Energy Harvesting in Unmanned Aerial Vehicle Networks with 3D Antenna Radiation Patterns

In this paper, we consider a UAV network consisting of UAVs operating at a certain altitude above ground. The locations of user equipments (UEs) are modeled as a Poisson cluster process (PCP), and the UAVs are assumed to be located at a certain height above the center of user clusters. Since UAVs are deployed in overloaded scenarios, locations of UAVs and UEs are expected to be correlated and UEs are more likely to form clusters. Hence, modeling the UE locations by PCP is more appropriate and realistic. Moreover, we consider that UWB antennas with doughnut-shaped radiation patterns are employed at both UAVs and UEs, and we study the effect of practical 3D antenna radiation patterns on energy harvesting from UAVs.

More specifically, our main contributions can be summarized as follows:

• •

  An analytical framework is provided to analyze energy coverage performance of a UAV network with clustered UEs. UE locations are assumed to be PCP distributed to capture the correlations between the UAV and UE locations.

• •

  We divide the network into two tiers: $0^{\text{th}}$ tier UAV and $1^{\text{st}}$ tier UAVs. $0^{\text{th}}$ tier UAV is the cluster center UAV around which the typical UE is located, while other UAVs constitute the $1^{\text{st}}$ tier.

• •

  Two different LOS probability functions, i.e., a high-altitude model and a low-altitude model, are considered in order to investigate and compare their impact on the network performance.

• •

  Different from the previous studies, more practical antennas with doughnut-shaped radiation patterns are employed at both UAVs and UEs to provide a more realistic performance evaluation for the network.

• •

  We first derive the complementary cumulative distribution functions (CCDFs) and the probability density functions (PDFs) of the path losses for each tier, then obtain the association probabilities by using the averaged received power UAV association rule.

• •

  Average harvested power expression is obtained. Then, total energy coverage probability is determined by deriving the Laplace transforms of the interference terms arising from each tier. Energy coverage is characterized for UAV deployments at a single height level and also at multiple heights.

The rest of the paper is organized as follows. In Section II, system model is introduced, and path loss, blockage, and 3D antenna models are described. Path loss and association probabilities are statistically characterized in Section III. In Section IV, energy coverage probability of the UAV network is determined. In Section VI, simulations and numerical results are provided, demonstrating the impact of several key parameters on the energy coverage performance of the network. Finally, Section VII concludes the paper. Proofs are included in the Appendix.

A transmitting UAV can either have a line-of-sight (LOS) or non-line-of-sight (NLOS) link to the typical UE. Consider an arbitrary link of length $r$ between a UE and a UAV, and define the LOS probability function as the probability that the link is LOS. Different LOS probability functions have been used in the literature. In this paper, we consider the two models proposed in [41] and [42], which are high-altitude and low-altitude models, respectively.

High-altitude model is widely used especially in satellite communications where the altitude is around hundred of kilometers. It has also been widely employed in UAV-assisted networks recently. LOS probability function for the high-altitude model is given as follows:

$$\mathcal{P}_{\text{L}}^{\text{high}}(r)=\left(\frac{1}{1+b\exp\left(-c\left(\frac{180}{\pi}\sin^{-1}\left(\frac{H}{r}\right)-b\right)\right)}\right),$$

(3)

where $r$ is the 3D distance between the UE and UAV, $H$ is the UAV height, $b$ and $c$ are constants whose values depend on the environment. It can be easily verified that the LOS probability in (3) increases as the UAV height, $H$, increases.

Since practical values for UAV height in certain applications is around $50\sim 100$ meters, a more realistic LOS probability function proposed for 3GPP terrestrial communications is employed also for UAV networks in [24]. The height of a macrocell base station is usually around 32 m, which is comparable to the practical UAV height. Therefore, employment of the LOS probability function for 3GPP macrocell-to-UE communciation is also reasonable for the UAV networks in such relatively low-altitude scenarios. For the low-altitude model, LOS probability function is expressed as

$$\mathcal{P}_{\text{L}}^{\text{low}}(r)=\min\left(1,\frac{18}{r}\right)\left(1-\exp\left(-\frac{r}{63}\right)\right)+\exp\left(-\frac{r}{63}\right).$$

(4)

Note that different from the high-altitude model, LOS probability function in (4) decreases with the increase in the 3D distance $r$, independent of the UAV height. In Fig. 2, LOS probability function is plotted using high-altitude and low-altitude models. Solid lines show the LOS probability as a function of the UAV height $H$ when the 2D distance to the UAV is fixed at $d=10$ m, and dashed lines display the LOS probability as a function of the 2D distance to the UAV $d$ when the UAV height is $H=50$ m. As shown in Fig. 2, LOS probability increases with increasing UAV height when the high-altitude model is used, and decreases when the low-altitude model is considered. We observe that the LOS probability decreases for both models as the 2D distance to the UAV increases. We also note that the analysis in the remainder of the paper is general and is applicable to any LOS probability function. Only in Section VI, we employ the LOS probability functions in (3) and (4) to obtain the numerical results.

In general, NLOS links experience higher path loss than the LOS links due to signals being reflected and scattered in NLOS propagation environments. Therefore, different path loss laws are applied to LOS and NLOS links. Thus, the path loss on each link in tier $k\in\{0,1\}$ can be expressed as

$$\begin{split}L_{k,\text{L}}(r)&=r^{\alpha_{\text{L}}}\\ L_{k,\text{N}}(r)&=r^{\alpha_{\text{N}}},\end{split}$$

(5)

where $\alpha_{\text{L}}$ and $\alpha_{\text{N}}$ are the LOS and NLOS path-loss exponents, respectively.

In this paper, we consider the analytical model developed in [37] for the effect of 3D antenna radiation patterns on the received signal. UWB transmitter and receiver antennas with doughnut-shaped radiation patterns centered at a frequency of 4 GHz are placed at the UAV and UE, respectively, and air-to-ground channel measurements are carried out in order to characterize the impact of the 3D antenna radiation pattern on the received signal for different antenna orientations in [37]. As a result of these measurements, transmitter and receiver antenna gains are modeled analytically for horizontal-horizontal (HH), horizontal-vertical (HV) and vertical-vertical (VV) antenna orientations as follows:

$$G_{k}(\theta)=G_{\text{TX}}(\theta)G_{\text{RX}}(\theta)=\begin{cases}\sin(\theta)\sin(\theta)&\text{for}\quad\text{HH}\\ \sin(\theta)\cos(\theta)&\text{for}\quad\text{HV},\\ \cos(\theta)\cos(\theta)&\text{for}\quad\text{VV}\end{cases}$$

(6)

where $\theta$ is the elevation angle between the transmitter at the UAV and the receiver at the UE on the ground. In this antenna model, radiation pattern is approximated by a circle in the vertical dimension, while it is assumed to be constant for all horizontal directions. In other words, antenna gains depend only on the elevation angle $\theta$, and are considered as independent of the azimuth angle between the transmitter at the UAV and the receiver at the UE. Approximated antenna radiation patterns of UAV and UE are shown in Fig. 3 for HH antenna orientation. They can be plotted for HV and VV orientations as well by rotating the transmitter and/or receiver antennas by $90^{\circ}$. Note that for HH antenna orientation $G_{\text{TX}}(\theta)=G_{\text{RX}}\rightarrow 0$ as $\theta\rightarrow 0$ which happens when the UEs are located far away from the cluster center, i.e. as the $\sigma_{c}$ increases, and $G_{\text{TX}}(\theta)=G_{\text{RX}}\rightarrow 1$ as $\theta\rightarrow 90^{o}$ which happens when the UEs get closer to the cluster center. Similar observations can be drawn for VH and VV antenna orientations. Effective antenna gain $G_{k}$ as a function of $r$ can be rewritten in terms of UAV height $H$ and the path loss on each link in tier $k\in\{0,1\}$ as

$$\small G_{k}(r)=\begin{cases}H^{2}L_{k,s}^{-\frac{2}{\alpha_{s}}}(r)&\text{for}\quad\text{HH}\\ H\left(\sqrt{L_{k,s}^{\frac{2}{\alpha_{s}}}(r)-H^{2}}\right)L_{k,s}^{-\frac{2}{\alpha_{s}}}(r)&\text{for}\quad\text{HV}\\ \left(L_{k,s}^{\frac{2}{\alpha_{s}}}(r)-H^{2}\right)L_{k,s}^{-\frac{2}{\alpha_{s}}}(r)&\text{for}\quad\text{VV}.\end{cases}\normalsize$$

(7)

In the rest of the analysis, we assume that the typical UE and all UAVs in the network have horizontal antenna orientation. Therefore, HH antenna orientation for the main link and interfering links are considered due to its analytical tractability. Moreover, UEs are considered to be clustered around the projections of UAVs on the ground and more UEs are encouraged to be associated with their cluster center UAV. As a result, the angle between the transmitter at the UAV and the receiver at the UE is expected to be large. Therefore, HH antenna orientation is more suitable than the other two orientations. However, in the numerical results section, simulation results for HV and VV orientations are also provided in order to compare their effect on the UAV association and energy coverage probabilities.

Finally, we note that a summary of notations is provided in Table I below.

In this section, we provide statistical characterizations by identifying the CCDF and the PDF of the path loss.

Lemma 1: The CCDF of the path loss from a typical UE to a $0^{\text{th}}$ tier UAV can be formulated as

	$\displaystyle\bar{F}_{L_{0}}(x)$	$\displaystyle=\sum_{s\in\{\text{L},\text{N}\}}\bar{F}_{L_{0,s}}(x)$
		$\displaystyle=\sum_{s\in\{\text{L},\text{N}\}}\int_{\sqrt{x^{2/\alpha_{s}}-H^{2}}}^{\infty}\mathcal{P}_{s}(\sqrt{d^{2}+H^{2}})f_{D}(d)\mathrm{d}d,$		(8)

where $f_{D}(d)$ is given in (1), and $\mathcal{P}_{s}(\cdot)$ is the LOS or NLOS probability depending on whether $s=\text{L}$ or $s=\text{N}$ ¹¹1For instance, LOS probability is given by (3) and (4) for the high-altitude and low-altitude models, respectively, and NLOS probability is $\mathcal{P}_{\text{N}}(\cdot)=1-\mathcal{P}_{\text{L}}(\cdot)$..

Proof: See Appendix -A. $\square$

Lemma 2: CCDF of the path loss from a typical UE to a $1^{\text{st}}$ tier UAV is

$$\bar{F}_{L_{1}}(x)=\prod_{s\in\{\text{L},\text{N}\}}\bar{F}_{L_{1,s}}(x)=\prod_{s\in\{\text{L},\text{N}\}}\exp\big{(}-\Lambda_{1,s}([0,x))\big{)},$$

(9)

where $\Lambda_{1,s}([0,x))$ is given by

$\displaystyle\Lambda_{1,s}([0,x))=2\pi\lambda_{U}\int_{H}^{x^{\frac{1}{\alpha_{s}}}}\mathcal{P}_{s}(r)r\mathrm{d}r.$

(10)

Proof: See Appendix -B. $\square$

Corollary 1: The PDF of the path loss from a typical UE to a $0^{\text{th}}$ tier LOS/NLOS UAV can be determined by applying the Leibniz integral rule as follows:

	$\displaystyle f_{L_{0,s}}(x)$	$\displaystyle=-\frac{d\bar{F}_{L_{0,s}}(x)}{dx}$
		$\displaystyle=\frac{1}{\sigma_{c}^{2}}\frac{x^{\frac{2}{\alpha_{s}}-1}}{\alpha_{s}}\mathcal{P}_{s}\left(x^{\frac{1}{\alpha_{s}}}\right)\exp\left(-\frac{1}{2\sigma_{c}^{2}}\left(x^{\frac{2}{\alpha_{s}}}-H^{2}\right)\right).$		(11)

Corollary 2: The PDF of the path loss from a typical UE to a $1^{\text{st}}$ tier LOS/NLOS UAV is

$$f_{L_{1,s}}(x)=-\frac{d\bar{F}_{L_{1,s}}(x)}{dx}=\Lambda_{1,s}^{\prime}([0,x))\exp\big{(}-\Lambda_{1,s}([0,x))\big{)},$$

(12)

where $\Lambda_{1,s}^{\prime}([0,x))$ is given by

$$\Lambda_{1,s}^{\prime}([0,x))=2\pi\lambda_{U}\frac{x^{\frac{2}{\alpha_{s}}-1}}{\alpha_{s}}\mathcal{P}_{s}\left(x^{\frac{1}{\alpha_{s}}}\right)$$

(13)

by again applying the Leibniz integral rule.

In the results above, we have determined the CCDFs and PDFs of the path loss for each tier. They depend on the key network parameters including the variance of the cluster process $\sigma_{c}^{2}$, UAV density $\lambda_{U}$, UAV LOS probability $\mathcal{P}_{s}(\cdot)$, UAV height $H$ and path loss exponents $\alpha_{s}$. In the following sections, these distributions are utilized in determining the association and energy coverage probabilities.

In this work, we assume that the UEs are associated with a UAV providing the strongest long-term averaged power. In other words, a typical UE is associated with its cluster center UAV, i.e., the $0^{\text{th}}$ tier UAV, if

$$P_{0}G_{0}(r)L_{0}^{-1}(r)\geq P_{1}G_{1}(r)L_{\text{min},1}^{-1}(r),$$

(14)

where $P_{k}$ and $G_{k}(r)$ denote the transmit power and antenna gain of the link, respectively, in tier $k\in(0,1)$. $L_{0}(r)$ is the path loss from the $0^{\text{th}}$ tier UAV, and $L_{\text{min},1}(r)$ is the path loss from $1^{\text{st}}$ tier UAV providing the minimum path loss. In the following lemma, we provide the association probabilities using the result of Lemmas 1, Lemma 2, Corollary 1 and Corollary 2.

Lemma 3: The association probabilities with a $0^{\text{th}}$ tier LOS/NLOS UAV and $1^{\text{st}}$ tier LOS/NLOS UAV are given, respectively, as

	$\displaystyle\mathcal{A}_{0,s}=\int_{H^{\alpha_{s}}}^{\infty}$	$\displaystyle\prod_{m\in\{\text{L},\text{N}\}}\bar{F}_{L_{1,m}}\left(\left(\frac{P_{1}}{P_{0}}l_{0,s}^{\frac{2}{\alpha_{s}}+1}\right)^{\frac{\alpha_{m}}{\alpha_{m}+2}}\right)$
		$\displaystyle\times f_{L_{0,s}}(l_{0,s})\mathrm{d}l_{0,s},$		(15)

	$\displaystyle\mathcal{A}_{1,s}=\int_{H^{\alpha_{s}}}^{\infty}$	$\displaystyle\sum_{m\in\{\text{L},\text{N}\}}\bar{F}_{L_{0,m}}\left(\left(\frac{P_{0}}{P_{1}}l_{1,s}^{\frac{2}{\alpha_{s}}+1}\right)^{\frac{\alpha_{m}}{\alpha_{m}+2}}\right)$
		$\displaystyle\times\bar{F}_{L_{1,s^{\prime}}}(l_{1,s})f_{L_{1,s}}(l_{1,s})\mathrm{d}l_{1,s},$		(16)

where $s,s^{\prime}\in\{\text{L},\text{N}\}$ and $s\neq s^{\prime}$.

Proof: See Appendix -C. $\square$

In the corollary below, we provide a closed-form expression for the association probability in a special case with which the effects of different parameters on association probability can be easily identified.

Corollary 3: Consider the same UAV network with $P_{0}=P_{1}$ and the LOS probability $\mathcal{P}_{\text{L}}(\cdot)=1$, i.e., all UAVs are LOS to the typical UE. Then, the association probabilities specialize to the following expressions (which also confirm the results in [43]):

	$\displaystyle\mathcal{A}_{0,L}$	$\displaystyle=\frac{1}{1+2\pi\lambda_{U}\sigma_{c}^{2}}$		(17)
	$\displaystyle\mathcal{A}_{1,L}$	$\displaystyle=\frac{2\pi\lambda_{U}\sigma_{c}^{2}}{1+2\pi\lambda_{U}\sigma_{c}^{2}}$		(18)

with $\mathcal{A}_{0,N}=\mathcal{A}_{1,N}=0$. According to the these results, a typical UE obviously prefers to connect to the $0^{\text{th}}$ tier UAV when the value of $\sigma_{c}$ is small, and connect to a $1^{\text{st}}$ tier UAV for higher values of $\sigma_{c}$ and $\lambda_{U}$.

The total power received at a typical UE at a random distance $r$ from its associated UAV in the $k^{\text{th}}$ tier can be written as

$\displaystyle P_{r,k}$

$\displaystyle=S_{k}+\sum_{j=0}^{1}I_{j,k}\quad\text{for}\quad k=0,1,$

(19)

where the received power from the serving UAV $S_{k}$ and the interference power received from the UAVs in the $j^{\text{th}}$ tier $I_{j,k}$ are given as follows:

	$\displaystyle S_{k}$	$\displaystyle=P_{k}G_{k}(r)h_{k,0}L_{k}^{-1}(r),$		(20)
	$\displaystyle I_{0,1}$	$\displaystyle=P_{0}G_{0}(r)h_{0,0}L_{0}^{-1}(r),$		(21)
	$\displaystyle I_{1,k}$	$\displaystyle=\sum_{i\in\Phi_{U}\setminus{\mathcal{E}_{k,0}}}P_{1}G_{i}(r)h_{1,i}L_{i}^{-1}(r),$		(22)

where $h_{k,0}$ and $h_{j,i}$ are the small-scale fading gains from the serving and interfering UAVs, respectively. Note that since only one UAV exists in the $0^{\text{th}}$ tier, $I_{0,0}=0$. $h$ denotes the small-scale fading gain and is assumed to be exponentially distributed. From the UAV association policy, when a typical UE is associated with a UAV whose path loss is $L_{k}(r)$, there exists no UAV within a disc $\mathcal{E}_{k,0}$ centered at the origin, which is also known as the exclusion disc. In this work, we also consider a linear energy harvesting model in which energy can be harvested if the received power is larger than zero. Therefore, the average harvested power at a typical UE is given in the following theorem.

Lemma 4: The average harvested power at a typical UE at a random distance $r$ from its associated UAV in the $k^{\text{th}}$ tier is given at the top of the next page in (23)

where

$\displaystyle\Psi_{I_{0,k}}(\mathcal{E}_{k,0})=\sum_{s^{\prime}\in\{\text{L},\text{N}\}}\int_{\mathcal{E}_{k,0}}^{\infty}P_{0}H^{2}x^{-\left(1+\frac{2}{\alpha_{s^{\prime}}}\right)}f_{L_{0,s^{\prime}}}(x)\mathrm{d}x,$

(24)

$\displaystyle\Psi_{I_{1,k}}(\mathcal{E}_{k,0})=\sum_{s^{\prime}\in\{\text{L},\text{N}\}}\int_{\mathcal{E}_{k,0}}^{\infty}P_{1}H^{2}x^{-\left(1+\frac{2}{\alpha_{s^{\prime}}}\right)}\Lambda_{1,s^{\prime}}^{\prime}([0,x))\mathrm{d}x.$

(25)

Proof: See Appendix -D. $\square$

The energy harvested at a typical UE in unit time is expressed as $E_{k}=\xi P_{r,k}$ where $\xi\in(0,1]$ is the rectifier efficiency, and $P_{r,k}$ is the total received power given in (19). Since the effect of additive noise power is negligibly small relative to the total received power, it is omitted [8]. The conditional energy coverage probability $\text{E}^{\text{C}}_{k}(\Gamma_{k})$ is the probability that the harvested energy $E_{k}$ is larger than the energy outage threshold $\Gamma_{k}>0$ given that the typical UE is associated with a UAV from the $k^{\text{th}}$ tier, i.e., $\text{E}^{\text{C}}_{k}(\Gamma_{k})=\mathbb{P}(E_{k}>\Gamma_{k}|t=k)$. Therefore, total energy coverage probability $\text{E}^{\text{C}}$ for the typical UE can be formulated as

$$\text{E}^{\text{C}}=\sum_{k=0}^{1}\sum_{s\in\{\text{L},\text{N}\}}\left[\text{E}^{\text{C}}_{k,s}(\Gamma_{k})\mathcal{A}_{k,s}\right],$$

(26)

where $\text{E}^{\text{C}}_{k,s}(\Gamma_{k})$ is the conditional energy coverage probability given that the UE is associated with a $k^{\text{th}}$ tier LOS/NLOS UAV, and $\mathcal{A}_{k,s}$ denotes the association probability. The following theorem provides our main characterization regarding the total energy coverage probability.

Proof: See Appendix -E. $\square$

Note that since $0^{\text{th}}$ tier consists of only one UAV, i.e., the cluster center UAV, Laplace transform expression $\mathcal{L}_{I_{0,0}}(\Gamma_{0},\mathcal{E}_{0,0})=1$. The total energy coverage probability of the network in Theorem 1 is obtained by first computing the conditional energy coverage probability given that a UE is associated with a $k^{\text{th}}$ tier LOS/NLOS UAV, and then summing up the conditional probabilities weighted with their corresponding association probabilities. In order to formulate the conditional energy coverage probabilities, Laplace transforms of the interference terms are determined. We also note that although the energy coverage probability approximation in Theorem 1 involves multiple integrals, we explicitly see the dependence of the energy coverage on, for instance, UAV heights, path loss distributions, path loss exponents, transmission power levels. Moreover, the integrals can be readily computed via numerical integration methods, providing us with additional insight on the impact of key system/network parameters, as demonstrated in the next section.

First, we address the effect of UE distribution’s standard deviation $\sigma_{c}$ on the association probability, average harvested power and the energy coverage probability using the LOS probability functions of high-altitude and low-altitude models of (3) and (4) in Figs. 4(a), 4(b) and 4(c). As the standard deviation increases, the UEs are spread more widely, resulting in generally larger distances between the cluster-center $0^{\text{th}}$ tier UAV and UEs. For example, for $\sigma_{c}=10$, the average link distance between cluster center UAV and UEs is around 12.5 meters, while it is around 115 meters for $\sigma_{c}=90$. Consequently, association probability with the $0^{\text{th}}$ tier UAV, $\mathcal{A}_{0}$, diminishes, while the association probability with $1^{\text{st}}$ tier UAVs, $\mathcal{A}_{1}$, increases for both models. Also, for a fixed height, LOS probability of cluster center UAV decreases for both models with the increasing cluster size, and hence association probabilities exhibit similar trends. Therefore, the average harvested power from the $0^{\text{th}}$ tier UAV, $P^{\text{avg}}_{0}$, increases while the average harvested power from the $1^{\text{st}}$ tier UAVs, $P^{\text{avg}}_{1}$, decreases as the cluster size grows in both models. On the other hand, the increase in $P^{\text{avg}}_{1}$ cannot compensate the decrease in $P^{\text{avg}}_{0}$, and therefore the total average harvested power $P^{\text{avg}}$ diminishes. In other words, smaller cluster size, i.e., more compactly distributed UEs results in a higher $P^{\text{avg}}$. The energy coverage probability in Fig. 4(c) exhibits a very similar behavior as the average harvested power in Fig. 4(b). Also note that, the association probability results of the Corollary 3 closely approximate the association probabilities of the high-altitude model. Finally, we note that simulation results are also plotted in the figure with markers and there is generally an excellent agreement with the analytical results, further validating our analysis.

Next, in Figs. 5(a) and 5(b), we plot the association probability and energy coverage probability as a function of UAV height considering the LOS probability functions of both high-altitude and low-altitude models. For the high-altitude model, since LOS probability increases with the increasing UAV height, association probability with the $0^{\text{th}}$ tier UAV increases slightly. On the other hand, LOS probability decreases as a result of the increase in the 3D distance with the increasing UAV height in the low-altitude model. Therefore, more UEs prefer to connect to $1^{\text{st}}$ tier UAVs (i.e., UAVs other than the cluster-center one) at higher values of the UAV height. Also note that the result of the Corollary 3 very closely approximates the association probabilities of the high-altitude model especially as the UAV height increases.

Energy coverage probability of the cluster center UAV, $\text{E}^{\text{C}}_{0}$, exhibits similar trends for both types of LOS functions. More specifically, $\text{E}^{\text{C}}_{0}$ increases first then it starts decreasing with the increasing UAV height. Since the effective antenna gain for HH antenna orientation is an increasing function of UAV height for a fixed cluster size, an initial increase in $\text{E}^{\text{C}}_{0}$ is expected. However, further increase in UAV height results in a decrease in $\text{E}^{\text{C}}_{0}$ of both high-altitude and low-altitude models due to the increase in the distance. Therefore, for a fixed cluster size, there exists an optimal UAV height maximizing the network energy coverage, $\text{E}^{\text{C}}$, for both models. On the other hand, optimal height maximizing the $\text{E}^{\text{C}}$ in the low-altitude model is lower and $\text{E}^{\text{C}}$ decreases faster than that in the high-altitude model because the LOS probability function of the low-altitude model is a decreasing function of distance while the LOS probability function of the high-altitude model is an increasing function of the UAV height (e.g., as seen in Fig. 2). Moreover, since UEs are more compactly distributed around the cluster center UAVs for $\sigma_{c}=10$, energy coverage probability of the $1^{\text{st}}$ tier UAVs, $\text{E}^{\text{C}}_{1}$, is relatively small and changes only very slightly for both models.

In Figs. 6(a), 6(b) and 6(c), we plot the association probability as a function of UAV height $H$ for different values of UAV density $\lambda_{U}$ for three different antenna orientations considering the high-altitude LOS probability model. Note that since the analysis for VV and HV antenna orientations seems to be intractable, only simulation results are plotted. Since effective antenna gain depends on the sine function of the angle between the UAVs and UEs for HH antenna orientation, UEs prefer to connect to their cluster center UAV, and hence $\mathcal{A}_{0}$ is much larger than $\mathcal{A}_{1}$ even when there is an increase in the number of UAVs (as seen when the UAV density is increased from $\lambda_{U}=10^{-5}$ to $\lambda_{U}=10^{-4}$) as shown in Fig. 6(a). Also note that since both antenna gain and LOS probability is an increasing function with UAV height, increase in them can compensate the increasing path loss and the association probabilities remain almost constant.

For the VV antenna orientation, effective antenna gain depends on the cosine of the angle between the UAVs and UEs. For larger values of UAV density, association probability with the $0^{\text{th}}$ tier UAV, $\mathcal{A}_{0}$, slightly increases with increasing UAV height at first as a result of the increase in both the LOS probability and the effective antenna gain. Subsequently, $\mathcal{A}_{0}$ starts decreasing because the increase in the LOS probability cannot compensate the rapid decrease in the effective antenna gain between the UE and the cluster center UAV. For a less dense network, UEs associate with the cluster-center UAV mostly at lower UAV heights. However, with the increasing height, antenna gain with the cluster-center UAV decreases and consequently, the association probability with $1^{\text{st}}$ tier UAVs, $\mathcal{A}_{1}$, increases.

Finally, for the HV antenna orientation, effective antenna gain is a function of both cosine and sine of the angle $\theta$. Association probabilities exhibit similar trends as in the case of VV orientation. However, different from the VV case, since the antenna gain depends on both the cosine and sine functions, association probability with the $0^{\text{th}}$ tier UAV, $\mathcal{A}_{0}$, is greater than that of the VV orientation.

We also plot the energy coverage probability for different UAV heights, antenna orientations, and UAV densities in Fig. 7. The performance with the HV and VV antenna orientations exhibit similar behaviors as that with the HH antenna orientation which is described in Section VI-B. For both higher-density (i.e., $\lambda_{U}=10^{-4}$) and lower-density UAV networks (i.e., $\lambda_{U}=10^{-5}$), HH orientation leads the best performance compared to the HV and VV cases. We also note that as the UAV density and UAV height are increased, energy coverage probability with the VV orientation starts exceeding that of the HV case. Therefore, energy coverage performance can be improved by varying the antenna orientations depending on the number of UAVs in the network and their height.

Furthermore, we display the average harvested power and the energy coverage probability as a function of the UAV density for three different antenna orientations considering the high-altitude LOS probability model in Figs. 8 and 9, respectively. Both the average harvested power and the energy coverage probability are increasing functions of the UAV density irrespective of the antenna orientation for a fixed UAV height. Expectedly, adding more UAVs to the network results in an increase in the total power received at the typical UE, and hence the average harvested power grows and the energy coverage performance of the network improves. We also note that HH antenna orientation generally leads to larger average harvested power and energy coverage probability. On the other hand, VV antenna orientation results in a higher average harvested power when the UAV density is sufficiently large due to the fact that one can harvest more power from the dense $1^{\text{st}}$-tier UAVs with smaller elevation angles when this antenna orientation is used.

In Fig. 10, we plot the energy coverage probability as a function of the UAV transmit power for three different antenna orientations considering the high-altitude LOS probability model. As expected, energy coverage probability is an increasing function of UAV transmit power. Moreover, HH antenna orientation generally results in a higher energy coverage probability than VV and HV antenna orientations.

In Fig. 11, we show the energy coverage probabilities of different tiers (i.e., $\text{E}^{\text{C}}_{0}$ and $\text{E}^{\text{C}}_{1}$) and the total energy coverage probability $\text{E}^{\text{C}}$ as a function of the energy outage threshold for both high-altitude and low-altitude models. As seen in Fig. 4(a) and Fig. 5(a), UEs are more likely to be associated with the $0^{\text{th}}$ tier UAV rather than $1^{\text{st}}$ tier UAVs in the high-altitude model when $\sigma_{c}=10$, and hence $\text{E}^{\text{C}}_{0}$ is much higher than $\text{E}^{\text{C}}_{1}$. On the other hand, for the low-altitude model, since association probabilities with each tier are not very different, more UEs can be covered by $1^{\text{st}}$ tier UAVs compared to the high-altitude model. However, $\text{E}^{\text{C}}_{0}$ is still greater than $\text{E}^{\text{C}}_{1}$ due to the relatively smaller distance to the cluster-center UAV. We also observe that as a general trend, energy coverage probabilities expectedly diminish with increasing energy outage threshold.

Finally, in Fig. 12, we plot the total energy coverage probabilities as a function of the energy outage threshold using the high-altitude LOS probability function model when UAVs are assumed to be located at different heights. In this setup, we use the same parameters given in Table II with some differences in UAV height and UAV density. More specifically, we consider $M=2$ groups of UAVs located at altitudes $H_{1}=50$ m and $H_{2}=80$ m with densities $\lambda_{U,1}=\lambda_{U,2}=\lambda_{U}/2$ and transmit powers $P_{1}=P_{2}=37$ dBm. Therefore, transmit power of the $0^{\text{th}}$ UAV is also equal to $P_{0}=37$ dBm. Similar to [24], we consider practical values for UAV heights which are around $50\sim 100$ meters.

In Fig. 12, solid lines plot the total energy coverage probabilities when the height is the same for all UAVs. Dashed lines display the coverage probabilities when half of the UAVs are located at height $H_{1}$ and the other half are located at height $H_{2}$, and the typical UE is clustered around a UAV at either height $H_{1}$ or $H_{2}$. As seen in the figure, energy coverage performance of the different-height UAV network is very similar to that of the same-height UAV. For example, if a typical UE is clustered around a UAV at height $H_{1}=50$ m, the performance of the different-height UAV network is almost the same with that of the same-height UAV for $H=50$ m. On the other hand, performance of the different-height network when the typical UE is clustered around a UAV at height $H_{1}=80$ m is slightly lower than that of the same-height UAV network with $H=80$ m as a result of growing impact of interferences from the UAVs at height $H_{2}=50$ m.

The CCDF of the path loss $L_{0,s}$ from a typical UE to a $0^{th}$ tier LOS/NLOS UAV is

	$\displaystyle\bar{F}_{L_{0,s}}(x)$
	$\displaystyle=\mathbb{E}_{r}\left[\mathbb{P}\left(L_{0,s}(r)\geq x\right)\mathcal{P}_{s}(r)\right]$
	$\displaystyle=\mathbb{E}_{d}\left[\mathbb{P}\left((d^{2}+H^{2})^{\alpha_{s}/2}\geq x\right)\mathcal{P}_{s}(\sqrt{d^{2}+H^{2}})\right]$		(32)
	$\displaystyle=\int_{0}^{\infty}\mathbb{P}\left(d\geq\sqrt{x^{2/\alpha_{s}}-H^{2}}\right)\mathcal{P}_{s}(\sqrt{d^{2}+H^{2}})f_{D}(d)\mathrm{d}d$
	$\displaystyle=\int_{\sqrt{x^{2/\alpha_{s}}-H^{2}}}^{\infty}\mathcal{P}_{s}(\sqrt{d^{2}+H^{2}})f_{D}(d)\mathrm{d}d$		(33)

$\text{for }s\in\{\text{L},\text{N}\}$ where $f_{D}(d)$ is given in (1), and $\mathcal{P}_{s}(\cdot)$ is the LOS or NLOS probability depending on whether $s=\text{L}$ or $s=\text{N}$. With this, the CCDF of the path loss $L_{0}$ from a typical UE to a $0^{\text{th}}$ tier UAV given in Lemma 1 can be obtained by summing up over $s$.

Intensity function for the path loss model from a typical UE to a $1^{\text{st}}$ tier UAV for $s\in\{\text{L},\text{N}\}$ can be formulated as follows:

	$\displaystyle\Lambda_{1,s}([0,x))$	$\displaystyle=\int_{\mathbb{R}^{2}}\mathbb{P}\left(L_{1}(r)<x\right)\mathrm{d}r$		(34)
		$\displaystyle=2\pi\lambda_{U}\int_{H}^{\infty}\mathbb{P}\left(r^{\alpha_{s}}<x\right)\mathcal{P}_{s}(r)r\mathrm{d}r$		(35)
		$\displaystyle=2\pi\lambda_{U}\int_{H}^{\infty}\mathbb{P}\left(r<x^{1/\alpha_{s}}\right)\mathcal{P}_{s}(r)r\mathrm{d}r$		(36)
		$\displaystyle=2\pi\lambda_{U}\int_{H}^{x^{1/\alpha_{s}}}\mathcal{P}_{s}(r)r\mathrm{d}r$		(37)

where (34) is due to the definition of intensity function for the point process of the path loss. CCDF of the path loss $L_{1}$ from a typical UE to a $1^{\text{st}}$ tier UAV given in Lemma 2 can be obtained by summing up $\Lambda_{1,s}([0,x))$ over $s$.

Association probability with a $0^{\text{th}}$ tier LOS/NLOS UAV can be expressed as

	$\displaystyle\mathcal{A}_{0,s}$	$\displaystyle=\prod_{m\in\{\text{L},\text{N}\}}\mathbb{P}(P_{0}G_{0}(r)L_{0,s}^{-1}\geq P_{1}G_{1}(r)L_{1,m}^{-1})$		(38)
		$\displaystyle=\prod_{m\in\{\text{L},\text{N}\}}\mathbb{P}\left(P_{0}\frac{H^{2}}{L_{0,s}^{\frac{2}{\alpha_{s}}}}L_{0,s}^{-1}\geq P_{1}\frac{H^{2}}{L_{1,m}^{\frac{2}{\alpha_{L}}}}L_{1,m}^{-1}\right)$		(39)
		$\displaystyle=\prod_{m\in\{\text{L},\text{N}\}}\mathbb{P}\left(L_{1,m}\geq\left(\frac{P_{1}}{P_{0}}L_{0,s}^{\frac{2}{\alpha_{s}}+1}\right)^{\frac{\alpha_{m}}{\alpha_{m}+2}}\right)$
		$\displaystyle=\int_{H^{\alpha_{s}}}^{\infty}\prod_{m\in\{\text{L},\text{N}\}}\bar{F}_{L_{m}}\left(\left(\frac{P_{1}}{P_{0}}l_{0,s}^{\frac{2}{\alpha_{s}}+1}\right)^{\frac{\alpha_{m}}{\alpha_{m}+2}}\right)$
		$\displaystyle\times f_{L_{0,s}}(l_{0,s})\mathrm{d}l_{0,s}$		(40)

where (38) uses the fact that LOS and NLOS links in the $1^{\text{st}}$ tier are independent, and (40) incorporates the definition of the CCDF of the path loss. Since the distance between UEs and UAVs is at least $H$, the lower limit of the integration is $l_{0,s}=H^{\alpha_{s}}$.