory, probabilistic inequalities and large deviation theory, leading to the following new findings.

• Consider the on-site harvester case. If the characteristic parameter is small () and the maximum energy density is large (), the outage probability monotonically decreases with increasing and in the form of with being the probability corresponding to a flat energy field and a constant. The result holds for both channel inversion and channel-independent transmissions. Despite this similarity, the former outperforms the latter by adapting transmission power to mobiles’ channels.

• Next, consider distributed harvesters and define the size of a harvester cluster as . As the cluster size increases, energy aggregation is shown to counteract the spatial randomness of the energy field and thereby stabilize the power supply for BSs. Specifically, the power it distributes to each BS converges to a constant proportional to the number of harvesters per BS. In other words, the energy field becomes a reliable power supply for the network.

• However, an insufficiently high voltage used by harvesters for power transmission to aggregators can incur significant energy loss. It is found that the loss can be regulated by increasing the voltage inversely with the aggregator density to the power of .

• Last, key results are extended to two variations of the energy field model characterized by 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 described 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 distributed 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 by concluding remarks in Section VIII.

Ii Models and Metrics

The spatial models for the energy field, energy harvesters, and cellular network are described in the subsections. The notations are summarized in Table I.

Ii-a Energy Harvester Model

Aggregation loss arises in the scenario of distributed harvesters, corresponding to the power loss due to transmissions from harvesters to aggregators over cables (see Fig. 2). It is impractical to assume high voltage transmission from distributed harvesters having small form factors and thus aggregation loss can be significant, which is analyzed in the sequel. On the other hand, each aggregator supplies power to BSs also over cables, assuming that is an integer for simplicity. Aggregators are assumed to be much larger than harvesters and thus can afford having relatively large transformers. Thus, high voltages can be reasonably assumed for power distribution such that the power loss is negligible. Consequently, the specific graph of connections between aggregators and BSs has no effect on the analysis except for the number of BSs each aggregator supports.

The energy field is represented by , which is determined by the function mapping a location 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 described below. Let represent the time-varying version of with denoting time. Harvesters are assumed to be homogeneous and time is partitioned into slots of unit duration. Let denote a constant combining factors such as the harvester physical configuration and conversion efficiency, referred to as the harvester aperture by analogy with the antenna aperture. Then the amount of energy harvested at in the -th slot is or equivalently .

Ii-B Cellular Network Model

Define an outage event as one that the receive SNR of a typical active mobile is below . The outage probability is the network performance metric and defined for the two transmission strategies as follows. First, consider channel-independent transmission, where a BS equally allocates supplied power for transmission to mobiles. Let and denote the propagation distance and the channel coefficient of a typical user in the typical cell, respectively. Then, the outage probability, denoted as , can be written as

 pidout =Pr(P0H0R−α0K0<θ) (1)

where given equal power allocation, the transmission power for each of the users in the typical cell is . Alternatively, the outage probability can be defined with respect to a set of users in the same cell as the probability that the received SNR for least one user is below the threshold. Following similar steps as in the subsequent analysis, one expect the resultant outage probabilities to have similar expressions as those in Propositions 1-4 that arise mainly from the distribution of the energy field instead of the the specific definition of outage probability.

Next, consider channel inversion transmission. Transmission power required for each mobile is computed by channel inversion in an attempt to ensure that the receive SNR is equal to . Specifically, the -th mobile in the typical cell is under coverage if the allocated transmission power is no smaller than , where and represent the corresponding propagation distance and channel coefficient, respectively. It is difficult to write down the outage probability for a typical active mobile, denoted as , since it depends on the channels of other active mobiles sharing the BS. However, it can be bounded using the union bound as

 pivout≤Pr(P0<θK0∑n=1Rα0,nH0,n). (2)

The upper bound in (2) is the probability 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

Iii-a Energy Field Model

The energy field refers to the set of energy intensities at different locations in the horizontal plane with the maximum denoted as . The PPP modeling for the energy centers and its density are denoted as and , respectively. The energy intensity function is defined for a given location as follows:

 g(X)=γmaxY∈Φef(|X−Y|) (3)

where the energy decay function is

 f(d)=e−d2/ν,d>0. (4)

Note that is a Boolean random function [27]. The positive parameter in (4) controls the shape of , thus called the shape parameter, and thereby determines the area of influence of an energy center. Then the characteristic parameter is defined as . A large value of 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 observed from the plots in Fig. 3, increasing or decreasing introduces more “ripples” in the energy field; the field is almost flat for a large characteristic parameter e.g., (or and ).

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 based on a different energy decay function is considered in Section VI-B and its effect on the network coverage is analyzed.

Iii-B Energy Field Properties

First, the distribution function of the energy intensity can be easily obtained by relating it to a Boolean model. To this end, define as the distance from an energy center to a location with the decayed energy intensity . Then can be obtained from the equation with in (4) as follows:

 r(x)=f−1(xγ)=√νlnγx. (5)

Moreover, let denote a disk in centered at and with a radius . The region of the energy field where energy intensities exceed a threshold corresponds to a Boolean model . Then for given and , the distribution function of can be written in terms of the model and obtained as

 Pr(g(X)≤x) =Pr(X∉⋃X∈ΦeB(X,r(x))) =e−πλer2(x) =(xγ)πψ (6)

where the last equality is obtained by substituting (5) and using the definition of . Next, the mean and variance of the energy intensity function can be directly derived using the distribution function in (6) as follows:

 E[g(X)] =πψγ1+πψ, (7) E[g2(X)] =πψγ22+πψ, var(g(X)) =πψγ2(2+πψ)(1+πψ)2.

As , is seen to converge to while diminishes inversely with . On the other hand, as , both quantities become proportional to .

Last, the joint distribution of the energy intensities and at two different locations and is derived. They are independent if :

 Pr(g(X1)≤x1,g(X2)≤x2) =Pr(g(X1)≤x1)× Pr(g(X2)≤x2) =(x1x2γ2)πψ. (8)

If ,

 Pr(g(X1)≤x1,g(X2)≤x2) (9) =(x1x2γ2)πνλeeλe|B(X1,r(x1))∩B(X2,r(x2))|

where the operator applied on a set gives its measure (area). For ease of notation, rewrite and as and and define . The area of the overlapping region between and is known to be given as

 |B(X1,r1)∩B(X2,r2)|=r21cos−1(r21+d2−r222dr1)+ +r22cos−1(r22+d2−r212dr2)−d ⎷r21−(r21+d2−r222d)2.

For the special case of (thus ),

 Pr(g(X1)≤x,g(X2)≤x)=(xγ)2πνλe(1−Δ(d/r)) (10)

where and

 Δ(y)=12πcos−1(y2)−y4π√4−y2.

Comparing (9) and (10), one can see that reducing the distance between two locations increases the joint cumulative distribution function of the corresponding energy intensities as they become more correlated.

Iv Network Coverage with On-Site Harvesters

Iv-a Network Coverage with Channel-Independent Transmission

To facilitate analysis, the outage probability is decomposed as with and defined as

 pa =Pr(P0H0R−α0K0<θ∣∣∣K0H−10Rα0≤ηγθ)× Pr(K0H−10Rα0≤ηγθ), (11) pb =Pr(K0H−10Rα0>ηγθ). (12)

The component is the probability that the energy intensity at the typical BS site is so low as to cause an outage event at the typical mobile even though the maximum intensity is sufficiently high for avoiding such an event. The other component represents the probability that the maximum energy intensity at the typical BS is insufficiently high for avoiding an outage event at the typical mobile, which occurs due to the combined effect of the number of simultaneous users being too large, the channel gain being too small, and the propagation distance being too long. In other words, and are the components of outage probability arising from the energy field spatial variation and the limit on the maximum harvested power, respectively. The outage probability is characterized by analyzing and separately.

To this end, it is useful to define a random variable as that gives the propagation distance of a mobile uniformly distributed in a cell of unit area. Note that a hexagon of unit area can be outer bounded by a disk with the area of . Let denote the distance from the center of the disk to a random point uniformly distributed in the disk. Then

 Pr(¯R0≤x) ≥Pr(D≤x) =3√32x2,0≤x≤√23√3. (13)

The probability is shown in the following lemma to be an exponential function of the energy field characteristic parameter , when is sufficiently small. The proof of the lemma is provided in Appendix -A.

Lemma 1.

The probability can be upper bounded as

 pa≤22+απψ⎛⎜⎝c2θE[H−10]λuγηλ1+α2b⎞⎟⎠min(πψ,1) (14)

where .

One can observe from (12) that is the tail probability of the product of multiple random variables and thus it is difficult to derive a closed-form expression for the probability. Following typical approaches, we apply a probabilistic inequality, namely Markov’s inequality, to obtain an upper bound on and large deviation theory to characterize its asymptotic scaling. The following lemma is obtained based on Markov’s inequality with the proof in Appendix -B.

Lemma 2.

The probability in (12) can be upper bounded as

 pb≤c3θλuE[H−10]ηγλ1+α2b, (15)

where the constant .

Given , combining the results in Lemmas 1 and 2 leads to the following first main result as follows.

Proposition 1.

Consider the scenario where BSs adopt channel-independent transmission and are powered by on-site harvesters. The outage probability can be bounded as

 pidout≤22+απψ⎛⎜⎝c2θE[H−10]λuλ1+α2bγη⎞⎟⎠min(πψ,1)+c3θλuE[H−10]λ1+α2bηγ.

Consider the scenario where the maximum harvested power is large and the characteristic parameter is small. Then the outage probability decreases monotonically with these parameters following the scaling law of

 pidout∝(γη)−πψ (16)

which is verified by simulation in the sequel.

Next, is analyzed using large deviation theory. To this end, it is necessary to consider a specific type of distribution for the channel coefficient . Let denote the complementary cumulative distribution function (CCDF) of the random variable (RV) in (12). Assume that is a regularly varying random variable defined by the following condition:

 limt→∞¯F(ζt)¯F(t)=ζ−ω (17)

where is the exponent of the distribution and . One example is that has the chi-squared distribution that is typical for wireless channels:

 Pr(H0

where is a positive integer and the distribution parameter and denotes the gamma function. Then is a regularly varying RV with the exponent . Based on the assumption and applying Breiman’s Theorem (see Lemma 7 in Appendix -C), an asymptotic bound on is obtained as shown below.

Lemma 3.

Suppose that the inverse of a channel coefficient is a regularly varying RV. As , the probability is bounded as222Two asymptotic relation operators, ”” and ””, are defined as follows. Two functions and are asymptotically equivalent, denoted as , if . The case of is represented by .

 pb⪯E[Kω0]Pr⎛⎜ ⎜ ⎜⎝H0≤(2λb3√3)α2θηγ⎞⎟ ⎟ ⎟⎠,ηγ→∞

where is given as

 E[Kω0]=ω∑m=1(λuλb)m1mm∑k=0(−1)m−k(mk)kω. (19)

In particular, if a channel coefficient is a chi-squared RV with parameter ,

 pb⪯E[Kω0]⎛⎜ ⎜ ⎜⎝ηγ(2λb3√3)α2θ⎞⎟ ⎟ ⎟⎠−ω+O((ηγ)−ω−1),ηγ→∞.

Remark:   It is possible to derive asymptotic upper bounds on for other types of channel coefficient distribution. For instance, [28, Theorem ] can be applied to obtain a bound for the sub-exponential distribution defined by the condition where ‘*’ denotes convolution. Such results will change the second but not the first term of the upper bound on in Proposition 2 presented shortly.

Combining the results in Lemmas 1 and 3 leads to the second main result as follows:

Proposition 2.

Consider the scenario where BSs adopt channel-independent transmission and are powered by on-site harvesters. Suppose that the inverse of a channel coefficient is a regularly varying RV, the outage probability can be bounded as as follows:

 pidout ⪯22+απψ⎛⎜⎝λ1+α2bγηc2θE[H−10]λu⎞⎟⎠−min(πψ,1)+ E[Kω0]⎛⎜⎝(2λb3√3)α2θ⎞⎟⎠ωPr(H0<1ηγ).

In particular, if channel coefficients follow i.i.d. chi-squared distributions, as ,

 pidout⪯22+απψ⎛⎜⎝λ1+α2bγηc2θΓ(ω−1)λu⎞⎟⎠−min(πψ,1)+ E[Kω0]Γ(ω+1)⎛⎜ ⎜ ⎜⎝ηγ(2λb3√3)α2θ⎞⎟ ⎟ ⎟⎠−ω+O((ηγ)−ω−1).

Comparing the results in Propositions 1 and 2, both upper bounds on have identical first term, which is dominant when is large. The second term in (2), proportional to , decays faster than the counterpart in Proposition 1, proportional to , when . This is due to a tighter bound on derived using large deviation theory with respect to that obtained using Markov’s inequality.

Iv-B Network Coverage with Channel Inversion Transmission

Following the same method as in the preceding subsection, the outage probability in (2) can be decomposed as where

 pc =Pr(P0<θK0∑n=1Rα0,nH0,n∣∣ ∣∣K0∑n=1Rα0,nH0,n≤ηγθ)× Pr(K0∑n=1Rα0,nH0,n≤ηγθ), (20) pd =Pr(K0∑n=1Rα0,nH0,n>ηγθ). (21)

Their key difference from those of their counterparts and in the preceding section is that the transmission power of a typical BS for the current case is a compound Poisson random variable. However, the expected transmission powers for both cases are identical due to the following equality:

 E[K0∑n=1Rα0,nH0,n]=E[K0]E[Rα0,n]E[H−10,n]. (22)

Given this equality, the bounds on in (11) and in (12) can be shown to hold for in (20) and in (21), respectively, yielding the following proposition.

Proposition 3.

Consider the scenario of on-site harvesters. The upper bound on the outage probability for the case of channel-independent transmission as given in Proposition 1 also holds for the case of channel inversion transmission.

Despite the identical upper bounds, the outage probability 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 number of mobiles in outage.

Next, is analyzed using large deviation theory. To this end, some useful definitions and results are introduced as follows. The distribution of a RV has a light tail if for some and a heavy tail if for all . The following result is from [29, Theorem ].

Lemma 4.

Let be a set of i.i.d. random variables having a heavy-tailed distribution and have a light-tailed distribution independent of those of . Then

 Pr(N∑m=1Xm>t)∼E[N]Pr(X1>t)% as t→∞.

To apply the result, the upper bound on in (21) is simplified using the inequality as

 pd≤Pr⎛⎜ ⎜ ⎜⎝K0∑n=1H−10,n>ηγ(2λb/3√3)α2θ⎞⎟ ⎟ ⎟⎠. (23)

One can see that the upper bound is the tail probability of a compound Poisson RV, . Note that the Poisson distribution of has a light tail since it decays faster than the exponential rate as observed from the following inequality (see e.g., [30, Theorem 5.4]):

 Pr(K0≥x)≤e−λm/λb(eλm/λb)xxx,if x>λm/λb.

Then the lemma below follows from (23) 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 distribution. As , the probability is bounded as

 pd⪯λuλbPr⎛⎜ ⎜ ⎜⎝H0≤(2λb3√3)α2θηγ⎞⎟ ⎟ ⎟⎠,ηγ→∞ (24)

In particular, if a channel coefficient is a chi-squared RV with parameter ,

 pd⪯λuλb⎛⎜ ⎜ ⎜⎝ηγ(2λb3√3)α2θ⎞⎟ ⎟ ⎟⎠−ω+O((ηγ)−ω−1),ηγ→∞.

As mentioned, the bound on in (11) holds for in (20). Combining this result, the bounds on in Lemma 5 and the inequality gives the following result.

Proposition 4.

Consider the scenario where BSs adopt channel inversion transmission and are powered by on-site harvesters. Suppose that the inverse of a channel coefficient is a heavy-tailed RV, the outage probability can be bounded as follows: as ,

 pivout ⪯22+απψ⎛⎜⎝λ1+α2bγηc2θE[H−10]λu⎞⎟⎠−min(πψ,1)+ λuλbPr⎛⎜⎝H0<(2λb3√3)α2θηγ⎞⎟⎠.

In particular, if channel coefficients follow i.i.d. chi-squared distributions, as ,

 pivout ⪯22+απψ⎛⎜⎝λ1+α2bγηc2θΓ(ω−1)λu⎞⎟⎠−min(πψ,1)+ λuΓ(ω+1)λb⎛⎜ ⎜ ⎜⎝ηγ(2λb3√3)α2θ⎞⎟ ⎟ ⎟⎠−ω+O((ηγ)−ω−1).

Comparing the results in Propositions 2 and 4, the outage probability 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, is smaller than , as also observed in the simulation results in the sequel.

V Network Coverage with Distributed Harvesters

V-a Network Coverage

In this section, it is shown that aggregating energy harvested by many distributed harvesters stabilizes the power supplied to the BSs by the law of large numbers. 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 . The design requirement under this constraint is analyzed in the next subsection. The harvester lattice partitions the plane into small hexagonal regions, each with area of . Whether each region contains an energy center can be indicated by a set of independent Bernoulli random variables with probabilities and for the values of and , respectively. Despite the independence of the numbers of energy center in different regions, it is important to note that the energy intensities at harvesters are correlated [see their joint distribution in (8) and (9)]. The energy intensity at each harvester is at least if the corresponding small region contains an energy center or otherwise takes on some positive value. Based on the above points, the transmission power of the typical BS is lower bounded as

 P0 ≥τγηλaλbM0∑n=1Qne(−23√3νλh) =τγηλhλb×λaM0λh×1M0M0∑n=1Qne−23√3νλh (25)

where represents the number of harvesters connected to the typical aggregator. Consider the scenario of sparse aggregators with each connected to many harvesters, corresponding to . As a result, and . Then it follows from (25) that

 limλa→0P0(λa)≥τγηλhλblimM0→∞1M0M0∑n=1Qne(−23√3νλh). (26)

The law of large numbers gives the following lemma.

Lemma 6.

With the harvester density fixed, as the aggregator density , the transmission power for the typical BS converges as

 limλa→0P0(λa)≥τγηλhλb(1−e−λeλh)e−23√3νλh,a.s. (27)

In other words, is lower bounded by a constant and thus its randomness due to energy spatial variation diminishes. If the energy centers are dense () and the shape parameter is large (), the transmission power approaches its upper bound .

The power stabilization by the spatial averaging of the energy field removes one random variable from the outage probability. Consider the case of channel-independent transmission and the corresponding outage probability in (1). Using Lemma 6 and applying Markov’s inequality, the outage probability for small aggregator density can be bounded as

 limλa→0pidout(λa)≤θλbE[H−1]E[K]E[Rα]τγηλh(1−e−λeλh)e−23√3νλh. (28)

Substituting and the result in (48) yields the result in Proposition 5 as given below. The result is proved to also hold for channel-inversion transmission using (2) and following a similar procedure.

Proposition 5.

With the harvester density fixed, as the aggregator density , the outage probabilities for both the channel-independent and channel inversion transmissions can be bounded as

 limλa→0pidout(λa),limλa→0pivout(λa)≤c3θE[H−1]λuτγηλα2bλh(1−e−λeλh)e−23√3νλh.

The results in Proposition 5 suggest that

 limλa→0pidout(λa),limλa→0pivout(λa)∝1γη×1λh/λb×λuλb×λ−α2b

where the factors represent in order the inverses of the maximum power generated by a single harvester, the number of harvesters per BS, the expected number of active mobiles õper cell, and the expected propagation loss.

V-B Energy Aggregation Loss

It is well known that the power line loss, , for transmitting the power of to a receiver is given as

 ΔP=βP2dV2 (29)

where is the voltage and is a constant depending on the power line characteristics, such as resistivity and cross-section area. In other words, the total transmission power is . Consider an arbitrary harvester in a typical cluster. Let , , and 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 by the harvester is . By the definition of transmission efficiency , the transmission loss, denoted as , should be no larger than the fraction of transmission power:

 ΔP0≤(1−τ)ηg0. (30)

Since the power effectively transmitted from the harvester to the aggregator is , it follows from (29) that

 ΔP0=β(τηg0)2V20d0. (31)

Combining (30) and (31) leads to a requirement on the voltage

 V0≥τ√βηg0d01−τ. (32)

The distance can be bounded as since the harvester lies in a hexagon centered at the connected aggregator and with an area of . Using this bound as well as , a sufficient condition for meeting the requirement in (32) is obtained as shown in the following proposition.

Proposition 6.

Consider distributed harvesters and aggregators with fixed densities. A sufficient condition on the harvester voltage for achieving a given transfer efficiency is

 V0=τ ⎷βηγ1−τ√23√3λ−14a. (33)

Given fixed , one can observe from Proposition 6 that as the size of harvester clusters increases by letting , it is sufficient for the harvester voltage to scale as multiplied by a constant, resulting in reliable power supply to all BSs.

It is also interesting to consider the case with fixed for which there exists a conflict between suppressing energy spatial randomness by energy aggregation and reducing the resultant power-transmission loss. Specifically, based on (33), decreasing the aggregator density reduces the transmission efficiency , which in turn increases the outage-probability upper bound in Proposition 5. This makes it necessary to optimize for minimizing the outage probability. Solving the problem is non-trivial and outside the scope of this paper but warrants further investigation.

Vi Extensions and Discussion

Vi-a Shot-Noise Based Energy Field Model

In this subsection, the energy field model proposed in Section III is compared with an alternative model described as follows. By replacing the max operator in the energy intensity function in (3) with a summation, the result, denoted as , represents spatial interpolation of the energy centers:

 ~g(X)=γ∑Y∈Φef(|X−Y|). (34)

One drawback of the resultant alternative model is that the maximum energy intensity of the field can no longer be retained as as it becomes a random variable with unbounded support, which is impractical. As another drawback, the random function in (34) is known as a shot-noise process and its distribution function has no simple form [31].

The expectation of the energy density in the alternative model in (34) can be obtained using Campbell’s theorem [21]:

 E[~g(X)] =γλe∫∞0e−r2ν2πrdr =πγψ. (35)

Hence, the expectation ratio . As is typically much smaller than , the ratio is close to one.

Using the result and the technique in [32] for bounding the tail probability of a shot-noise process, it can be obtained that

 Pr(~g(X)>x) =Pr(g(X)>x)+Pr(g(X)≤x)× Pr(∑Y∈Φ∩¯B(X,f−1(x/γ))f(|X−Y|)>xγ) ≤Pr(g(X)>x)+Pr(g(X)≤x)× E[∑Y∈Φ∩¯B(X,f−1(x/γ))f(|X−Y|)]x/γ =Pr(g(X)>x)+Pr(g(X)≤x)×πλeν.

The inequality can be rewritten as

 Pr(~g(X)>x)−Pr(g(X)>x) ≤πλeν×Pr(g(X)≤x) ≤πλeνϵ (36)

where the constant is related to the maximum outage probability of the renewable powered network and is close to zero in the operating regime of interest. It follows that the tail probabilities for both energy field models are similar.

Vi-B Energy Field Model with Power Law Energy Decay and Its Effect on Network Coverage

In the preceding sections, the outage probability is analyzed based on the energy field model characterized by the exponential energy decay function of distance in (4). In this section, the analysis is extended to the alternative power law function defined as

 f′(d)=(1+d2ν)−1 (37)

where if is large. Note that as and as , having the desirable 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 distribution with parameter . A similar procedure can be followed to obtain results for other cases, however the key insights are identical to those from the current analysis. Therefore, the details are omitted for brevity.

Recall that the outage probability can be decomposed as with and defined in (11) and (12). From their definitions, the change on the energy field model by modifying the energy decay function affects only but not . The resultant is analyzed as follows. By a slight abuse of notation, let the distance function in (5) and the energy intensity function in (3) also denote their counterparts corresponding to in (37). Then

 r(x)=f′−1(x/γ)=√ν(γ/x−1). (38)

This results in the following distribution of the energy intensity for an arbitrary location :

 Pr(g(X)≤x) =e−πλer2(x) =e−πψ(γx−1).

Then from (11), can be upper bounded as

 pa ≤Pr(g(X)H0R−α0K0≤θη) =eπψE[exp(−πψγηH0θK0Rα0)] =(θπψηγ)