Energy Conversion and Management 76 (2013) 986–995

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Energy Conversion and Management

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The equivalence of gravitational potential and rechargeable battery forhigh-altitude long-endurance solar-powered aircraft on energy storage

0196-8904/$ - see front matter Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2013.08.023

⇑ Corresponding authors. Address: No. 109, Deya Street, Kaifu District, ChangshaCity, Hunan Province 410073, PR China. Tel./fax: +86 0731 84573189 (X.-Q. Chen).

E-mail addresses: hzx@nudt.edu.cn (Z.-X. Hou), chenxiaoqian@nudt.edu.cn(X.-Q. Chen).

Gao Xian-Zhong, Hou Zhong-Xi ⇑, Guo Zheng, Fan Rong-Fei, Chen Xiao-Qian ⇑College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, PR China

a r t i c l e i n f o

Article history:Received 8 April 2013Accepted 11 August 2013

Keywords:Gravitational potentialRechargeable batteryHigh-altitude long-enduranceSolar-powered aircraft

a b s t r a c t

Applying solar energy is one of the most promising methods to achieve the aim of High-altitude Long-endurance (HALE) flight, and solar-powered aircraft is usually taken by the research groups to developHALE aircraft. However, the crucial factor which constrains the solar-powered aircraft to achieve theaim of HALE is the problem how to fulfill the power requirement under weight constraint of rechargeablebatteries. Motivated by the birds store energy from thermal by gaining height, the method of energystored by gravitational potential for solar-powered aircraft have attracted great attentions in recentyears. In order to make the method of energy stored in gravitational potential more practical in solar-powered aircraft, the equivalence of gravitational potential and rechargeable battery for aircraft onenergy storage has been analyzed, and four kinds of factors are discussed in this paper: the duration ofsolar irradiation, the charging rate, the energy density of rechargeable battery and the initial altitudeof aircraft. This work can provide some governing principles for the solar-powered aircraft to achievethe unlimited endurance flight, and the endurance performance of solar-powered aircraft may be greatlyimproved by the application of energy storage using gravitational potential.

Crown Copyright � 2013 Published by Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Background

Great efforts have been paid for the design of the High-altitudeLong-endurance (HALE) aircrafts in recent years [1–3]. This air-crafts are developed to operate at stratospheric altitudes and usedto support tactical battlefield operations, communications, atmo-spheric monitoring, precise agricultural, wild fire monitoring,surveillance, and any other missions requiring satellite-basedinfrastructure or high resolution imagery [4].

As everyone known that applying solar energy is one of themost promising method to achieve the aim of HALE flight [5], sosolar-powered aircraft is always taken by the research groups todevelop HALE aircrafts [6–8]. Many pioneering experiments havebeen done for the development of HALE aircraft, such as ERASTand Zephyr [9,10], but none of aircraft realizes HALE flight in realsense until nowadays. The main reason is rooted in the fact thatthe solar power is not an ideal energy source, as pointing out inRef. [11], the solar cell panels in aircraft can only generate powerat certain times of the day, and then how to keep the aircraft aloftat other times is the most important consideration. The most

common form of the energy storage for the stand alone solar-pow-ered system is rechargeable battery [12]. While under currenttechnological level, in order to achieve the aim of HALE flight,the weight of the rechargeable batteries needs to occupy around30% of the total mass of solar-powered aircrafts [13,14]. Althoughmore battery can provide more energy during night, the weight ofadditional batteries needs more energy to sustain continuous flightat the same time. So, as shown in Ref. [15], it is really hard to de-velop a solar-powered HALE aircraft under current technologicallevel.

1.2. Motivation

While, in the nature, birds can stay in air for a long time withoutany additional power and gain height by circling in thermals withwing spread until the desired height is reached, then a more or lessstraight advancing but sinking phase follows until the next thermalis reached [16]. This indicates that birds can store the energy inthermals by height, i.e. gravitational potential, to achieve a power-less long-endurance flight [17]. By analogy, the solar-poweredaircraft can also store the surplus energy by gravitational potentialwhen the solar power is available. Not as the method to store en-ergy by rechargeable battery, the gravitational potential can storethe energy without any weight penalty. Thus, it is a very promisingtechnological route to achieve the HALE flight for solar-poweredaircraft.

Nomenclature

a attack anglel pitch anglegA efficiency of airscrewgB efficiency of batterygBM efficiency of battery ManagergMPPT efficiency of MPPTgPC efficiency of power conversiongSC efficiency of solar cell panelq the air densitys the transmittance factorAm the air mass ratioAR aspect ratioCD drag coefficientsCD0 parasitic drag coefficientsCL lift coefficientsD drag forceH altitude of the aircraftKp endurance flight factorL lift forceLspan span lengthPlevel the power consumed during level flightPGP the required power by using gravitational potential to

store energyPRB the required power by using rechargeable battery to

store energyPs the available solar power

QB the electric quantity of batteryQs the total available energy from solar powerRKP the ratio of endurance flight factorSC area of solar cellsSW the wing areaT thrust forceTBD the time of battery dischargingTs the duration of solar powerTFE the flight enduranceV airspeedk the induced drag factorke the local extinction coefficient,m the mass of aircraftmb the mass of battery�mb the energy density of batterymstruct the mass of structureq the charging rate of batteryqmax the maximum rate of chargex horizontal ordinate of the aircraft

AbbreviationsEMS Energy Management StrategyEFF Endurance Flight FactorHALE High-altitude Long-enduranceMPPT Maximum Power Point Tracking

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1.3. Relative works

The phenomenon of energy stored in gravitational potential hasbeen discovered for a long time since 1870 [16]. The systematic re-search about the theory of energy stored in gravitational potentialcan be found in Ref. [17], in which the author proposes the conceptof ‘energy height’ to describe the stored energy which could lift theanimal or aircraft against the prevailing gravity.

Prof. Sachs is the first one to propose that the method of energystored in gravitational potential is one of means to achieve anunlimited endurance performance of solar-powered aircraft [18].He has demonstrated in the paper that with an appropriate trajec-tory control, it is possible for the solar-powered aircraft to stayaloft during night with minimum or even no solar energy to bestored in rechargeable battery. Motivated by his ideas, the proper-ties of the maximum endurance path of gravitational gliding havebeen studied in Ref. [19], and a new energy management strategyhas been proposed in Ref. [20].

1.4. Contributions

Significant progresses have been made over the past decadeson understanding the method of energy stored by gravitationalpotential. However, the application of this method in aerospaceengineering has not been sufficiently discussed yet. This statusin quo may be caused by the reason that the storage method ofenergy extracted from environment is not the crucial constraintfor the fuel-powered aircraft. However, with the rapid develop-ment of solar-powered aircraft, how to store the energy extractedfrom solar power becomes more and more important, and willplay a dominant role to decide whether the aircraft can achievea HALE flight. In order to make the method of energy stored ingravitational potential more practical in solar-powered aircraft,there are still numerous works need to do. One of curial problems

is to analyzed the equivalence of gravitational potential andrechargeable battery for aircraft on energy storage, i.e., in whichsituation it is advantageous to use the method of energy storedin gravitational potential, and in which situation the other meth-od is preferred. Moreover, it is also important to analyze theinfluence of the performance of rechargeable battery, such ascharging rate and energy density to the equivalence of gravita-tional potential and rechargeable battery.

The main aim of this paper is to answer these questions fromthe energy storage’s point of view. In order to make the discussionmore clear, the system and dynamic model of solar-powered air-craft are described firstly. Secondly, the energy management strat-egy (EMS) for rechargeable battery and gravitation potential tostore energy are briefly reviewed. Thirdly, the equivalence of grav-itational potential and rechargeable battery is discussed with thedefinition of the Endurance Flight Factor (EFF). The discussionsabout the influences of solar power, charging rate, the mass ofrechargeable battery and the initial altitude of flight on the perfor-mance of energy stored by gravitational potential and rechargeablebattery are presented at last. This work has a significant sense forthe application of energy storage using gravitational potential, andmay provide some governing principles for the solar-powered air-craft to achieve the unlimited endurance flight with minimal oreven zero rechargeable battery.

The rest of the paper is organized as follows: The system anddynamic model of solar-powered aircraft are described in Section 2.The EMS for rechargeable battery and gravitation potential to storeenergy are introduced in Section 3. The methodology to analyzethe equivalence of gravitational potential and rechargeable batteryis presented in Section 4. The influence of solar power, chargingrate, the mass of rechargeable battery and the initial altitude offlight on the performance of energy stored by gravitational poten-tial are discussed in Section 5, and the concluding remarks aremade in Section 6.

Fig. 1. The system of solar-powered aircraft.

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2. System description and dynamic model of solar-poweredaircraft

The main character of solar-powered aircraft is rooted on themethod to extract energy from environment, it can be completelypowered by solar, and the energy source can be considered inex-haustible [21]. The 1:4 scaled solar-powered aircraft designed byour group is shown in Fig. 1. The energy of this aircraft comes fromthe solar panels which are composed of silicon arrays covered onthe surface of the wing. The key component to manage theon-board energy is named as energy management system. Thefunction of this component can be described as follows: Duringdaytime, it divides the electrical energy converted by the silicon ar-rays from the sunlight into two parts: one part is used to supplythe motor and on-board electronics, the other part is charged intothe rechargeable battery, which can be used to power the aircraftduring nighttime [22]. Of course, in PV systems, maximum powerpoint tracking (MPPT) is essential because it enables the extractionof maximum available energy from the system [23], and it is al-ways integrated in the energy management system of the solar-powered aircraft.

The efficiencies of all the components in Fig. 1 are listed in Ta-ble 1. The charging efficiency and discharging efficiency of batter-ies are not distinguished, instead, the efficiency of batteries gB isused to describe how much energy can be discharged when thecharging energy is 1.

The airscrew efficiency in Table 1 is both the propeller efficiencyand motor efficiency. Since the propeller efficiency is changed withaltitude for the variation of Reynolds numbers, in order to conquerthis problem, the adjustable blade is always adopted in the propel-ler of HALE aircraft. It is also true that the well-known large solarpowered aircraft are all use constant pitch propeller. However,even if the efficiency of propeller varies in climb, level flight andgliding, the advantage of climb to store gravitational potential doesexist. Because the advantage of energy stored by gravitational po-tential is rooted in the fact that gravitational potential can store thesurplus solar energy without weight penalty. Thus, it is reasonable

Table 1The efficiencies of the components in energy management system.

Component Symbol and efficiency of value

Solar cell panel gSC = 0.2MPPT gMPPT = 0.95Battery manager gBM = 0.99Battery gB = 0.95Power conversion gPC = 0.98Airscrew gA = 0.7

to assume the aircrew efficiency is constant in basic conceptualanalysis.

In order to analyze the motion of solar-powered aircraft duringflight, a mathematical model that consists of the equations of mo-tion for three-dimensional flight is adopted. The dynamic model ofaircraft is given as follows, where the speeds are modeled in a air-speed reference frame and the positions are modeled in an earthfixed frame.

m _V ¼ T cos a� D�mg sin l ð1Þ

mV _l ¼ T sinaþ L�mg cosl ð2Þ

_x ¼ V cosl ð3Þ

_H ¼ V sin l ð4Þ

where the m is the mass, the V is airspeed, T is the thrust, a is theangle of attack, D is the drag force, l is the pitch angle, L is the liftforce, x is the level position and H is the altitude.

The lift and drag force are expressed as following

L ¼ 12qSW CLV2 ð5Þ

D ¼ 12qSW CDV2 ð6Þ

where the q is the air-density, SW is the reference area of aircraft, CL

and CD are the lift coefficient and drag coefficient, respectively. Thedrag coefficient CD depends on the lift coefficient CL, yielding

CD ¼ CD0 þ kC2L ð7Þ

where the CD0 is the parasitic drag coefficient and k is the induceddrag factor.

The methods to study the aerodynamic characteristics of thestudied aircraft are detailed in Ref. [24], in fact, the values of CD0

and k are not constant for solar-powered aircraft, because the Rey-nolds number is varied with altitude and velocity. The variation ofaerodynamic efficiency in different altitude are also considered inthe aerodynamic model. For simplification, this is not discussedin this paper again.

The basic parameters of studied solar-powered UAV are listed inTable 2, which is mainly referred from Zephyr [25].

It should be noted that the simplified 2 DoF dynamic model ofthe aircraft as shown in Eqs. (1)–(4) in this paper does not take intoaccount the rotation of the apparatus about the center of mass.This approach is tolerable for primary qualitative estimations only,and change of the pitch angle due to the aerodynamic momentabout the center of mass could influence the results considerably.Some improvements to the model of aircraft will be brought andanalyzed in the future.

Because the solar radiation tightly depends on the altitude,especially at 10–30 km, so the model to calculate the transmittancefactor s in Ref. [20] must make a small modification to comprise thefactor of altitude. In our previous model the transmittance factors sis computed by the following equation:

Table 2The basic parameters of studied solar-powered UAV.

Parameter Value Unit Description

mstruct 37 kg Mass of structuremb 16 kg Mass of batteryLspan 22.5 m Span lengthAR 25 – Aspect ratioSW 20.25 m2 Wing areaSC 16.2 m2 Area of solar cells

Fig. 3. The EMS for the gravitational potential to store energy.

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s ¼ e�keAm ð8Þ

where ke is the local extinction coefficient, Am is the air mass ratio;in ChangSha, China (28.2N, 112.6E), k = 0.149, Am = 1.5. Here, the Am

is calculated by the following equation:

Am ¼ Am0PðHÞ

P0ð9Þ

where the Am0 = 1.5, P0 is the atmosphere pressure at sea level, P(H)is the atmosphere pressure at altitude of H.

3. EMS for rechargeable battery and gravitational potential tostore energy

The energy management strategy (EMS) which is embedded inthe energy management system is the algorithm used to controlthe power flow. The main aim of the solar-powered aircraft’ EMSis to keep aircraft aloft for long-endurance missions without en-ergy replenishment. Once the energy storage method of aircraftis determined, the corresponding EMS must be designed.

The EMS for the rechargeable battery and gravitational poten-tial to store energy have been designed in [20]. For the aim of sim-plification, it will not be described again here. However, in order todiscuss the equivalence of gravitational potential and rechargeablebattery for solar-powered aircraft on energy storage, the efficiencyof gravitational potential on energy storage without rechargeablebattery must be considered, which means that, comparing to theEMS designed in Ref. [20], the corresponding EMS needs to take aminor modification. Thus, it is necessary to give a brief descriptionto the EMS for the rechargeable battery and gravitational potentialto store energy.

As shown in Fig. 2, the EMS for the rechargeable battery to storeenergy is elaborated as follows: The aircraft always keep the bal-ance between lift force and weight with the minimal power factorin level flight. When the power converted from solar is greater thanthe power consumed by drag, the surplus power will be charged tobattery, otherwise, the insufficient part of required power must besupplied by discharging the rechargeable batteries.

While, as shown in Fig. 3, the EMS for the gravitational potentialto store energy is in another way: When the power converted fromsolar is greater than the power consumed by drag, all the surpluselectric power will be supplied to generate thrust and climb to ahigh altitude. Once the solar radiation can no longer sustain the le-vel flight, the aircraft will descend with a minus pitch angle and bepowered by gravitational potential. The control law of descendinghas been designed in Ref. [19] to make the endurance time sus-tained by the unit altitude difference as long as possible.

The greatest difference between the EMS for the rechargeablebattery and the gravitational potential is the manner how to dealwith the surplus power. The former one charges this power to

Fig. 2. The EMS for the rechargeable battery to store energy.

rechargeable battery, and discharges power when the solar radia-tion is insufficient; the latter one stores the surplus power to grav-itational potential, and keep the aircraft aloft by gravitationalgliding when the power converted from solar is smaller than thepower consumed by drag. By the comparison it is can be found thatthe aircraft can only store solar energy in rechargeable batteries bythe former EMS. Since the rechargeable battery is limited, and themass of battery will further aggravate the power consumed bydrag, it is not efficient to utilize solar power by this kind of EMS.However, although all the surplus powered can be stored in thegravitational potential by the latter EMS, the endurance flight timesustained by the unit vertical distance is little at high altitude be-cause of the rare air density. In this situation, the utilization of so-lar power is not efficient, either. So it is important to find theequivalence of gravitational potential and rechargeable battery atdifferent conditions.

4. The equivalence of gravitational potential and rechargeablebattery

4.1. The definition of Endurance Flight Factor (EFF)

The biggest problem for the solar-powered aircraft to achievethe aim of high-altitude long-endurance flight is how to keep aloftduring night without solar irradiation. The common method todeal with this problem is to store the surplus power during day-time by rechargeable battery [11]. So the index to evaluate the per-formance of rechargeable battery can be set as how long theaircraft can be sustained to fly in a certain altitude by the batterywith given electrical power. Thus, it is reasonable to define the fol-lowing factor Kp to find the equivalence of gravitational potentialand rechargeable battery:

Kp ¼TFE

Q sð10Þ

Here, TFE is the flight endurance, Qs is the total available energyfrom solar power, which is equivalent to the integration of Ps in thetime interval of Ts, the Ps is the available solar power and Ts is theduration of solar power. The Kp is named as Endurance Flight Fac-tor (EFF), which indicates how long the aircraft can be sustained tofly in a certain altitude with given solar power. Because the Kp isdifferent with the different method to store electrical power, it isappropriate to be used to judge the performance of the energy stor-age method.

Our motivation to define the Kp is to compare the fly enduranceof aircraft in different method to store energy. The unit of Kp in thedefinition of Eq. (10) is W�1, and it is better if a dimensionless var-iable can be used. However, it is not convenient to find a reference

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variable in aircraft whose unit is W�1 and can be selected to com-pose a dimensionless variable.

Fig. 4. The compare of altitude.

Fig. 5. The compare of power required.

4.2. The method to compute EFF

After the definition of EFF, the next issue is how to compute it.For the method of energy stored by rechargeable battery, the air-craft always keep level flight when the solar power is sufficient,and the battery will be charged when the electric quantity of bat-tery is not full. When the solar power is insufficient, the batterywill be discharged until the available electric quantity of batteryis empty. So, the required power PRB during level flight can be com-puted as follows:

PRB ¼Plevel þ q Ps > Plevel & Q B 6 mb �mb

Plevel Q B > mb �mbð1� gBÞ0 Ps 6 Plevel & Q B 6 mb �mbð1� gBÞ

8><>: ð11Þ

where q is the charging rate of battery, QB is the electric quantity ofbattery, mb and �mb are respectively the mass and energy density ofbattery, Plevel is the power consumed by drag, which is defined asfollows to include the efficiency of propulsion systems:

Plevel ¼ThVgA¼ DV

gA cos a¼ 1

gACL=CD

Lcos a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2L

qCLSW

s

¼ 1gA

CD

C3=2L

mg � Th sinacos a

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðmg � Th sin aÞ

qSW

sð12Þ

The flight endurance TFE for this method is the sum of Ts and thetime of battery discharging TBD, which can be calculated as follows:

TBD ¼mb �mbð1� gBÞ

Plevelð13Þ

For the method of energy stored by gravitational potential, allthe electric power converted from solar power will be suppliedto generate thrust and climb to a high altitude until the solar radi-ation is below the power required to maintain level flight. By thisway, the additional solar energy can be stored by gravitational po-tential. After that, the aircraft will fly at the manner of gravitationalgliding. So, the required power PGP can be computed as follows:

PGP ¼gAgPCgMPPTgSCPs Ps P Plevel=gPCgMPPTgSC

0 Ps < Plevel=gPCgMPPTgSC

�ð14Þ

The methodology to compute the flight endurance TFE for thismethod has also been described in Ref. [20].

4.3. The case study

To illustrate the methodology to compute the equivalence ofgravitational potential and rechargeable battery, a case and simu-lation results are presented in this section. The simulation usedin this study is an solar-powered HALE aircraft with elevator actu-ators and electric propulsion, and the flight model is depicted inSection 2. The efficiencies of the components in energy manage-ment and the parameters of HALE aircraft are listed in Tables 1and 2, respectively.

Here, the flight altitude is supposed as H = 10 km, the solarpower is Ps = 1000 Wh/m2 (In fact, we are not supposing that thesolar power is constant in all the day. We just want to comparethe flight performance of aircraft in different methods to storeenergy under this standard test condition), the duration of solarirradiation is Ts = 0.1 h, i.e., 360 s, the maximum charging rate ofrechargeable battery is qmax = 960 W, and the energy density ofrechargeable battery is �mb ¼ 350 Wh/kg.

For the method of rechargeable battery to store energy, theaircraft always keeps level flight with a zero pitch angle at thealtitude of 10 km. Whereas, for the method of gravitation potentialto store energy, the aircraft will climb to a high altitude when thesolar power is sufficient and descend to 10 km by gravitationalgliding when the solar power is not sufficient. The compares of alti-tude H, power required between two method to store energy areshown in the Figs. 4 and 5, respectively.

It can be seen from Fig. 4 that the solar-powered aircraft can flyTEF|GP = 3720 s with the method of energy stored by gravitationalpotential, whereas, it can only fly TEF|RB = 1577 s with the methodof energy stored by rechargeable battery. So, it can be concludedthat the efficiency of energy stored by gravitational potential isgreater than that of energy stored by rechargeable battery. As de-fined in Eq. (10), the quantitative analysis of the efficiency can bedescribed by the ratio of EFF, i.e.:

RKp ¼KPjGP

KpjRB¼ TEF jGP

TEF jRB¼ 3720

1577¼ 2:36 ð15Þ

The Eq. (15) indicates that the efficiency of energy stored bygravitational potential is 2.36 times of energy stored by recharge-able battery. Thus, from this point of view, it is advantageous tostore energy by gravitational potential at the altitude of 10 km, un-der the parameters’ constraints of Tables 1 and 2.

The reason why the efficiency of gravitational potential is great-er than that of rechargeable battery can be interpreted by Fig. 5,which shows the power required during the flight of solar-pow-ered aircraft. Although power consumed in gliding should be con-sidered, the power required for propeller system is zero during

Fig. 7. The compare of pitch angle with different duration of solar irradiation.

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gliding. so, as shown in Fig. 5, the required power in gliding is zero.It can be seen that during the aircraft powered by solar irradiation,the power required by the method of gravitational potentialachieves the maximum available power, while the power requiredby the method of rechargeable battery is the sum of power con-sumed by drag and the power charging to battery, which cannotachieve the maximum available power. During the flight endur-ance without solar irradiation, the method of rechargeable batterymust discharge power from battery to keep aircraft aloft, whereas,the aircraft requires no additional power to take gravitational glid-ing by the method of energy stored by gravitational potential.

5. Discussion

Although the performance and conceptual parameters shouldact as the main input parameters for analysis, as shown in Ref.[24], the influence of season and latitude can be treated as theduration of solar irradiation. Since the main difference among eachseason and latitude is the duration of solar irradiation. For the gi-ven airplane, the cruise velocity, wing load, and correspondingvelocity and propulsion efficiency are also determined. So theinfluences of these factors on the performance of airplane are moresuitable to be analyzed in the concept design stage rather thanoperational stage. In fact, the most important factors to affect theoperational performance of solar-powered aircraft are the opera-tional time, place and altitude (which are incorporated in the dura-tion of solar irradiation and the initial altitude), the operationalperformances of rechargeable battery (which are determined bythe charging rate and the energy density). So the discussed factorsare selected as the duration of solar irradiation, charging rate, theenergy density of rechargeable battery, and the initial altitude inthis section.

5.1. The influence of the duration of solar irradiation

In the discussion of Section 4.3, the duration of solar irradiationis chosen as Ts = 360 s, and it can be expected that the longer theduration of solar irradiation is, the higher the solar-powered air-craft can achieve to be. However, how about the influence of theduration of solar irradiation on the ratio of EFF is still unknown,and it needs to make a further discussion.

It is also quite reasonable to derive the results from the param-eters of season, time, latitude and so on. However, the influence ofseason time, and latitude can be treated as the duration of solarirradiation, since the main difference among each season and lati-tude is the duration of solar irradiation. Thus, it is better to pick up

Fig. 6. The compare of altitude with different duration of solar irradiation.

the duration of solar irradiation as the solo variable to instead theparameters of season, time, latitude and so on.

Here, the durations of solar irradiation are supposed to be1800 s, 3600 s, 5400 s, 7200 s, respectively. The altitude profilesof aircraft are shown in Fig. 6 in each case. It can be seen thatthe aircraft can achieve to be at altitude of 16.4 km, 22.1 km,26.4 km and 28.9 km when the durations of solar irradiations are1800 s, 3600 s, 5400 s and 7200 s, respectively. However, as every-one known that the higher the aircraft flies to be, the rarer the airdensity is, therefore, the smaller the pitch angle of aircraft is underthe same solar power. As shown in Fig. 7, at the end of 1800 s,3600 s, 5400 s and 7200 s, the maximum pitch angle are about9.7�, 5.1�, 2.5� and 0.9�, respectively. It means that the less and lessvertical distance can be increased for the aircraft, i.e., the incre-ment of altitude is smaller and smaller with the increase of dura-tion of solar irradiation by the method of gravitational potential.Consequently, the increment of flight endurance is also smallerand smaller, as shown in Fig. 6, the increment of flight endurancefrom 3600 s to 1800 s, from 5400 s to 3600 s and from 7200 s to5400 s are 8467 s, 4631 s and 2827 s, respectively.

Whereas, for the method of rechargeable battery, because theflight altitude is a constant, so both the electrical quantity storedby battery and the flight endurance are in linear direct proportionto the duration of solar irradiation, as shown in Fig. 8, the flightendurance of aircraft are 7886 s, 15,768 s, 23,660 s and 31,500 swhen the durations of solar irradiations are 1800 s, 3600 s,5400 s and 7200 s, respectively.

Fig. 8. The compare of electrical quantity with different duration of solarirradiation.

Fig. 9. The changes of EEF with different duration of solar irradiation.

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By the above analysis, it can be concluded that the ratio of theefficiency of gravitational potential to that of rechargeable batterywill be lower and lower with the increase of the duration of solarirradiation. The changes of EFF with different duration of solar irra-diation are shown in Fig. 9, it can be seen that the RKp are 1.92,1.50, 1.20 and 0.98 when the durations of solar irradiations are1800 s, 3600 s, 5400 s and 7200 s, respectively. All of them are low-er than the RKp = 2.36 with the durations of solar irradiations of360 s. When the durations of solar irradiations is 7200 s, the RKp

is even less than 1, which indicates that the efficiency of gravita-tional potential is lower than that of rechargeable battery.

5.2. The influence of charging rate

The charging rate determines how long time will be taken forthe rechargeable battery from empty to fully charge. As shown inFig. 1, the energy used to charge rechargeable batteries comes fromthe energy management system, which can typically charge bat-tery in two to five hours from empty to fully charge, dependingon the properties of rechargeable battery. Of course, the energymanagement system must have multiple ways of detecting whenthe rechargeable battery reaches full charge, such as changes interminal voltage, temperature, etc., to stop charging before harm-ful overcharging or overheating occurs (http://en.m.wikipe-dia.org/wiki/Rechargeable_battery#section_2).

Battery charging rates are often discussed by referencing a Crate of current. The C rate is that which would theoretically fullycharge the battery in 1 h. In order to discuss the influence of charg-ing rate to the ratio of EFF conveniently, here, the unit of charging

Fig. 10. The compare of electrical quantity with different charging rate.

rate is set as W, which is coherent with the power consumed bydrag. In general, the higher the charging rate relative to batterycapacity, the worse the effective storage capacity and overall lifeof the battery will be, but the lower the charging rate, the fewer en-ergy can be stored in battery during certain duration.

Here, the duration of solar irradiation is set as 5 h, and thecharging rates of rechargeable battery are set as 400 W, 800 W,1200 W, 1600 W, respectively. The changes of electrical quantityover time in the battery are shown in Fig. 10 in each case. It canbe seen that the flight durations of aircraft are respectively12.04 h, 19.09 h, 20.73 h and 20.73 h. It means that the higherthe charging rate, the longer time the aircraft can be sustained tofly with the rechargeable battery. However, the flight duration ofaircraft will not increase if the electrical quantity achieves themaximum value, as shown in the blue line of Fig. 10. So it is naturalto conclude that the ratio of the efficiency of gravitational potentialto that of rechargeable battery will be lower and lower with the in-crease of the charging rate, as shown in Fig. 11, the RKp are 0.72,0.45, 0.35 and 0.35 when the charging rates are 400 W, 800 W,1200 W and 1600 W, respectively.

5.3. The influence of the energy density of rechargeable battery

The rechargeable battery is the most important component forthe solar-powered aircraft to keep aloft during night. The keyparameter to evaluate the rechargeable battery is the gravimetricenergy density, which is defined as how much energy can be pro-vide for 1 kg of rechargeable battery. The lithium-ion battery is oneof the most popular electrochemical storage systems due to highenergy density, high operating voltage and low self-discharge.However, the gravimetric energy density is known to be limitedto 250 Wh/kg at the cell level, which is not enough to meet therequirements of aircraft for extended range. The lithium–sulfurbattery is a promising energy storage system due to its high theo-retical specific capacity of about 2600 Wh/kg [26]. In practical, theaverage discharge potential is around 2.1 V, and the complete lith-ium/sulfur system should allow to reach a gravimetric energy den-sity close to 500 Wh/kg [27]. So, here, the lithium–sulfur battery ischosen as the studied rechargeable battery.

Because it is impossible for a lithium-sulfur battery with a sul-fur cathode to discharge fully [28], until nowadays, the demon-strated specific energy levels at cell level are still 350 Wh/kg. So,in Section 4.3, the energy density of rechargeable battery is chosenas 350 Wh/kg. In order to analyze the influence of energy density ofrechargeable battery on the performance of solar-powered aircraft,here, the energy densities are supposed to be �mb ¼ ½350;450;550;650�Wh/kg, the duration of solar irradiation is also set as 5 h, the

Fig. 11. The changes of EEF with different charging rate.

Fig. 12. The compare of electrical quantity with different energy density.

Fig. 13. The changes of EEF with different energy density.

Fig. 14. The compare of altitude with different initial altitude.

X.-Z. Gao et al. / Energy Conversion and Management 76 (2013) 986–995 993

charging rate of rechargeable battery is 1600 W and the otherparameters in simulation are the same with Section 4.3. As dis-cussed in Section 5.1, it can be expected that the RKp is lower than1 when the duration of solar irradiation is greater than 2 hours. Be-cause the total energy needed to be stored is a constant, the greaterthe energy density is, the less the mass of rechargeable battery isrequired, and the fewer power is consumed during flying. Theinfluence of energy density to the performance of aircraft is shownin Figs. 12 and 13.

It should be noted that the wing load of solar-powered airplanewould change with different energy density of rechargeable bat-tery, especially in the case of high energy density. The lower thewing load, the less the power is consumed, and the aerodynamicefficiency would change correspondingly. All of these factors mustbe considered in simulation and it can be embodied by the methodto calculate the Plevel in Eq. (12), which is coupled with the wingload and aerodynamic efficiency.

It can be seen from Fig. 12 that the electrical quantity stored inbattery is not reached the maximum available electrical quantitywhen the energy densities are 350 Wh/kg and 450 Wh/kg becausethe maximum mass of battery is set as 16 kg. However, when theenergy densities are 550 Wh/kg and 650 Wh/kg, both of casesreach the maximum available electrical quantity, and the requiredmass of battery are 14.5 kg and 12.3 kg, respectively. It also can befound that, the improvement of energy density has two kinds ofinfluences on electrical quantity, the first one is to enhance theelectrical quantity stored in battery when the maximum availableelectrical quantity is not reached; the second one is to reduce the

discharging rate of electrical quantity with the less required massof battery.

As it is expected in the former, all of the RKp are lower than 1when the duration of solar irradiation is set as 5 h, as shown inFig. 13, the RKp are 0.48, 0.39, 0.34 and 0.32 when the energy den-sities are 350 Wh/kg, 450 Wh/kg, 550 Wh/kg and 650 Wh/kg,respectively. It also can be found that the RKp descends morequickly from 350 Wh/kg to 450 Wh/kg than that from 550 Wh/kgto 650 Wh/kg, which indicates that the first influence is moregreater than the second one on the EFF of solar-powered aircraft.

5.4. The influence of the initial altitude

The last and the most important factor to influence the EEF isthe initial altitude, because the lower the initial altitude, the higherthe air density will be, and the higher air density will improve theaerodynamic efficiency which directly enhances the flight endur-ance of aircraft. Thus, the equivalence of gravitational potentialand rechargeable battery for solar-powered aircraft will be greatlyimpacted by the initial altitude.

In order to discuss the influence of the initial altitude on the ra-tio of EFF, here, the effect of wind is ignored. The object of this pa-per is to study the high-altitude long-endurance solar-poweredaircraft, which are developed to operate at stratospheric altitudes,such as Zephyer7 and Helios. The cruise altitude of Zephyer7 isabout 20 km, and the highest altitude of Helios is about 29.7 km.So it is appropriate to assume the aircraft can be operated at thealtitude from 5 km to near 30 km.

The initial altitudes are supposed to be hinit ¼ ½5;10;15;20� km,the duration of solar irradiation is set as 2 h, and the other param-eters in simulation are the same with Section 4.3, the influences ofthe initial altitude on the performance of aircraft are shown inFigs. 14 and 15.

It can be seen from Fig. 14 that the higher initial altitude, theless flight endurance can be achieved under the same duration ofsolar irradiation for the method of energy stored by gravitationalpotential. The total flight endurance of solar-powered aircraft are12.8 h, 8.6 h, 5.6 h and 3.7 h when the initial altitude are 5 km,10 km, 15 km and 20 km, respectively. The reason can also befound from the changes of pitch angles, as shown in Fig. 15. Themaximum pitch angles are about 25.3�, 17.6�, 11.1� and 6.6� foreach case at the beginning of solar irradiation, and the maximumminus pitch angles are about �2.79�, �3.27�, �3.49�, and �3.60�for each case at the beginning of no solar irradiation. These meanthat the less and less vertical distance can be increased for the air-craft under the same duration of solar irradiation, and the more

Fig. 15. The compare of pitch angle with different initial altitude.

Fig. 16. The changes of EEF with different initial altitude.

994 X.-Z. Gao et al. / Energy Conversion and Management 76 (2013) 986–995

and more vertical distance will be descended without solar irradi-ation under the same length of time.

However, for its counterpart, the method of energy stored byrechargeable battery, the total flight endurance of solar-poweredaircraft are 11.2 h, 8.8 h, 6.5 h and 4.7 h when the initial altitudeare 5 km, 10 km, 15 km and 20 km, respectively, according to thecalculation of Eq. (13). The changes of the ratio of EEF, i.e., RKp withthe initial altitude are shown in Fig. 16. It can be found from thisfigure that the RKp is greater than 1 when the initial altitude is

Fig. 17. The phase-plane of the duration of solar irradiation and initial altitude.

lower than about 10 km, and less than 1 when the initial altitudeis higher than 10 km, which mean that the efficiency of energystored by gravitational potential is only greater than that of energystored by rechargeable battery when the initial altitude is lowerthan 10 km.

As analyzed in Section 5.1, the duration of solar irradiation alsohas a great influence on the ratio of EFF, thus, it is valuable to de-scribe the region where the RKp is greater than 1 in the phase planeof the duration of solar irradiation and initial altitude, as shown inFig. 17. This figure give an intuitionistic description about the influ-ence of the duration of solar irradiation and initial altitude on theratio of EFF, RKp. It can be found that the longer the duration ofsolar irradiation and the higher the initial altitude, the smallerthe ratio of EFF, as analyzed above. The red line in Fig. 17 indicatesthe dividing line of the region where the ratio of EFF is greater than1 or less than 1. On the left side of the red line, the ratio of EFF isgreater than 1, and on the right side of the red line, the ratio ofEFF is less than 1. This figure can provide a useful guidance forthe design of an appropriate method to store energy.

6. Conclusions

The main aim of this paper is to discuss the equivalence ofgravitational potential and rechargeable battery for solar-poweredaircraft from the energy storage’s point of view, which have asignificant sense for the application of energy storage usinggravitational potential.

In order to answer the question in which situation it is advan-tageous to use the method of energy stored in gravitational poten-tial, and in which situation the other method is preferred. Fourkinds of factors have been discussed to compare the two methodof energy storage: the duration of solar irradiation, the chargingrate, the energy density of rechargeable battery and the initial alti-tude of aircraft.

The simulation results show that the efficiency of method tostore energy by gravitational potential is lower and lower withthe increase of the duration of solar irradiation, the charging rate,and energy density. It is only preferred when the initial altitudeis low and the solar irradiation is short.

This work can provide some governing principles for the solar-powered aircraft to achieve the unlimited endurance flight withminimal or even zero rechargeable battery. It can be expected thatthe endurance performance of solar-powered aircraft will begreatly improved by the application of energy storage using grav-itational potential.

There also needs a careful balance to comprehensively considerthe influence of the charging rate, gravitational potential storedand the weight of rechargeable batteries, since each factor areinternally coupled in practical, for example, the higher the charg-ing rate, the more weight of energy management system, and theaircrafts have to pay more energy to lift it. However, this problemis out of the scope of this paper, since the main aim of this paper isto investigate the equivalence of gravitational potential andrechargeable battery. This issue will be carefully considered inthe future.

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