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On the Performance of Renewable Energy-PoweredUAV-Assisted Wireless Communications

Silvia Sekander, Hina Tabassum, and Ekram Hossain

Abstract—We develop novel statistical models of the harvestedenergy from renewable energy sources (such as solar and windenergy) considering harvest-store-consume (HSC) architecture.We consider three renewable energy harvesting scenarios, i.e.(i) harvesting from the solar power, (ii) harvesting from the windpower, and (iii) hybrid solar and wind power. In this context,we first derive the closed-form expressions for the probabilitydensity function (PDF) and cumulative density function (CDF)of the harvested power from the solar and wind energy sources.Based on the derived expressions, we calculate the probability ofenergy outage at UAVs and signal-to-noise ratio (SNR) outage atground cellular users. The energy outage occurs when the UAVis unable to support the flight consumption and transmissionconsumption from its battery power and the harvested power.Due to the intricate distribution of the hybrid solar and windpower, we derive novel closed-form expressions for the momentgenerating function (MGF) of the harvested solar power andwind power. Then, we apply Gil-Pelaez inversion to evaluate theenergy outage at the UAV and signal-to-noise-ratio (SNR) outageat the ground users. We formulate the SNR outage minimizationproblem and obtain closed-form solutions for the transmit powerand flight time of the UAV. In addition, we derive novel closed-form expressions for the moments of the solar power and windpower and demonstrate their applications in computing novelperformance metrics considering the stochastic nature of theamount of harvested energy as well as energy arrival time. Theseperformance metrics include the probability of charging the UAVbattery within the flight time, average UAV battery chargingtime, probability of energy outage at UAVs, and the probabilityof eventual energy outage (i.e. the probability of energy outagein a finite duration of time) at UAVs. Numerical results validatethe analytical expressions and reveal interesting insights relatedto the optimal flight time and transmit power of the UAV as afunction of the harvested energy.

Index Terms—UAV-assisted wireless communications, energyharvesting, energy outage, SNR outage, battery charging time,eventual energy outage probability

I. INTRODUCTIONThe unmanned aerial vehicles (UAVs) will play an impor-

tant role in fulfilling the communication requirements of thenext generation wireless networks [1]. The primary benefits ofUAV networks include the operation in dangerous and disas-trous environments, on-demand relocation, improved coveragedue to higher line-of-sight (LOS) connections with the groundcellular users, and an extra-degree of freedom due to three-dimensional (3D) movements [2]. Despite various potentialbenefits, the energy consumption at UAVs is a primary bot-tleneck due to their mechanical and communication power

S. Sekander and E. Hossain are with the Department of Electrical andComputer Engineering, University of Manitoba, Canada (Emails: sekan-des@myumanitoba.ca, Ekram.Hossain@umanitoba.ca). H. Tabassum is withthe Lassonde School of Engineering, York University, Canada (Email:Hina.Tabassum@lassonde.yorku.ca). This work was supported by the NaturalSciences and Engineering Research Council of Canada (NSERC).

requirements. Energy consumption at UAVs can potentiallylimit the endurance time and communication performanceof UAVs. To enhance the sustainability and endurance timeof UAVs, energy harvesting from renewable energy sources(e.g. solar, wind, electromagnetic radiations) is a low-costalternative where a device can harvest from free energysources. However, renewable energy sources are intermittentand uncertain and thus the amount of energy harvested israndom [3]. For deployment and operation of the renewableenergy-powered UAV-assisted wireless communications sys-tems, it will be useful to mathematically model the dynamicsof energy harvested from a variety of renewable energy sourcesand analyze the communication performance as a function ofthe parameters such as time of the day, wind speed, etc.

A. Background Work

Several studies in the existing literature have dealt with theenergy efficiency of UAV based wireless communications. Theauthors in [4] investigated the power minimization problem ina UAV network such that the cell boundaries and locationsof UAVs can be optimized iteratively. Given a certain cellboundary, the locations of the UAVs were derived using afacility location framework. In [5], the authors discussed anenergy-efficient UAV deployment method to collect informa-tion from internet-of-things (IoT) devices in uplink. Usingthe K-means clustering approach, the ground devices arefirst clustered and then served by the UAV. The authors in[6] proposed an optimal placement algorithm for UAV BSssuch that the coverage to a ground BS and the networkenergy efficiency can be maximized. In [7], authors haveproposed a coverage model considering multi-UAV systemto achieve energy-efficient communication. They have solvedthe problem in two steps: coverage maximization and powercontrol and proved that both the problems fall in the categoryof exact potential games (EPG). Finally, they have devised analgorithm to do energy-efficient coverage deployment usingspatial adaptive play (MUECD-SAP) method.

Nonetheless, the aforementioned research works have notexplored the benefits of energy harvesting for UAV-assistedwireless communications. A preliminary study in [8] discussedthe concept of Energy Neutral Internet of Drones (enIoD) toachieve enhanced connectivity by overcoming energy limi-tations for longer endurance and continuous operation. Theauthors considered wireless power transfer to energize theUAVs and thus reducing the gap in harvested and consumedenergy. They have also conceptualized special UAVs whichare able to carry energy from one charging station to theother using the concept of opportunistic charging (OC). The

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authors in [9] proposed the radio energy harvesting at theUAV to improve the endurance time of the UAV. Dirty pa-per coding and information-theoretic uplink-downlink channelduality was considered to maximize the network throughput.In [10], UAV-based relaying was considered with energyharvesting capability at the UAV. The outage probability wasderived considering different urban environment parameters.Here, the harvested energy comes from the ground basestation (GBS). In [11], the authors investigated the resourceallocation problem for UAV-assisted networks, where the UAVprovides radio frequency energy to the device-to-device (D2D)pairs. A resource allocation problem to maximize the averagethroughput of the UAV-assisted D2D network was formulatedas a non-convex optimization problem considering the energycausality constraints. In [12], a UAV-enabled two-user wire-less power transfer system was considered where the UAVcharges multiple energy receivers for specific time period. Viaoptimization of the UAV trajectory subject to its maximumspeed constraints, the authors have minimized the transferredenergy to the intended receivers. Authors in [13] have solvedthe similar problem considering multi-user UAV system.

The aforementioned research works are focused on con-sidering energy harvesting through radio frequency sourcesinstead of renewable energy sources. The primary benefits ofrenewable energy harvesting over the distance-dependent wire-less powered networks include (i) the reduced consumptionof network resources (e.g., transmission channel, transmissiontime, and transmit power) and (ii) power transfer between theenergy transmitter and the energy receiver is independent ofthe distance between them.

Recently, in [14], the authors considered maximizing thesum throughput over a given period of time for a solar-powered UAV systems. A mixed-integer non-convex optimiza-tion problem was formulated considering energy harvesting,aerodynamic power consumption, finite energy storage system,and quality-of-service (QoS) requirements of the users. Theyhave used monotonic optimization to solve the non-convexproblem and attain the optimal 3D-trajectory along with re-source allocation. In [15], an energy management frameworkwas proposed for cellular heterogeneous networks (HetNets)supported by solar powered drones to jointly find the optimaltrips of the UAVs and the ground BSs that can be turnedoff to minimize the total energy consumption of the network.UAVs are able to charge their batteries either at a chargingstation or from harvested solar energy. Another research work[16] formulated a framework for energy management in UAV-assisted HetNets. The UAVs have the provision for solarenergy harvesting as well as energy charging from fixedcharging stations. They have studied the optimal deploymentof UAVs to minimize energy consumption in the network.

B. Motivation and Contributions

The aforementioned research works do not incorporate ac-curate models to characterize the renewable energy harvestedfrom solar and wind sources (as a function of specific solarand wind parameters) and therefore rely on assumptions (e.g.solar energy is modeled as a Gamma random variable in [16])

for tractability reasons. A plethora of research works considersolar energy harvesting in wireless sensor networks [17]–[19];however, to the best of our knowledge, there are no concretestatistical models for the solar or wind harvested energy orpower and their applications to communication networks areunknown. In addition, the aforementioned research works arefocused mainly on the numerical optimization or simulation-based studies. Subsequently, the impact of the network pa-rameters such as time of the day, velocity of the wind, solarradiance cannot be captured on the energy outage at UAVsand transmission outage at ground users.

The contributions of this paper are summarized as follows:• We develop novel statistical models for the amount of

harvested energy considering three renewable energy har-vesting scenarios, i.e. (i) solar power, (ii) wind power, and(iii) hybrid solar and wind power. Based on the derivedmodels, we calculate the probability of energy outage atthe UAV and signal-to-noise ratio (SNR) outage at groundcellular users.

• We derive the closed-form expressions for the probabilitydensity function (PDF) and cumulative density function(CDF) of the harvested power from the solar and windenergy sources. Due to the intricate distribution of thehybrid solar and wind power, we derive the closed-formexpressions for the moment generating function (MGF)of the harvested solar power and wind power. Then, weapplied Gil-Pelaez inversion to evaluate the energy outageat the UAV and SNR outage at users.

• We formulate the SNR outage minimization problem andobtain closed-form solutions for the transmit power andflight time of the UAV.

• We derive the closed-form expressions for the momentsof the harvested wind power and solar power and demon-strate their applications in computing new performancemetrics considering the scenario when both the amountof energy as well as the energy arrival time is stochastic.That is, not only the amount of energy is random butalso the time of energy arrival is random. These per-formance metrics include the probability of harvestingenergy within the flight duration, average battery chargingtime, probability of energy outage at the UAV, andthe probability of eventual energy outage (which is theprobability of energy outage in a finite duration of time)at the UAV are analyzed.

• Numerical results validate the analytical expressions byproviding a comparison with the Monte-Carlo simulationsand exhibit interesting insights related to the optimalflight time and transmit power of the UAV as a functionof the harvested energy.

Notations: Γ(a) =∫∞

0xa−1e−xdx represents the Gamma

function, Γu(a; b) =∫∞bxa−1e−xdx denotes the upper in-

complete Gamma function, Γl(a; b) =∫ b

0xa−1e−xdx denotes

the lower incomplete Gamma function and Γ(a; b1; b2) =Γu(a; b1) − Γu(a; b2) =

∫ b2b1xa−1e−xdx denotes the gener-

alized Gamma function [20]. 2F1[·, ·, ·, ·] denotes the Gauss’shypergeometric function. P(A) denotes the probability ofevent A. f(·), F (·), and L(·) denote the probability density

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TABLE I: Mathematical notations

Notation Description Notation DescriptionTf ;Tb − Tf Flight time; Hover & transmission duration Pb;Pd Back-up battery power; Downlink transmit power

I(t) Radiation intensity Kc Threshold radiation intensityId(t) Deterministic fundamental intensity ∆I(t) Stochastic attenuationηc PV system efficiency c; k Scale & shape parameter of Weibull random variable

Vci;Vr;Vco Cut-in; rated; cut-off wind velocity A; ρ Rotor area; Air densitynp;rp Number of propellers; Propeller radius P ;Pw Harvested solar power; Harvested wind powerEc Energy consumed from battery Et Energy required for transmissionEf Energy required for flight hd UAV altitudeλ Energy arrival rate Ai Inter-arrival times of the energy packets

U(t) Accumulated energy in the battery Xi Energy packet sizeu0 Initial battery energy τ Battery recharge time

φ(u0) Eventual energy outage probability r∗ Adjustment coefficient

function (PDF), cumulative distribution function (CDF), andLaplace Transform, respectively. Finally, U(·), δ(·), and E[·]denote the unit step function, the Dirac-delta function, and theexpectation operator, respectively. erf(x) is the error function

expressed as erf(x) = 2√π

x∫0

e−t2

dt and erfc(x) = 1− erf(x)denotes the complementary error function [20, 8.25/4]. A listof important variables is presented in Table 1.

The rest of the paper is organized as follows. The systemmodel and assumptions are stated in Section II. The energyoutage and the signal-to-noise ratio (SNR) outage probabilitiesare evaluated in Sections III and IV, respectively. Section Vanalyzes the moments of the harvested power and presentsseveral applications of these moments including evaluation ofprobability of battery charging, average battery charging time,and eventual energy outage probability. Section VI presentsthe numerical results before the paper is concluded in SectionVII.

II. SYSTEM MODEL AND ASSUMPTIONS

In this section, we describe the network model, the air-to-ground (AtG) channel propagation model, the UAV energyconsumption model, and the harvested energy models for solarand wind energy sources.

A. Network Model

We consider a UAV-enabled with solar and wind energyharvesting capability that serves ground cellular users onorthogonal transmission channels. The users are distributeduniformly in a circular region of area A = πR2, where Rrepresents the radius of the considered circular region. TheUAV harvests energy from the solar and/or wind energy de-pending on the energy harvesting model. The UAV operates intwo states: (i) traveling to the desired location for transmissionwhile harvesting energy, (ii) hovering and transmitting to thecellular users or traveling back to the charging station (whichis located at the origin) to charge itself if needed. A durationof Tb is considered in which the UAV travels for durationTf and hovers at the destination for a duration Tb − Tfto perform downlink transmission given there is no energyoutage. Since the maximum distance a UAV can travel is Rm

Charging

Station

C

Solar Energy

Wind Energy

Harvesting Transmission

Backup

Battery

Fig. 1: System model.

in a straight line trajectory from the charging station, Tb isset as Tb = Rm/vd. The UAV travels with the speed vd (inm/sec) and is equipped with the fixed back-up battery powerPb to support the UAV flight (in case if the harvested poweris not enough). The UAV performs data transmission to thecellular users in the downlink using transmit power Pd for thetime duration Tb − Tf given that there is no energy outage.

B. Air-to-Ground (AtG) Channel Model

The RF signals generated by the UAV first travels throughthe free space until they reach the man-made urban environ-ment, where additional losses (referred to as excessive path-loss) occur due to foliage and/or urban environment. Theexcessive path-loss is random in nature and cannot be charac-terized by a well-known distribution. As such, the mean valueof excessive path-loss η� obtained from empirical distributionfitting is typically considered. The RF transmissions from agiven UAV fall into three propagation groups, Line-of-Sight(LOS) propagation, non-LOS (NLOS) propagation via strongreflection and refraction, and a very limited contribution (lessthan 3% as reported in [21]) by the deep fading resultingfrom consecutive reflections and diffraction. As such, the thirdgroup has been discarded in most of the relevant researchstudies. Since the excessive path-loss depends largely onthe first two propagation groups, η� can be considered as aconstant that can be obtained by averaging all samples in

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a certain propagation group. The values of η� are listed forvarious frequencies and urban environments in [21, Table II].

The AtG path-loss can thus be defined as follows [22]:

PL� = FSPL + η�, (1)

where � ∈ {LOS,NLOS} and free space path-loss (FSPL) canbe evaluated using the standard Friis equation, i.e. FSPL =20log10

(4πfcdc

), where fc is the carrier frequency (Hz), c

is the speed of light (m/s), and d is the distance between theUAV and the receiving user. The probability of having LOSfor a user i depends on the altitude hd of the serving UAV andthe horizontal distance between the UAV and ith user, whichis ri =

√(xD − xi)2 + (yD − yi)2. The ith user is located at

(xi, yi, 0) and the UAV is located at (xD, yD, hd). The LOSprobability is thus given by:

PLOS(hd, ri) =1

1 + aexp(−b(

arctan(hdri

)− a)) , (2)

where arctan(hdri ) is the elevation angle between the UAV andthe served user (in degrees). Here a and b are constant valuesthat depend on the choice of urban environment (high-riseurban, dense urban, sub-urban, urban). They are also known asS-curve parameters as they are obtained by approximating theLOS probability (given by International TelecommunicationUnion (ITU-R) [22], [23]) with a simple modified Sigmoidfunction (S-curve). Subsequently, the approximate LOS prob-ability can be given for various urban environments whilecapturing the buildings’ heights distribution, mean number ofman made structures, and percentage of the built-up land area,of the considered urban environment. The NLOS probabilityis given as PNLOS(hd, ri) = 1− PLOS(hd, ri).

The path-loss expression can then be written as [24]:

L(hd, ri) = 20log

(√h2d + r

2i

)+APLOS(hd, ri) +B, (3)

where A = ηLOS−ηNLOS, B = 20log(

4πfcc

)+ηNLOS, ηLOS

and ηNLOS (in dB) are, respectively, the losses correspondingto the LOS and non-LOS reception depending on the envi-ronment. The considered AtG propagation model can capturevarious environments (such as high-rise urban, dense urban,sub-urban, urban) [4], [5], [22], [24], [25].

C. Harvested Power Model

1) Solar Power Model: The output power from the PhotoVoltaic (PV) system depends on the solar radiation intensity,the solar cell temperature, and the PV system efficiency. Theoutput power of the PV system P at time t can therefore bemodeled as follows [26]:

P =

{ηcKcI(t)2, 0 < I(t) < Kc

ηcI(t), I(t) ≥ Kc,

where I(t) denotes the radiation intensity and Kc is a thresh-old for the radiation intensity beyond which the efficiency ηccan be approximated as a constant. Note that I(t) can be cal-culated as a sum of deterministic fundamental intensity Id(t)and stochastic attenuation ∆I(t) due to weather effects as well

as clouds occlusion, i.e. I(t) = Id(t)+∆I(t). Generally, Id(t)depends on the time of a day and the months/seasons of a yearand can be found from the following equation [26]:

Id(t) =

{Imax(−1/36t2 + 2/3t− 3), if 6 ≤ t < 180, otherwise

.

The distribution of ∆I follows a standard normal distribution,i.e. ∆I ∼ N(0, 1). Then the distribution of I(t) is a normaldistribution with shifted mean as expressed below:

fI(I) =1√2π

exp

((I − Id)2

2

). (4)

2) Wind Power Model: To estimate the wind energy poten-tial, the velocity of wind is a primary variable and is typicallymodeled by the Weibull distribution as shown below [27]:

fV (v) =k

c

(vc

)k−1e−(v/c)

k

, (5)

where v is the speed of wind, c and k denote the scale andshape parameter of the Weibull random variable, respectively.Physically speaking, c indicates the wind strength of theconsidered location and k is the peak value of the winddistribution. The CDF of Weibull variable is given by

FV (v) = 1− exp[−(v/c)k], (6)

where k = (σµ )−1.086 and c = µΓ(1+1/k) .

The energy produced by a wind turbine generator can beobtained by means of its power curve, where the relationshipbetween the wind speed and the delivered power can beestablished as shown below [27]:

Pw =

q(v), Vci < v ≤ VrPr, Vr < v ≤ Vco0, elsewhere

,

where Pw and v denote the output power of the wind turbineand the wind velocity, respectively. Vci, Vr and Vco representthe cut-in wind velocity, rated wind velocity, and cut-off windvelocity, respectively. The non-linear part of the power curvecan be defined as follows [28]:

q(v) =1

2ρACv3, (7)

where C is a constant equivalent to the power coefficient, Ais the rotor area, and ρ denotes the air density.

D. UAV Energy Consumption Model

Based on the momentum disk theory and blade elementtheory, the power consumption model for the UAV in hoverstate can be defined. This model considers the power consump-tion due to thrust T which is defined as a force to move anaircraft through the air. During hover, it can be assumed thatthe thrust is approximately the same as the total weight forceW in Newton, i.e. T = W . In the case of a hovering aircraft,reaction force is approximately equal to the gravitational force.As a result, we can assume W = mg, where m is theUAV mass (in kg) and g denotes earth gravity (in m/s2). The

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mechanical power Phov of a system that exerts a force T onan object moving with velocity vd can be defined as [29]:

Phov = Tvd = mgvd =

√(mg)3

2πr2pnpρ, (8)

where A denotes the rotor disk area and ρ is air density inkg/m3, A = πr2pnp, np and rp denote the number of propellersand propeller radius, respectively.

During transmission period Tb− Tf , the UAV has to spendthe power for hovering as well as for transmitting to the desiredusers. On the other hand, during flight the UAV needs to hoverand spend some energy to keep the UAV active. The energyconsumption at UAVs can thus be modeled as follows:• Energy consumed during transmission (while hovering),Et = (Phov + Pd)(Tb − Tf ).

• Energy consumed during return flight (no transmission),Ef = (Phov + γd)Tf .

where γd denotes the activation energy, Tf denotes the flighttime, and Pd denotes the transmit power of the UAV.

E. Actual Energy Consumption from the UAV Battery

If the harvested solar energy Eh = PTf or harvested windenergy Eh = PwTf is high enough to support the Ef and Et,no energy needs to be consumed from the battery. Therefore,the net energy consumption (or energy cost) from the batteryEc is zero. Another scenario can happen where the harvestedenergy levels are not sufficient. However, after combining withthe UAV backup battery, the transmission and return flight canstill be supported. In this case, after spending all the harvestedenergy, we need to consume the deficit energy from the UAVbattery. The net energy consumption becomes Ec = Et+Ef−Eh, where Et is the energy required for transmission, Ef isthe energy required for the return flight, and Eh denotes theharvested energy. Finally, if the harvested energy along withthe battery energy cannot support transmission, the UAV willnot transmit and fly back to the charging station and consumeonly the energy for return flight Ef . Finally, there could bea scenario when the harvested energy and battery energy areinsufficient to support the return flight, however, we considerthat the back-up battery is designed to support the emergencyreturn flight. The net energy consumption is as follows:

Ec =

0, Eh > Et + Ef

Et + Ef − Eh, Eh + Eb > Et + EfEf − Eh, Eh + Eb > Ef

. (9)

III. CHARACTERIZATION OF THE ENERGY OUTAGE

In this section, we characterize the energy outage probabilityof the UAV considering (i) solar power, (ii) wind power, and(iii) hybrid solar and wind power. We derive the PDF andCDF of the harvested energy from the solar and wind powersources and then characterize their respective energy outages.For hybrid solar and wind power, we derive the LaplaceTransforms of the solar and wind harvested power and useGil-Pelaez inversion to characterize the energy outage.

The UAV is declared to be in energy outage if the batteryenergy along with the harvested energy is not enough to sup-port the energy consumption of the return flight and the datatransmission. Mathematically, the energy outage probabilityEout can be defined as follows:

Eout = P[PbTb +HTf < Et + Ef ],

= P[H <

Et + Ef − PbTbTf

],

= P [H < θ] , (10)

where θ = Et+Ef−PbTbTf and H is the harvested power andH = P if the source of energy is solar, H = Pw if the sourceof energy is wind, and H = P + Pw in case of harvestingfrom both solar and wind energy.

A. Harvested Power - Solar and Wind

The harvested power from solar energy is a function of thetime of the day and months of the year, solar radiance, andefficiency of the photo-voltaic system. The distribution of theharvested solar power P (t) can be written as in the following:

Theorem 1 (Distribution of the Harvested Solar Power). ThePDF of the harvested solar power can be given as follows:

fP (p) =

12√

Kcηcp

fI

(√Kcpηc

), p < ηcKc

1ηcfI(p/ηc), otherwise.

Proof. See Appendix A. �

On the other hand, the distribution of the harvested powerfrom wind energy can be derived as follows:

Theorem 2 (Distribution of the Harvested Wind Power).Letting a = 12ρAC, the PDF of the harvested wind powercan be derived as follows:

fPW (pw) =

kpwk3−1exp

(− p

k3w

ak3 ck

)3cka

k3

U(aV 3ci < pw < aV 3r )+

δ(pw − Pr)(F (Vco)− F (Vr)) + δ(pw)(F (Vci) + 1− F (Vco)).

Essentially, the last two terms represent the case when theharvested power becomes equal to the maximum constantrated power and the case when the harvested power is zero.

Proof. See Appendix B. �

B. Energy Outage

1) Solar Power: In order to characterize the probability ofenergy outage, first we derive the CDF FP (p) =

∫ p0fP (t)dt

of the harvested solar power using Theorem 1 as follows:

FP (p) =

12

(erf[Id√

2

]− erf

[Id−

√Kcpηc√

2

]), p < ηcKc

12

(erf[Id−Kc√

2

]+ erf

[p−Idηc√

2ηc

]), p > ηcKc

.

(11)

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Using the CDF of the harvested power in (11) and thedefinition of energy outage event given in (20), we can evaluatethe energy outage as follows:

Eout = FP (θ) . (12)

2) Wind Power: Similarly, the probability of energy outagewith wind harvesting can be derived by first determining theCDF of the harvested wind power using Theorem 2 as:

FPw(pw) =

[γ

(1,

pk3w

ak3 ck

)− γ

(1,

Vk3

ci

ak3 ck

)]+

U(pw)[F (Vci) + 1− F (Vco)] + U(pw − Pr)[F (Vco)− F (Vr)].(13)

Using the CDF of the harvested power in (13) and thedefinition of energy outage event given in (20), we can evaluatethe energy outage as Eout = FPw (θ).

3) Hybrid Solar-Wind Power: Evaluating the PDF of thesum of the harvested solar and wind power is not tractabledue to the convolution of the distributions of the solar andwind powers. Therefore, we utilize an MGF-based approachto evaluate the energy outage. That is, we propose to use Gil-Pelaez inversion theorem to characterize energy outage. Notethat the solar and wind powers are independent random vari-ables. Therefore, the Laplace Transform of the total harvestedpower can be given by the product of the Laplace Transformsof the solar power and wind power. For this, we first derivethe Laplace Transforms (or MGF) of the solar power and windpower and then determine the energy outage using Gil-Pelaezinversion theorem.

Theorem 3 (Laplace Transform of Solar Power P andWind Power Pw). The Laplace transform MP (s) =∫∞

0e−sP fP (p)dp of the harvested power from solar energy

can be derived as follows:

LP (s) =1

2esη(

sη2 −Id)erfc

[Kc − Id + sηc√

2

]+

e− Id

2

2 +I2d

2+4sηcKc

(erf

[Id√

2+ 4sηcKc

]+ erf

[Id−Kc−2sηc√

2+ 4sηcKc

])2√

2s+Kc/ηc.

(14)

The Laplace transform of the harvested power from windenergy can be given as follows:

LPw(s) =∞∑n=0

(−sac3)n

n!

(3n

k

)!.

Proof. See Appendices C and D. �

The Laplace Transform of the total power harvested fromsolar and wind sources can be given as follows:

LT (s) = LPw(s)LP (s). (15)

The characteristic function φT (s) of the total power harvestedfrom solar and wind sources can then be derived by substitut-ing s = jω in the expressions derived in Theorem 3 and then

substituting them into (15). The energy outage Eout can thenbe given by applying the Gil-Pelaez inversion theorem as:

Eout =1

2+

1

π

∫ ∞0

Im[φT (jω))e−jθω]

ωdω, (16)

where Im[·] represents the imaginary operator.

IV. SNR OUTAGE ANALYSIS

When the signal-to-noise ratio (SNR) of a ground cellularuser falls below a certain target threshold SNRth, the user issaid to be in SNR outage. Mathematically, the SNR outagecan be defined as follows:

Sout = P(SNR ≤ SNRth), (17)

where SNRth is the SNR threshold for a specific ground userso that the user can achieve minimum data rate as

Rth =Tb − TfTb

log2(1 + SNRth). (18)

Subsequently, SNRth can be calculated as SNRth = 2TbRthTb−Tf −

1. From the path-loss model in Section II, SNR is given as:

SNR =L(hd, ri)Pdχ

N0, (19)

where χ denotes Gamma distributed channel fading. Weconsider Nakagami-m fading since it provides a generalizedmodel that mimics various fading environments [30]. Thus,conditioned on the distance of the ground user from the UAV,we can derive the SNR outage as follows:

Sout(r) = Er [P (SNR ≤ SNRth)] ,

= Er[P(L(hd, ri)Pdχ

N0≤ SNRth

)],

= Er[P(χ ≤ SNRthN0

L(hd, ri)Pd

)],

= Er

[γ(m, SNRthN0ΘL(hd,ri)Pd )

Γ(m)

],

(20)

where m and Θ are shape and scale parameters of Gammadistribution, respectively. For Rayleigh fading channels, wecan simplify the SNR outage probability as follows:

Sout(r) = 1− e− SNRthN0L(hd,ri)Pd , (21)

where L(hd, ri) (in dB) can be expressed as follows:

L(hd, ri) =c2

y(4πfc)2(h2d + r2i )

(yx

)PLOS(hd,ri), (22)

and x = 10ηLOS

10 , y = 10ηNLOS

10 , and PLOS(hd, ri) =(1 + aexp(−b(arctan(hdri )− a)))

−1. Here arctan(hdri

)is the

elevation angle between the UAV and the served user (indegrees). Also, a and b are constant values based on the choiceof the urban environment.

We now consider the problem of SNR outage minimizationto optimize the transmit power of the UAV Pd. The problem

7

can be formulated as follows:

minimizePd

γ(m, SNRthN0ΘL(hd,ri)Pd )

Γ(m),

subject to Pd ≤(PbTb +HTf )− (Phov + γd)Tf

Tb − Tf,

Tf < Tb.

The first constraint imposes a restriction on the transmit powerof the UAV. We know that, the total available UAV energy isthe sum of battery energy and harvested energy in time Tf , i.e.PbTb + HTf . And the total available UAV energy should begreater than the energy required during the flight as well as thetransmission time (Pd +Phov)(Tb− Tf ) + (Phov + γd)Tf . Asa result, we have an upper bound on the transmit power of theUAV. The second constraint denotes the fact that a UAV cannot have a flight time which exceeds the observation period Tb.Since the lower incomplete Gamma function is monotonicallydecreasing function with respect to increasing Pd, the optimalsolution for Pd will lie at the boundary, i.e.

P ∗d =(PbTb +HTf )− (Phov + γd)Tf

Tb − Tf. (23)

The first constraint imposes a restriction on the transmit powerof the UAV. We know that, the total available UAV energy isthe sum of battery energy and harvested energy in time Tf , i.e.PbTb + HTf . And the total available UAV energy should begreater than the energy required during the flight as well as thetransmission time (Pd +Phov)(Tb− Tf ) + (Phov + γd)Tf . Asa result, we have an upper bound on the transmit power of theUAV. The second constraint denotes the fact that a UAV cannot have a flight time which exceeds the observation period Tb.Since the lower incomplete Gamma function is monotonicallydecreasing function with respect to increasing Pd, the optimalsolution for Pd will lie at the boundary, i.e.

P ∗d =(PbTb +HTf )− (Phov + γd)Tf

Tb − Tf. (24)

We now consider the problem of SNR outage minimizationto optimize the flight time of the UAV Tf . The problem canbe formulated as follows:

minimizeTf

γ

(m, (2

TbRthTb−Tf −1)N0

ΘL(hd,ri)Pd

)Γ(m)

,

subject to Pd ≤(PbTb +HTf )− (Phov + γd)Tf

Tb − Tf,

Tf < Tb.

Since the lower incomplete Gamma function is monotonicallyincreasing with respect to increasing Tf , the optimal solutionfor Tf will be based upon the minimum value of the Tfobtained from constraints, i.e.

T ∗f = min

(Tb,

(Pd − Pb)TbH − (Phov + γd) + Pd

). (25)

The distance between the origin and an arbitrary user whoselocation is uniformly distributed within a circular region of

radius R has the PDF given by:

fR(r) =2r

R2, (26)

where 0 ≤ r ≤ R. By averaging over the distribution of r, theSNR outage of a given user can thus be calculated as follows:

Sout =

R∫0

Sout(r)2r

R2dr, (27)

in which Sout(r) is given in (20) and (21) for Gamma fad-ing and Rayleigh fading channels, respectively. For Rayleighfading channels, the SNR outage can be computed as follows:

= 1−R∫

0

e−C(h2d+r

2)zPLOS(hd,r) 2r

R2dr, (28)

where C = SNRthN0y(4πfc)2

c2Pdand z = xy . The integral can be

computed numerically using standard mathematical softwarepackages such as Mathematica, Matlab and Maple.

V. MOMENTS OF THE HARVESTED POWER AND THEIRAPPLICATIONS

In this section, we first derive the moments of the har-vested solar and wind power. Based on these moments, wecharacterize important metrics such as eventual energy outageprobability, probability of charging the UAV battery in a finitetime, average battery charging time, probability of eventualenergy outage, and energy outage probability considering thatthe energy arrival follows a discrete stochastic process whereboth the amount of energy received at a certain time as wellas the inter-arrival times are random.

A. Moments of the Harvested Solar Power

The i-th moment of the harvested solar power can be cal-

culated by applying the definition E[P i] =∞∫0

pifP (p)dp. The

closed-form expressions for the i-th moment of distributionof harvested power from solar energy Mi can be derived asfollows:

E[P i] =

ηcKc∫0

pi−0.5e−

(√Kcpηc−Id

)22

2√

2πηc/Kc+

∞∫ηcKc

pie−( pηc −Id)

2

2

ηc√

2π,

(a)=

Kc−Id∫−Id

(x+ Id)2ie−

x2

2 dx(Kcηc

)i√2π︸ ︷︷ ︸

I1

+

∞∫Kc−Id

(y + Id)iηice

− y2

2 dy

√2π︸ ︷︷ ︸I2

,

(29)

where (a) results from substituting√

Kcpηc− Id = x and

pηc− Id = y in the first and second integral, respectively.

8

The integral in I1 can be solved in closed-form as follows:

I1 =

η4ce− I

2d2

(2m′ − 2eIdKc−

K2c2 A+ e

I2d2 d√

2πν

)K4c 2√

2π,

where m′ = Id(279 + 185I2d + 27I4d + I

6d), a = (7 + I

2d)(15 +

18I2d + I4d), b = Id(87 + 22I

2d + I

4d), c = (35 + 18I

2d + I

4d),

d = 105 + 420I2d + 210I4d + 28I

6d + I

8d , A = m

′ + aKc +bK2c + cK

3c + Id(13 + I

2d)K

4c + (7 + I

2d)K

5c + IdK

6c +K

7c .

Similarly, the integral in I2 can be solved as follows:

I2 =η2c√2π

(Id +Kc)e− (Id−Kc)

2

2 +η2c2

(1 + I2d)A′,

where A′ = 1 + erf(

Id−Kc√2Sign(Id−Kc)

)sign[Id −Kc]3.

Subsequently, the first moment can be simplified in theclosed-form as in the following Lemma.

Corollary 1 (Mean Harvested Solar Power). The closed-formexpression for the first moment of harvested solar power E[P ]can be obtained by substituting i = 1 in (29) as shown below:

M1 =1

2ηcIdτ −

Idηc(e− (Id−Kc)

2

2 − e−I2d2 )√

2πKc− 1

2

ηcKc

(1 + I2d)ν,

where ν = τ − 1− erf(Id√

2

)and τ = 1 + erf

(Id−Kc√

2

).

Corollary 2 (Second Moment of Harvested Solar Power). Thesecond moment of harvested solar power can be obtained inclosed-form as follows:

M2 =η2ce− I

2d2 Id(5 + I

2d)

K2c√

2π+η2c (1 + I

2d)τ

2− η

2c (3 + 6Id + I

4d)ν

2K2c

+η2c√2πe−

(Id−Kc)2

2

(Id −

I3dK2c− 3Kc− I

2d

Kc− 5IdK2c− 1),

where ν = τ − 1− erf(Id√

2

)and τ = 1 + erf

(Id−Kc√

2

).

B. Moments of the Harvested Wind Power

The i-th moment of the harvested wind power can be calcu-

lated by applying the definition E[P iw] =∞∫0

piwfPw(pw)dpw.

The closed-form expressions for the i-th moment of winddistribution Mwi can be derived as follows:

Mwi =k

3ak/3ck

aVr3∫

aVci3

xk3 +i−1exp

(− x

k3

ak3 ck

)dx.

This can be solved using the identityu∫0

xme−βxn

dx =

γ(v,βun)nβv , v =

m+1n [20]. where u > 0, Re v > 0, Re

n > 0 and Re β > 0. Hence, we can solve Mwi consideringm = k3 + i−1, n =

k3 and β = a

− k3 c−k. Thus v = 1+ 3ik +1.

We also know thatb∫a

f(x)dx =b′∫0

f(x)dx −a′∫0

f(x)dx. Here

b′ = aVr3 and a′ = aVci3. Finally after simplification Mwi

can be expressed as:

Mwi = aic3i

[γ

(1 +

3i

k,

(Vrc

)k)− γ

(1 +

3i

k,

(Vcic

)k)].

(30)Subsequently, the first and second moments can be simpli-

fied in the closed-form as in the following Lemma.

Corollary 3 (First and Second Moments of Harvested WindPower). The first and second moment of harvested wind powercan be obtained in closed-form, respectively, as follows:

Mw1 = ac3

[γ

(1 +

3

k,

(Vrc

)k)− γ

(1 +

3

k,

(Vcic

)k)],

Mw2 = a2c6

[γ

(1 +

6

k,

(Vrc

)k)− γ

(1 +

6

k,

(Vcic

)k)].

C. Applications of the Moments of the Harvested Power

To compute the probability of charging the UAV batteryin a finite time, average battery charging time, eventualenergy outage probability, and average energy outage, wemake the following assumptions. The energy arrivals occur inbursts (i.e. as “energy packets”) and the size of the energypackets be X1, X2, · · · , Xi arriving at time t1, t2, · · · , ti,respectively. Here, packet size X is a random variable whichfollows the distribution of solar or wind harvested power. LetA1, A2, · · · , Ai denote the inter-arrival times of the energypackets such that tn = A0+A1+· · ·+An. Due to the memory-less property of the Poisson process, each Ai is exponentiallydistributed with parameter λ. The mean and variance of A arefinite. Since X and A are independent random variables, theaccumulated energy in the battery at any time t is given as:

U(t) =

NA(t)∑i=1

Xi, (31)

where NA(t) denotes the number of packets arrived in time t.To connect the above assumptions with the system model as-

sumed before, note that, in the previous sections, the harvestedpower H has been a random variable whose value variesacross the flight intervals and whose distribution is given usingTheorem 1 and Theorem 2 for solar power (H = P ) andwind power (H = Pw), respectively. Here, in this subsection,we consider that the harvested power H is equivalent to theharvested energy per unit time, i.e. X = Ht, where X isthe energy packet size. Assuming t = 1 for simplicity, theenergy packet size is a random variable whose PDF can begiven using Theorem 1 and Theorem 2 for solar and wind,respectively. For other values of t, the distribution of X can beobtained by scaling the distribution of H using single variabletransformation. Furthermore, the energy packet arrival followsa Poisson process, i.e. the inter-arrival time Ai is exponentiallydistributed with arrival rate λ.

1) Probability of Charging UAV Battery During FlightTime: The time needed to charge the battery up to a desiredlevel, u0 > 0 can be formulated as a first passage time problemand can be derived as follows: τ(u0) = inft{t : U(t) > u0},where τ(u0) denotes the battery recharge time. Since the

9

charging process is a pure jump process for an ideal battery, theevent U(t) > u0 is equivalent to the event τ(u0) < t whichresults in P(U(t) > u0) = P(τ(u0) < t). Now to find thedistribution of U(t), we condition on the number of packetsand represent the sum of random variables Xi by convolvingtheir distributions. Finally, we average over the distribution ofthe number of packets arrived in time t to obtain:

P(U(t) ≤ u0) = e−λt∞∑n=0

(λt)n

n!F

(n)X (u0), (32)

where F (n)X (x) denotes the n-fold convolution of FX(x) whereF

(i)X (x) =

∫F

(i−1)X (x − t)dF (t). F

(0)X (x) is the unit step

function at the origin. The probability of UAV battery chargingin time Tf can thus be given as follows [31]:

P(τ(u0) ≤ Tf ) = 1− e−λTf∞∑n=0

(λTf )n

n!F

(n)X (u0), (33)

where F (n)X (u0) can be approximated using central limittheorem as Φ

(u0−nX̄σX√n

)[31], where Φ(.) denotes the CDF

of the standard normal distribution.2) Average Charging Time of UAV Battery: Subsequently,

the expected charging time for UAV battery can be derived as:

E[τ(u0)] =

Tf∫0

P(τ(u0) ≥ t)dt,

=

∞∑n=0

Φ

(u0 − nX̄σX√n

)1

n!

Tf∫0

(λt)ne−λtdt.

Finally, the average charging time can be given as:

E[τ(u0)] =∞∑n=0

Φ

(u0 − nX̄σX√n

)1

n!

Tf∫0

(λt)ne−λtdt,

=Γl(1 + n, Tfλ)

n!λ

∞∑n=0

Φ

(u0 − nX̄σX√n

),

where X̄ and σX denote the mean and standard deviation ofthe solar and wind energy, u0 is the required level of energyup to which we need to charge the battery, τ(u0) is the timerequired to reach the level of energy u, λ denotes the energyarrival rate, and n is the total number of energy packets arrived.

3) Eventual Energy Outage: The eventual energy outageφ(u0) is defined as the energy outage that occurs within afinite amount of time while the initial battery energy is u0 andthe power consumed is either Phov + γd during the flight andPhov + Pd during the hover time. If the energy packet arrivalcan be modeled as a Poisson process, the energy surplus atany time t can be modeled as follows:

U1(t) =

u0 − (Phov + γd)t+

∑N(t)i=1 Xi,

for t < Tfu0 − Phovt− Pd(t− Tf )− γdTf +

∑N(Tf )i=1 Xi,

for t > Tf

.

Here u0 ≥ 0 denotes the initial battery energy and N(t) isthe total number of energy packets harvested within time t.The energy outage that occurs within finite amount of time

making the battery energy to zero is known as eventual energyoutage φ(u0). The first time when battery energy goes to zerois denoted as first time to outage τ . τ = inf{t > 0 : U1(t) ≤0, U1(0) = u0}. If τ is less than Tf or if Tf ≤ τ ≤ Tb, wesay that eventual outage has occurred. Thus, the probabilityof eventual outage within duration Tf and Tb − Tf can bemodeled, respectively, as follows:

φ(u0) = P (τ < Tf |u0, Phov + γd), (34)

φ(u0) = P (Tf < τ < Tb|u0+PdTf−γdTf , Phov+Pd). (35)

The eventual energy outage probability φ(u0) within durationTf and Tb− Tf can be given using the approach described in[31], respectively, as follows:

φ(u0) =

(1− r

∗(Phov + γd)

λ

)e−r

∗u0 , (36)

φ(u0) =

(1− r

∗(Phov + Pd)

λ

)e−r

∗(u0+PdTf−γdTf ), (37)

where r∗ denotes the adjustment coefficient. The value r∗ 6= 0is said to be the adjustment coefficient of X if E[exp(r∗X)] =∫er∗xdFX = 1. The adjustment coefficient must satisfy

the condition E[exp(r∗X)] =MX(r∗). Considering Poissonenergy arrival, we have:

1− pr∗

λ=MX(−r∗),

where p = Phov +γd during the flight time and p = Phov +Pdduring the transmission time. A simple expression for r∗

can be obtained by making a formal power expansion ofMX(−r) in terms of moments of X up to second order termasMX(−r) ≈ 1−X̄r+ X̄

2

2 r2. As such, a simple generalized

approximation of r∗ can be given as [31]:

r∗ ≈ 2pλX̄2

(λX̄

p− 1), (38)

where X̄ is given in Corollary 1 and Corollary 3 for solarand wind power, respectively.

4) Energy Outage: From Proposition 10 of [31], we knowthat for any harvest store consume system, if there exists a non-negative probability of eventual energy outage, the averageenergy outage can be calculated as follows:

Eout =

{1− λX̄Phov+γd , t < Tf1− λX̄Phov+Pd , t > Tf

. (39)

VI. NUMERICAL RESULTS AND DISCUSSIONSIn this section, we first describe the simulation parameters

and then present selected numerical and simulation results. Wevalidate the accuracy of the derived expressions for the energyoutage probability of UAV and the SNR outage of a grounduser. Also, we extract design insights related to the impactof key UAV parameters such as flight time, transmit power,altitude, and time of the day on the energy outage of UAV andSNR outage of the ground user.

A. Simulation ParametersUnless stated otherwise, we use the simulation parameters

as listed herein. We consider the maximum sunlight intensity

10

Fig. 2: Distribution of the harvested power from solar energyas a function of daytime.

Imax = 2000. The threshold for radiation intensity I isKc = 150. The efficiency of the solar cell beyond Kc isηc = 0.02. The random attenuation amount ∆I follows anormal distribution with mean µ = 0 and variance σ2 = 1. Themass of a UAV is taken as 0.75 kg. The number of propellersand propeller radius are, respectively, 4 and 0.2m. The airdensity is ρ = 1.225 kg/m3 and g = 9.8 m/s2 is the gravityof the earth. The activation energy of the UAV is γd = 2.9 W.We consider the initial battery power Pb = 2 + Phov + γd.Also, fc = 2.5 × 109 is the carrier frequency (Hz) andc = 3× 108 m/s is the speed of light. The UAV altitude is setat h = 200 m. The UAV power is assumed to be Pd = 40 W.The UAV flies at a speed v = 10 m/s. The block duration isset to Tb = Rmax/vd, where Rmax = 200 m. The thresholdrate requirement for a cellular user is set to Rth = 2 bps. TheS-curve parameters for the UAV are a = 12.08 and b = 0.11.Also, ηLOS = 1.6 and ηNLOS = 23 (in dB) are, respectively,the losses corresponding to the LOS and non-LOS receptiondepending on the environment. The flight time Tf is set to0.2Tb, unless specified otherwise.

B. Results and Discussions

1) Distribution of the Harvested Power: Fig. 2 depicts thelevel of the harvested power from solar energy at differenttimes of the day. The Monte-Carlo simulation results matchperfectly with the analytical results as we derive an exactexpression for the harvested solar power. It can be seen thatthe harvested power is maximum at noon. Along similar lines,Fig. 3 depicts the level of the harvested power from thewind energy. Since the PDF of wind power is a mixture ofcontinuous and discrete random variables, we can observe twodiscrete probabilities when harvested wind power is zero andwhen the wind power becomes equal to the rated power.

2) Impact of the Time of the Day: In Fig. 4, we quantifythe energy outage probability of the UAV and the SNR outageprobability of the user as a function of time of the day. Our

Fig. 3: Mixed probability mass/density function of the har-vested power from wind energy.

7 8 9 10 11 12 13 14 15 16 17

Time of the Day (Hours)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Out

age

Pro

babi

lity

Energy Outage - SimulationsSNR Outage - SimulationsSNR Outage - AnalysisEnergy Outage - Analysis

Pd = 10 W

Pd = 15 W

Fig. 4: SNR outage of the user and energy outage at the UAVas a function of the harvesting time.

simulations match well with the derived expressions. Sincethe harvested solar energy varies significantly with the time ofharvesting, the energy outage of the UAV and the SNR outageof the user vary accordingly. For example, the probability ofenergy outage is high in the morning due to lower harvestedpower and continues to increase until noon. After noon, theharvested power reduces and so the energy outage as expected.Subsequently, the SNR outage probability of a user reducesin first half of the day and becomes fixed for a specific timeduration in which energy outage does not happen. On the otherhand, during the time from noon to evening, the SNR outageagain increases due to frequent energy outages resulting intransmission failures to the ground user. In addition, it canbe observed that reducing the transmission power from Pd =15 W to Pd = 10 W will reduce the time duration in whichthe energy and SNR outages are expected. That is, with Pd =

11

7 8 9 10 11 12 13 14 15 16 17

Time of the Day (Hours)

0

100

200

300

400

500

600

700E

nerg

y (J

oule

s)

Battery Energy Consumption, Pd = 15 W

Harvested EnergyTotal Available EnergyBattery Energy Consumption, P

d = 10 W

Fig. 5: Harvested and consumed energy of UAV as a functionof the harvesting time.

10 W the energy outage is not expected from 8:00 to 16:00hours compared to when Pd = 10 W where energy outage isexpected from 7:00 to 10:00 and 14:00 to 17:00 hours.

In Fig. 5, we analyze the consumed energy from the batteryas a function of the time. That is, the more harvested energy wehave, less consumption from the battery occurs. This is evidentfrom the case when Pd = 10 W. However, for both cases, wecan see that the battery consumption was low initially and thenraised significantly which is counter-intuitive. The reason isthat when the harvested energy is below a certain threshold, theUAV cannot support transmission and, subsequently, resumesreturn flight which consumes relatively less power compared tothe hovering and transmission consumption. However, as soonas the harvested energy plus UAV battery becomes sufficientto support the transmission, the UAV transmits for Tb − Tfduration resulting in the increased power consumption. Forthe case of Pd = 15 W, the harvested energy level needsto be significantly high compared to Pd = 10 W to supporttransmission. Note that, the harvested energy does not varysignificantly from 10 am to 2 pm, therefore, a significant powerconsumption reduction cannot be observed as can be seen inPd = 10 W scenario.

3) Impact of the UAV Transmit Power: Fig. 6(a) and (b)depict the energy outage probability of the UAV and the SNRoutage probability of the ground user considering both thesolar and wind energy sources as a function of the transmissionpower of the drone, respectively. Analytical results match wellwith the simulations. We can observe that the increase inthe transmit power increases the energy outage due to theincreased energy consumption. On the other hand, the SNRoutage at UAV initially reduces due to increased SNR values;however, there is a maximum transmit power threshold beyondwhich the energy outage at UAV becomes more evident andin turn SNR outage happens. That is, an increase in transmitpower increases the energy consumption from the UAV batteryand at some point the UAV battery combined with the har-

10 15 20

Transmission Power of the UAV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Out

age

Pro

babi

lity

SNR Outage (Solar)AnalysisEnergy Outage (Solar)

0 50 100

Transmission Power of the UAV

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Out

age

Pro

babi

lity

SNR Outage (Wind)AnalysisEnergy Outage (Wind)

Fig. 6: Energy outage and rate outage as a function of thetransmission power of the UAV Pd and the flight duration Tf .

10 12 14 16 18 20 22 24 26 28 30Transmission Power of the UAV

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Out

age

Pro

babi

lity

SNR Outage (Solar)SNR Outage (Hybrid)SNR Outage (Wind)

Fig. 7: SNR outage as a function of the transmission powerof the UAV Pd.

vested energy will not be able to support cellular transmissionand ultimately the UAV will fly back to its charging station.Finally, both the solar and wind harvested powers show similartrends. However, the energy outage trends are sharp for solarenergy compared to the wind energy. The reason is that thesolar energy is nearly deterministic for a given time of theday, whereas the wind energy is random.

In Fig. 7, the SNR outage probabilities are plotted againstthe transmit power of the UAV with three configurations. Theenergy outage for hybrid configuration follows harvested solarpower as solar energy is more deterministic with higher har-vested levels compared to the wind energy (especially aroundnoon). Another observation is that with an increase in transmitpower, the change in wind SNR outage is almost nearlysame as the transitions in wind energy are less significantcompared to the solar energy. Finally, we observe an improvedSNR outage performance with the hybrid (solar and wind)harvesting model.

In Fig. 8 (a) and (b), the SNR outage probability for a useris plotted against the altitude of the UAV. It can be observed

12

100 200 300 400 500

Altitude [meters]

0.4

0.5

0.6

0.7

0.8

0.9

1

SN

R O

utag

e P

roba

bilit

y

Tf = 0.4 T

b

Tf = 0.6 T

b

Tf = 0.2 T

b

100 200 300 400 500

Altitude [meters]

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Tf = 0.2 T

b

Tf=0.4 T

b

T-f =0. 6 Tb

Fig. 8: Energy outage and rate outage as a function of thealtitude of the UAV h and the flight duration Tf .

2 4 6 8 10 12 14

Flight Time of the UAV

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Out

age

Pro

babi

lity

SNR OutageEnergy OutageAnalysis

Fig. 9: Energy outage and rate outage as a function of thetransmission power of the UAV Pd and the flight duration Tf .

that an increase in the altitude of the UAV first reduces theSNR outage due to an increased probability of LOS and thenincreases the SNR outage due to a higher distance from theground. With an increase in the flight duration, the probabilityof outage increases due to the reduction of transmission time.

4) Impact of the Flight Time: In Fig. 9, the energy outageand the SNR outage probabilities are plotted against the flighttime of the UAV. We observe that with an increase in the flighttime, the energy outage probability reduces since a longerflight time implies longer harvesting and shorter transmissiondurations which in turn reduces energy consumption. On theother hand, with increasing flight time, the SNR outage firstdecreases and after a point it starts to increase. That is, anoptimal flight time exists. If the flight time is really small,then the harvesting time becomes limited and most of theenergy consumption occurs during transmission, which resultsin energy outage, and subsequently, in SNR outage. On theother hand, if the flight time is long, the transmission timebecomes limited resulting in an SNR outage.

5) Battery Recharge Time: Fig. 10 depicts the time tocharge the battery of the UAV as a function of the level uptowhich the battery needs to be charged. Here, the energy packet

50 100 150 200 250 300 350 400 450 500 550

Initial battery energy level (u0)

0

10

20

30

40

50

60

Ave

rage

Bat

tery

Rec

harg

e T

ime

Simulations Analysis - 12 (noon)SimulationsAnalysis - 10 am

Fig. 10: Energy outage and rate outage as a function of thetransmission power of the UAV Pd and the flight duration Tf .

450 500 550 600 650 700 750 800 850 900 950

Initial Battery Energy (u0)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Eve

ntua

l Ene

rgy

Out

age

Pro

babi

lity

Tf=0.4 T

b

Tf=0.8 T

b

Fig. 11: Eventual energy outage as a function of the transmis-sion power of the UAV and the initial UAV battery.

size follows distribution of harvested power from solar energywith finite mean and variance. Here the inter arrival time isexponentially distributed due to Poisson arrival consideration.The results from the simulations match closely with thetheoretical prediction. We can observe that the more the levelof energy, the more time it takes for the UAV battery torecharge which is intuitive. However, battery charging time isalso affected by the harvesting time of the day. We can observethat, at noon, the harvested solar energy is comparativelyhigher, and therefore, the time to charge the battery to thesame level is reduced.

In Fig. 11, the eventual energy outage is plotted againstthe initial battery energy of the UAV. It is evident that, if weincrease the initial battery energy, the eventual outage reducesdue to the energy consumption required for supporting flightand transmission. We evaluate the eventual energy outagewhere time is greater than or equal to Tf . Also, we can see thatthe eventual energy outage probability decreases significantlyif Tf increases due to more time devoted for energy harvestingand thus accumulating more energy packets. Also, the energyconsumption during flight is less than the energy consumptionduring hover time as discussed earlier.

In Fig. 12, the energy outage is plotted against the transmitpower of the UAV. For t > Tf , if we increase the transmit

13

Transmission Power of the UAV10 15 20 25 30 35 40

Ene

rgy

Out

age

Pro

babi

lity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t > Tf (Wind)

t < Tf (Wind)

t > Tf (Solar)

t < Tf (Solar)

Fig. 12: Energy outage as a function of the transmission powerof the UAV [Vci = 1, Vr = 8].

power, energy consumption from the UAV battery increaseswhich in result increases the energy outage. We observe thatboth solar and wind energy outage follows similar pattern. Forany time t < Tf , the energy outage is constant as there is notransmission during the flight. Also, the energy consumptionduring flight is less than the consumption during transmissiontherefore we observe reduced energy outage probability.

Note that the energy outage in Fig. 12 is different from theenergy outage in Fig. 6. In Fig. 12, both the harvested powerand energy arrival times are random whereas in Fig. 6 onlythe harvested power is random. However, it can be noticedthat the energy outage tends to be higher in Fig. 12 as theharvested power (or equivalently energy packets) are arrivingintermittently rather than continuously throughout the flightduration as is the case in Fig. 6. It can also be noted that,since solar energy is more deterministic compared to wind,the average energy outage in Fig. 6 has a sharp transition. InFig. 12, the cumulative solar energy during flight time is morestochastic due to not only the amount of harvested power butalso the arrival times. As such, the energy outage increasesgradually rather than sharply.

VII. CONCLUSION

We have derived the exact and novel statistical models forthe three renewable energy harvesting scenarios, i.e. harvestedsolar power, harvested wind power, and hybrid solar and windpower, which have been verified by the simulations. Basedon the developed models, we have derived the energy outageprobability of the UAV and the SNR outage probability ofthe user. The impacts of the transmit power and flight timeon the SNR and energy outage have been observed to obtainuseful design guidelines for the optimal transmit power andflight duration of the UAVs. In addition, we have shown thenovel applications of the moments of the harvested solar andwind power in characterizing performance measures such as

the probability of charging UAV battery in the flight time, theaverage battery charging time, and the eventual energy outageprobability. Numerical results demonstrate that an optimaltransmit power or flight time can be determined to minimizethe energy and SNR outage, respectively. Also, the resultsdemonstrate the distinct stochastic behavior of solar and windharvested power as the solar energy is relatively more deter-ministic compared to wind energy. The developed statisticalmodels for harvested solar and wind power can as well beused for analyzing the performances of other aerial wirelesscommunications systems as well as integrated terrestrial-aerialcommunications systems using energy harvesting.

APPENDIX A: PROOF OF THEOREM 1

Given I(t) is a shifted Gaussian random variable, theharvested solar power of the PV system P at time t in SectionII can be rewritten as:

P =ηcKc

I2U (Kc − I) + ηcIU (I −Kc) ,

(a)=

ηcKc

I (I −Kc)U(Kc − I) + ηcI,

where (a) utilizes the property U (Kc − I) = 1−U (I −Kc).For notational simplicity, we skip the argument t, i.e. weconsider I(t) = I . Note that P is a function of I; therefore,we define P = g(I). Since Dirac delta functions can be usedto evaluate the PDF of the transformed random variables, wecan derive fP (p) as follows:

fP (p) =

∫ ∞0

δ (p− g(I)) f(I)dI,

=∫∞

0δ(p− ηcI(I−Kc)Kc U(Kc − I)− ηcI

)f(I)dI. (A.1)

Note that the integral in (A.1) can be decomposed into twointegrals as shown below:

fP (p) =

∫ Kc0

δ

[p− ηc

KcI [I −Kc] + ηcI

]f(I)dI

+

∫ ∞Kc

δ(p− ηcI)f(I)dI.(A.2)

The first part of (A.2) can be simplified as follows:∫Kc0

δ[p− ηcKc I

2]f(I)dI,=

∫Kc0

δ[ηcKc

(I2 − Kcηc p

)]f(I)dI,

(a)=∫Kc

0Kcηcδ[(I2 − Kcηc p

)]f(I)dI,

(b)= Kc2ηc

∫Kc0

√ηcpKc

(δ(I +

√pKcηc

) + δ(I −√

pKcηc

))f(I)dI,

(c)= Kc2ηc

∫Kc0

√ηcpKc

δ(I −√

pKcηc

)f(I)dI,

= 12∫∞

0

√Kcpηc

δ(I −√

pKcηc

)U(Kc − I)f(I)dI,(d)= 12

√Kcpηc

U(Kc −√

pKcηc

)f(√

pKcηc

),

where (a) is obtained by applying the properties of Dirac deltafunctions δ(ax) = a−1δ(x) and (b) is obtained by applying thedelta function property δ(x2− a2) = 12a [δ(x− a) + δ(x+ a)].Note that radiation intensity I cannot be negative, therefore,we discard the negative root and (c) can thus be obtained.

14

Finally, the integral can be solved using the properties ofDirac-delta function as shown in (d).

The second part of (A.2) can be rewritten as follows:∫ ∞0

δ(p− ηcI)U(I −Kc)f(I)dI, (A.3)

(a)=

1

ηcf

(p

ηc

)U(

p

ηc−Kc), (A.4)

where (a) is obtained by solving the integral. The distributionof the harvested solar power can thus be derived by combiningthe results in (A.3) and (A.4) as is shown in Theorem 1.

APPENDIX B: PROOF OF THEOREM 2

Similar to the proof of Theorem 1, the distribution of outputpower in the range Vci < v ≤ Vr can be derived using thesingle random variable transformation, as follows:

fPW (pw) =

∫ VrVci

δ(pw − av̄3

)fV̄ (v̄), (B.1)

where a = 12ρAC. It is noteworthy that fV (v) has beentruncated first over the range Vci < v ≤ Vr yielding fV̄ (v̄)and then is used in the aforementioned expression. That is,

fV̄ (v̄) =fV (v)

FV (Vci)− FV (Vr). (B.2)

Now we apply the Dirac-delta function property in (B.1)

δ(f(x)) =∑i

δ(x− ai)| dfdx (ai)|

,

and rewrite (B.1) as follows:∫ VrVci

δ(v̄ − v1)32ρACv̄

2fV̄ (v̄)dv̄, (B.3)

where v1 = (pwa )1/3 is the only real root. After solving the

integral, the PDF of output power harvested from wind energycan be given as follows:

fP (p) =

2fV̄ (v1)3ρACv12

, aV 3ci < pw ≤ aV 3rδ(p− Pr), aV 3r < pw ≤ aV 3coδ(p), otherwise

,

The PDF of the harvested power from wind energy can thusbe given as in Theorem 2.

APPENDIX C: PROOF OF THEOREM 3: PART (A)

Using Theorem 1, the PDF of the solar power fP (p) can

be given. The Laplace transform LP (s) =∞∫0

e−spfP (p)dp can

thus be expressed as follows:

LP (s) =

∞∫ηcKc

e−sp−( pηc −Id)

2

2

ηc√

2π+

ηcKc∫0

√1pe−sp−

(√Kcpηc−Id

)22

2√

2πηc/Kc.

(C.1)

Using [20, Eq. 3.322/1], i.e.∞∫u

e−x2

4β−γxdx =√πβeβγ

2

[1 −

erf(γ√β + u

2√β

)] and taking β = η2c/2 and γ = s − Idηc , thefirst integral in (C.1) can be solved in closed-form as follows:

1

2e−

Id2

2 +12 (Id−sηc)

2

erfc

[−Id +Kc + sηc√

2

]. (C.2)

The second integral in (C.1) can be rewritten by changingvariables

√p = x as shown below:√

Kc2πηc

e−Id

2

2

√ηcKc∫0

e−( Kc2ηc+s)x

2+√KcηcxIddx. (C.3)

We note thatu∫0

f(x)dx =∞∫0

f(x)dx−∞∫u

f(x)dx; therefore, we

split (C.3) into two integrals. Then we use∞∫0

e−x2

4β−γxdx =

√πβeβγ

2

[1 − erf(γ√β)] [20, Eq. 3.322/2] as well as [20,

Eq. 3.322/1] to solve the integrals by taking β = ηc2(Kc+2ηcs)and γ = −

√KcηcId. The closed-form for (C.3) can thus be

expressed as follows:

e− Id

2

2 +I2d

2+4sηcKc

(erf

[Id√

2+ 4sηcKc

]+ erf

[Id−Kc−2sηc√

2+ 4sηcKc

])2√

2s+Kc/ηc.

(C.4)Combining (C.2) and (C.4), the MGF can be expressed asshown in Theorem 3.

APPENDIX D: PROOF OF THEOREM 3: PART (B)

The Laplace transform of the harvested wind power can bederived using fPw(pw) given in Theorem 2 as follows:

LPw(s) =ka−

k3

3ck

∞∫0

e−sppwk−3

3 exp

(− pw

k3

ak3 ck

)dp,

(a)=

k

3

∞∫0

e−sac3xe−x

k3 x

k3−1dx,

(b)=

∞∫0

e−sac3y

3k e−ydy,

(c)=

∞∫0

∞∑n=0

(−sac3y 3k )n

n!e−ydy,

(d)=

∞∑n=0

(−sac3)n

n!

(3n

k

)!,

where (a) follows by taking x = pac3 , (b) follows by sub-stituting y = xk/3, (c) follows by using the identity ex =∞∑0

xn

n! , and (d) follows by interchanging the summation and

integration and solving the integral using∫∞

0xme−µxdx =

m!µ−m−1 from [20, 3.351/2].

15

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