Power Coverage Analysis of Cellular Networks With Energy Harvesting and Microwave Power Transfer-Based Power Sharing

This paper studies the performance of hyperdense large-scale cellular networks with energy harvesting. To combat the randomness of renewables, the base stations (BSs) with the harvested power larger than the predefined power target are assumed to cooperatively share their surplus power to the BSs with insufficient energy via directed microwave power transfer (MPT). To evaluate the performance of the proposed cooperative power sharing scheme, a new performance metric, namely the power coverage probability, which is defined as the probability that a BS satisfies its respective power target, is introduced and then characterized by applying tools from stochastic geometry. It is shown that in the hyperdense regime, the power coverage probability of cellular networks with energy harvesting and MPT based power sharing depends on the statistical mean of the harvested renewables, the power target, the power sharing distance, and the MPT path-loss exponent. Simulations are provided to validate our analysis.


I. INTRODUCTION
Hyper densification of cellular network deployment is consider to be the key enabling technology for 5G system to meet the 1000x data rate challenge [1]- [3]. With densely deployed base stations (BSs), the reduction of cell size considerably increases the spatial reuse of the cellular network while decreases the transmission distance of each data link. Though deploying more cells significantly improves the spectrum efficiency, the corresponding energy consumption grows dramatically with the increased BSs. Renewable energy, envisioned as a promising alternative to the traditional fossil fuel based power generation, provides an effective way of reducing energy costs in dense cellular networks [4]. One of the most challenging issues of cellular networks powered by renewable energy lies in the stochastic and intermittent nature of the renewable sources [5]. In this paper, motivated by the law of large numbers, we smooth out such fluctuations by enabling cooperative power sharing among the BSs.
Point process theory [6], [7] has been widely applied in the study of large-scale wireless networks with energy The associate editor coordinating the review of this manuscript and approving it for publication was Zhenyu Zhou . harvesting. In [8], Huang characterized the tradeoff between the energy arrival rate and the spatial throughput of mobile ad hoc networks with energy harvesting. In [9], Che et al. investigated the optimization problem of bidirectional energy harvesting and information transmission in large-scale communication networks with wireless charging. In [10], Lee et al. studied the performance of opportunistic energy harvesting in large-scale cognitive radio networks and maximized the throughput of secondary network subject to outage constraints. In [11], Dhillon et al. developed a tractable model for K-tier downlink heterogeneous networks with energy harvesting and characterized the availability region for a set of general uncoordinated BS operational strategies. In [12], Huang considered deploying power beacons in cellular networks to enable microwave power transfer (MPT) for mobile recharging and derived the tradeoffs between the network parameters. In [13], Sakr et al. investigated the performance of device to device (D2D) communications with energy harvesting and characterized the outage probabilities for both D2D and cellular users. It is worth noting that in the above prior works [8]- [13], with energy storage units, the randomness of the harvested energy was smoothed out by exploiting the time diversity of the 1 In the following, due to the fact that power is the amount of energy consumed per unit time, we simply denote energy as power by abuse of notation.
to validate our analysis. The main contributions of this paper are summarized as follows: • We consider a hyperdense large-scale cellular network with BSs powered by energy harvesting, where the locations of BSs are modeled as a Poisson point process (PPP) on R 2 . To cope with the randomness of the harvested renewables, a MPT based power sharing scheme is proposed. Particularly, it is designed that the BSs with the harvested power larger than the predefined power target cooperatively share their surplus power to the BSs with the harvested power lower than the predefined power target via directed MPT.
• To analyze the performance of the proposed MPT based power sharing scheme, a novel framework is developed with stochastic geometry. We first derive the mean and variance of the received shared power via MPT at a BS with the respective harvested power lower than the predefined power target. Then, based on the results, we further derived the upper and lower bounds on the power coverage probability by applying the Cantelli's inequality.
• Extensive simulations are provided to verify our analysis. An implication of our analytical and simulated results is that: for cellular networks with energy harvesting and MPT based power sharing, it is more beneficial to increase the density of BSs than expanding the power sharing distance to counteract the randomness of the renewables. The remainder of this paper is organized as follows. The system model and performance metrics are described in Section II. The power coverage probability of the cellular network with energy harvesting and MPT based power sharing is characterized in Sections III. Simulation results are presented in Section IV. Finally, we conclude our paper in Section V.

II. MODEL AND METRIC A. SYSTEM MODEL
We consider a hyperdense large-scale cellular network powered by renewable energies on R 2 , where the locations of BSs are modeled as a HPPP with density λ. Time is assumed to be slotted. In each time slot, the BSs are designed to collect and store ambient renewable energies for data transmissions in the subsequent time slot. It is worth mentioning that compared with the traditional cellular network with stable power supply, the cellular network with energy harvesting suffers considerable performance loss due to unreliable and spatially fluctuating renewable sources. To combat such randomness of renewables, in this paper, motivated by the law of large numbers, we enable the BSs to cooperatively share their surplus power via directed MPT.
Let P k be the renewable power harvested by the k-th BS in time slot t. To capture the fluctuating effect of renewables, it is assumed that P k follows an exponential distribution with mean given by P. Further, to simplify the analysis, the renewable power harvested by BSs at different locations are assumed to be independent. 2 The power targets, i.e., the transmission powers, for BSs in the cellular network with energy harvesting are assumed to be the same and denoted by η b . Let b be the set of BSs with the harvested power larger than the predefined power target η b and let b be the corresponding locations. In addition, let c b be the set of BSs with the harvested power less than the predefined power target η b and let c b be the corresponding locations. Then, with coloring theorem [7], it can be easily verified that b and c b follow two independent HPPPs with densities The power P k harvested by the k-th BS is expected to exceed its respective power target η b such that the transmission outage constraints at the intended mobile users are satisfied. It is worth noting that due to the randomness of the renewables, P k is unstable and thus may falls below the predefined power target η b . To make all the BSs reach their respective power target η b , in this paper, it is assumed that the i-th BS in b located at X i equally share its surplus denotes the set of BSs in c b within a distance of d from X i , and d denotes the power sharing distance of cellular networks with energy harvesting. Particularly, denoting M i as the counting measure where M i is a Poisson distributed random variable with density given by λ m = λ c b πd 2 . It is worth noting that in each time slot, for BSs in b , all the remaining power η i b will be utilized for data transmission in the next time slot.
The propagation loss of directed MPT is modeled as [12] l where r denotes the power sharing distance, 2 < α < 4 denotes the directed MPT path-loss exponent, and Let Y j be the location of the j-th BS in c b . In addition, let b (Y j , d) be the set of BSs in b within a distance of d from Y j as illustrated in Fig.1(b). Then, based on (1) and (2), we obtain the total shared power received at the j-th BS in c b as where X i denotes the location of the i-th BS in b (Y j , d).
Let N j be the counting measure of b (Y j , d). Then, it can be easily verified that N j follows a Poisson distribution with density given by λ n = λ b π d 2 . It can be also verified that With (4), it thus follows that the total power (the harvested power plus the shared power) received by the j-th BS in c b at the tagged time slot is given by where P j denotes the renewable power harvested by Y j .
To simplify the analysis, it is worth noting that in each time slot, if T j ≥ η b , the j-th BS in c b will use η b for data transmission in the next time slot and discard the remaining power.

B. PERFORMANCE METRIC
To evaluate the performance of the proposed cooperative power sharing scheme, we introduce a new performance metric, namely the power coverage probability, for cellular networks with energy harvesting and MPT based power sharing, which is specified as follows.
Power Coverage Probability: For cellular networks with energy harvesting and MPT based power sharing, the power coverage probability, denoted by τ , is defined as the probability that a BS satisfies its respective power target. Particularly, denoting P as the harvested power of the BS, η as the corresponding power target, and S as the total shared power received at the BS, the power coverage probability for cellular networks with energy harvesting and MPT based power sharing is given by where 1(A) denotes the indicator function.

III. CHARACTERIZATION OF S j
In this section, we characterize the mean and variance of S j . Lemma 1: For cellular networks with energy harvesting and MPT-based power sharing, the mean value of S j received at the j-th BS in c b is given by where Proof: See Appendix A. Remark 1: It can be easily verified that E S j is an increasing function of β b when β b is small, while a decreasing function of β b when β b is large.
Remark 2: It can be also verified that E S j is a decreasing function of d, which is intuitively expected since a long range of power transfer via MPT may suffer a considerable loss of energy.
Remark 3: Further, it can be verified that E S j is an increasing function of λ, which is due to the fact that the energy loss of MPT based power transfer decreases with the transmission d and thereby λ.
Lemma 2: For cellular networks with energy harvesting and MPT-based power sharing, the variance of S j received at the j-th BS in b can be approximated by Proof: See Appendix B. Remark 4: It can be easily verified that Var S j is a decreasing function of d, which is due to the fact the fluctuation of Var S j decreases with the energy loss of MPT.
Remark 5: It can be also verified that Var S j is a decreasing function of λ, which is due to the law of large number.
With Lemmas 1 and 2, in the following section, we characterize the power coverage probability of the cellular networks with MPT-based power sharing.

IV. POWER COVERAGE PROBABILITY
In this section, we analyze the power coverage probability of the cellular networks with MPT based power sharing. It is worth noting that thanks to the homogeneousness of the studied cellular network with energy harvesting and MPT based power sharing, the average power coverage performance of the cellular network can be characterized by the coverage performance experienced at the j-th BS. With this fact and (6), we obtain the power coverage probability of cellular networks with energy harvesting and MPT-based power sharing as where Remark 6: It can be easily verified that the power coverage probability τ contains two parts. The first part calculates the power coverage probability of BSs with the harvested energy less than the predefined target η. In this case, the respective BSs may satisfy the power target η b if and only if P j + S j ≥ η b . The second part calculates the power coverage probability of BSs with the harvested energy lager than the predefined target η, which is straight forward from the proposed MPT based power sharing scheme. In the following, we first derive β b and ϕ b , and then based on which characterize τ .
Lemma 3: For cellular networks with energy harvesting and MPT-based power sharing, we obtain β b as Remark 7: It can be verified that β b is a decreasing function of η b while an increasing function of P.
Proof: See Appendix C. In the following, we first characterize Pr S j ≥ η b , and then based on which characterize ϕ b .
Lemma 4: For cellular networks with energy harvesting and MPT-based power sharing, Pr S j ≥ η b is upper and lower bounded by Proof: See Appendix D. Remark 8: It is worth noting that only the mean and variance of S j can be derived. As such, only upper and lower bounds on Pr S j ≥ η b can be obtained.
Remark 9: It is also worth noting that the derived upper and lower bounds on Pr S j ≥ η b are piecewise functions, where the segmentation point depends on the values of E S j and η b .
Based on Lemma 4, we characterize ϕ b in the following lemma.
Lemma 5: For cellular networks with energy harvesting and MPT-based power sharing, ϕ b is upper and lower bounded by Proof: See Appendix E. Based on Lemmas 3 and 5, we characterize τ in the following theorem.
Theorem 1: For cellular networks with energy harvesting and MPT-based power sharing, τ is upper and lower bounded by where and Proof: Based on Lemmas 3 and 5, with (9), (16) can be immediately obtained. This thus completes the proof of Theorem 1.
Based on Theorem 1, we characterize lim λ→∞ τ in the following corollary.
Corollary 1: For cellular networks with energy harvesting and MPT-based power sharing, we have where Proof: See Appendix F.

V. NUMERICAL RESULTS
In this section, to validate our analytical results and demonstrate the benefit of MPT based multi-cell cooperation, extensive simulations are presented. Throughout this section, unless specified otherwise, we set P = 5, η b = 0.5, and α = 3. Further, the power sharing distance d is measured in meter.   2 plots the analytical and simulated results on the mean value of S j for λ = 0.01, 0.1, 1, respectively. It is observed that the simulation results match with analytical results. 3 It is also observed that the mean value of S j is a decreasing function of the power sharing distance d, which is intuitively expected since the energy transformed via MPT may suffer a considerable loss due to the long range transmission. Further, it is observed that the mean value of S j is an increasing function of λ, which is due to the fact that the total energy loss of MPT based power transfer decreases with d and thereby λ. Fig. 3 plots the analytical and simulated results on the variance of S j for λ = 0.01, 0.1, 1, respectively. It is observed that the simulation results match with analytical results. It is also observed that the variance of S j is a decreasing function 3 It worth noting that the simulation results are fluctuating around the analytical results, which is due to the the central limit theorem.  of the power sharing distance d, which is due to the fact that the fluctuation of S j vanishes as d. Further, it is observed that the variance of S j is decreasing function of λ, which is due to the law of large number.
Figs. 4, 5, 6 shows the analytical and simulated results on power coverage probability of cellular networks with energy harvesting and MPT based power sharing versus the power sharing distance d, for λ = 0.01, 0.1, 1, respectively. It is observed that the derived upper and lower bounds on power coverage probability are valid. It is also observed that the power coverage probability is a decreasing function of d, which is due to the fact the energy loss via MPT increases dramatically with d. Fig. 7 compares the simulated values of power coverage probability of cellular networks with energy harvesting and MPT based power sharing for λ = 0.01, 0.1, 1, ∞. It is observed that the power coverage probability of cellular networks with energy harvesting and MPT based power sharing is an increasing function of λ, which is mainly due to the fact that the distance of the power transfer decreases   with λ. Particularly, the tight upper bound on the power coverage probability is a piecewise function of d, which can be achieved by increasing λ to ∞. Further, the segmentation VOLUME 8, 2020 point of the power coverage probability depends on the mean value of S j and η b . An implication of the above observations is that: for cellular networks with energy harvesting and MPT based power sharing, it is more beneficial to increase the density of BSs than expanding the power sharing distance d to counteract the randomness of the renewables.

VI. CONCLUSION
This paper has studied the performance of hyperdense large-scale cellular networks with energy harvesting. To combat the randomness of renewables, the BSs with the harvested power larger than the predefined power targets are assumed to cooperatively share their surplus power to the BSs with the harvested power lower than the predefined power targets via directed MPT. By applying tools from stochastic geometry, we first derived the mean and variance of S j to capture the statistics of the shared power. Then, based on the obtained results, we characterize the power coverage probability of the studied large-scale cellular network with energy harvesting and MPT based power sharing. Finally, extensive simulations are provided to verify our analysis. An implication of our analytical and simulated results is that: for cellular networks with energy harvesting and MPT based power sharing, it is more beneficial to increase the density of BSs than expanding the power sharing distance to counteract the randomness of the renewables. It is hoped that the results in this paper will provide new insights to the practical design of power sharing schemes via MPT in cellular networks with energy harvesting.

APPENDIXES APPENDIX A PROOF OF LEMMA 3.1
Proof: For cellular networks with energy harvesting and MPT-based power sharing, it can be easily verified that where (a) follows from the Campbell's Theorem, (b) follows from the fact that and (c) follows from the fact that This thus completes the proof of Lemma 1.

APPENDIX B PROOF OF LEMMA 3.2
Proof: For cellular networks with energy harvesting and MPT-based power sharing, it can be easily verified that where (a) follows from [20], and (b) follows from the fact that It is worth noting that where λ m = λ (1 − β b ) πd 2 , (a) follows from [21], s(n, k) denotes the Stirling numbers of the first kind as s(n, k) = (−1) n−k n k , and (b) follows from the fact that s(1, 1) = 1. Then, with (24) and (26), (8) is immediately obtained. This thus completes the proof of Lemma 2.

APPENDIX C PROOF OF LEMMA 4.1
Proof: For cellular networks with energy harvesting and MPT-based power sharing, given that the power targets for different BSs are equal to the same constant η b , it can be easily verified that This thus completes the proof of Lemma 3.

APPENDIX D PROOF OF LEMMA 4.2
Proof: For cellular networks with energy harvesting and MPT-based power sharing, given that the power targets for different BSs are equal to the same constant η b , it can be easily verified that for E S j ≥ η b , while for E S j < η b , where (a) and (b) follow from the Cantelli's inequality [22] that This thus completes the proof of Lemma 4.

APPENDIX E PROOF OF LEMMA 4.3
Proof: For cellular networks with energy harvesting and MPT-based power sharing, given that the power targets for different BSs are equal to the same constant η b , it can be easily verified from (10) and Lemma 4 that for E S j ≥ η b , On the other hand, for E S j < η b , we have where (a) follows from the property of convex functions that Then, for E S j < η b , based on (10) and Lemma 4, it can be easily verified that where (a) follows from the fact that This thus completes the proof of Lemma 5.

APPENDIX F PROOF OF COROLLARY 4.1
Proof: Based on Lemmas 1 and 2, for η b ≤ E S j , it can be easily verified that Thus, with Theorem 1, for η b ≤ E S j , we obtain that On the other hand, for η b > E S j , based on Lemma 4, we have As such, for η b > E S j , it follows that lim λ→∞ Pr S j < η b = 1.
Further, based on the proof of Lemma 5, it can be easily verified that lim λ→∞ Pr P j + S j ≥ η b P j < η b , S j < η b where (a) follows from Lemma 2 that As such, with (9), (10) and (37), we obtain that This thus completes the proof of Corollary 1.