Applied Mathematical Sciences, Vol. 4, 2010, no. 76, 3761 - 3778

Periodic Optimal Flight of Solar Aircraft

with Unlimited Endurance Performance

G. Sachs, J. Lenz and F. Holzapfel

Institute of Flight System Dynamics Technische Universität München

Boltzmannstr. 15, 85747 Garching, Germany sachs@tum.de

Abstract

Solar aircraft offer the possibility of an unlimited endurance. This can be achieved if the energy required for the flight at night is stored in batteries. The goal of an unlimited endurance performance is treated as a trajectory optimization problem. This is a problem of periodic optimal flight because the trajectory is repeated after a complete day-night cycle. A trajectory optimization is performed with the objective of minimizing the required battery capacity. Thus, the weight penalty due to the batteries can be kept as small as possible. Realistic mathematical models of the energy management system, the available solar radiation and the dynamics of the aircraft are applied. With the use of an efficient optimization method, solutions of the periodic trajectory optimization problem are generated. Keywords: unlimited endurance, solar aircraft, periodic trajectory optimization

Nomenclature

DC drag coefficient LC lift coefficient

E solar flux D drag g acceleration due to gravity h altitude J cost function L lift m mass N day number n load factor, interval number n vector

3762 G. Sachs, J. Lenz and F. Holzapfel P power

BQ battery charging state

Bq battery charging rate q dynamic pressure, 2)2/( Vq ρ= S reference area s range T thrust t time

iU control estimation ur control function V speed xr state function

Perx period

Sunz solar zenith angle

Sunθ sun elevation angle γ flight path angle δ control variable, solar declination angle η efficiency factor μ angle ρ density τ time, extinction factor ω hour angle 1 Introduction Solar aircraft have attained significant interest in recent years, and there are considerable research and development efforts in this field. A description of the history of solar aircraft and an outlook is given in [1, 2]. Various vehicle concepts and aircraft types have been investigated or developed and realized, e.g. [3 - 7]. These range from small-size configurations in terms of model airplanes to very large aerial vehicles [2, 8]. Solar-powered flight poses challenging problems in different areas, like solar and battery technologies with the goal to improve the efficiency or materials and structures with the goal to achieve extreme lightweight constructions [1, 5]. The progress in these technologies contributes to the increasing interest in solar-powered airplanes. The flight performance of solar-powered aircraft has been successively improved from the beginning of the existence of this type of aerial vehicle [2, 9, 10]. After the capability of level flight of a manned solar airplane has been achieved, other successes relating to solar-powered vehicles were attained [2, 11]. Vehicles for different applications and purposes were constructed and built, like solar-powered motor gliders, high altitude long endurance platforms, etc. (e.g. [12, 13]).

Periodic optimal flight of solar aircraft 3763 The solar airplane type of interest in this paper is a vehicle which offers the possibility of an unlimited flight endurance. Various configurations of this type are subject of development and realization efforts. Unmanned aerial vehicles aiming for a long endurance flight capability were planned or have been constructed [14 - 16]. Concerning manned vehicles, an airplane enabling a flight capability for a flight around the world is under construction [17]. For the purpose of an unlimited flight endurance, the solar-powered vehicle is considered to be equipped with batteries which can store enough energy for the flight in the night when no solar radiation is available, e.g. [2, 8]. During daytime, the energy from the sun radiation is used to propel the aircraft as well as to charge the batteries with solar energy required for the flight at night. The described endurance flight method can be regarded as a problem of periodic optimal control. This is because after completing a day-night cycle a new one with the same properties follows the next day. Therefore, it is sufficient to optimize a single day-night cycle which forms the basic constituent of periodic optimal flight yielding an unlimited endurance performance. An important aspect for solar-powered aircraft of the type under consideration is the required capacity of the batteries for storing electrical energy, e.g. [17]. The reason is that the batteries yield a weight penalty. An appropriate control of the trajectory is a means to contribute to the goal of keeping the weight penalty of the batteries as small as possible. This is an issue of trajectory optimization [18]. It is the purpose of this paper to present results on the described periodic optimal flight problem, using an efficient optimization procedure. The optimization goal is to minimize the required capacity of the batteries and, thus, to provide a contribution for keeping the related weight penalty as small as possible. 2 Energy Management Solar-powered aircraft basically show an aerodynamic configuration which comprises components (wing, fuselage, etc.) comparable to those of other aerial vehicles. But, there are also unique features, posing new technological challenges. This is relating to the energy management system which uses solar power from the sun radiation for propelling the vehicle and for supplying other elements requiring electrical power [1, 8]. The energy management and the propulsion systems are schematically shown in Fig. 1, yielding the following elements: – Solar cells (20 % efficiency) – Maximum power point tracker MPPT (95 % efficiency) – Electric lines (99.5 % efficiency) – Converter (98.5 % efficiency) – Battery manager (99.5 % efficiency) – Batteries (96.5 % efficiency)

3764 G. Sachs, J. Lenz and F. Holzapfel – Propulsion system (72 % efficiency) The solar cells are essential for the energy management system because they collect the energy from the sun radiation. Efficiency achievable with solar cells is a technological issue which is subject of significant research efforts [8]. For the present investigation, an efficiency of 20 % is assumed. For the batteries, a high energy density is desirable. This is also an issue of current technological development efforts [8]. Modelling of the batteries for the trajectory optimization problem involves several points. The batteries are charged at a constant rate until 90 % of their capacity is reached. The final 10 % are charged at a decreasing rate. Furthermore, it is assumed that the batteries cannot be completely discharged. Rather, there is a remainder which cannot be used. The following relation is applied for the modelling the charging process

)( BBB QqQ =& (1) The propulsion system consists of an electrical motor and a propeller. Modelling of this system, involving the power and the efficiency factors of the motor and the propeller, yields for the thrust

VPT P

MPδηη max= (2)

The maximum power which the electrical motor provides is regarded as constant for the speed and altitude ranges of interest. The efficiencies of the motor and the propeller are assumed to have the following values – 93 % efficiency for motor – 77 % efficiency for propeller In addition to the electrical motor, there are other components of the aircraft consuming electrical power. For these components, a constant power supply is supposed to be required which is taken from the solar cells and/or from the batteries. This is also included in the mathematical model of the energy management system. 3 Sun Radiation Modelling The available solar energy is dependent on a variety of factors, including the time of day, the altitude, the latitude, the longitude and the season. These effects are accounted for in the sun radiation model which was developed with reference made to [19 - 21].

Periodic optimal flight of solar aircraft 3765 As a starting point, the average extraterrestrial solar flux avexSolE ,, is introduced which

amounts to 2,, W/m367.1==avexSolE . Due to the Earth’s elliptical orbit around the Sun, the

extraterrestrial solar flux depends on the day of the year, yielding

{ })]8,82(sin[034,01,,, NxEE PeravexSolexSol −⋅+⋅= (3) where 25,365/2π=Perx is the period and N the day-number. At a given position on the surface of the Earth, the Sun appears under an elevation angle,

Sunθ . The elevation angle which is dependent on the time of day is measured with reference to the horizontal plane, yielding 0=Sunθ at sunrise and sunset. It is related to the solar zenith-angle SunΖ (Fig. 2) according to

SunSun Ζπθ −= 2/ (4) Using the quantities δ , ω and μ depicted in Fig. 3, the elevation angle can be expressed as

ωδμδμΖθ coscoscossinsincossin ⋅⋅+⋅=⋅== horSunSunSun nn vv (5)

There are various effects on these quantities, concerning the period, the day-number etc. They are appropriately accounted for. In a next step, the extinction of the solar radiation due to Rayleigh-scattering and absorption by water vapour are determined. This yields the total extinction factor

WaterRayleightotal τττ ⋅= (6) where both factors depend on the air mass related to the path of a light ray through the atmosphere. The final outcome is the radiation on a horizontal surface

SunexSolTotalSol,hor EE θτ sin,= (7) This is the sun radiation available for a solar aircraft. A numerical evaluation showing the effects of the time of day and the altitude on the available sun radiation is presented in Fig. 4. The data pertain to a place in the Northern hemisphere and to a summer day. The effects of the time of day and the latitude are graphically presented for a summer day in Fig. 5.

3766 G. Sachs, J. Lenz and F. Holzapfel 4 Vehicle Dynamics Modelling For describing the motion of the vehicle, a model based on point mass dynamics can be used, yielding (Fig. 6)

γγ

γγ

γ

cossin

cos

sin

VsVh

Vg

mVL

gm

DTV

==

−=

−−

=

&

&

&

&

(8)

The aerodynamic forces which are the drag and lift are modelled as

SVCL

SVCD

L

D2

2

)2/(

)2/(

ρ

ρ

=

= (9)

where

)( LDD CCC = (10) Vehicle data relevant for the performance characteristics are given in Table 1.

Data

Mass m [kg] 1650.0 Wing Loading Sm / [kg/m2] 0.119 Maximum Lift-to-Drag Ratio max)/( DL CC 32.9 Lift Coefficient Associated with max)/( DL CC 0.82 Motor Power maxP [kW] 25.0

Table 1 Vehicle data

5 Formulation of Periodic Optimal Control Problem 5.1 Characteristics of Optimal Unlimited-Endurance Flight The optimization goal is to determine the periodic optimal flight path which yields the minimum of the weight penalty due to the batteries storing electrical energy necessary for the flight in the night. For this purpose, the required capacity of the batteries is considered a measure for their weight, yielding for the cost function

Periodic optimal flight of solar aircraft 3767

BQJ = (11) The periodic optimal flight path shows a form which is schematically depicted in Fig. 7 There are three phases which can be regarded as characteristic elements:

1) Climb to maximum altitude 2) Descent to minimum altitude 3) Flight at lowest altitude

Phase 1 begins at a time when the sufficient radiation of the sun becomes available in the morning. Since the sun radiation is continually growing until reaching its maximum (as shown in Figs 4 and 5), the solar power available for the aircraft is increasing correspondingly. This power is used for the electrical motor to propel the vehicle and for all other electrical components of the aircraft as well as for charging the batteries. The optimal distribution of the available solar power between the motor, the other electrical components and the batteries is subject of the optimization in phase 1. The maximum altitude is supposed to be limited due to breathing requirements and safety considerations for the pilot wearing a pressure suit. After having reached the maximum altitude, phase 2 begins where the aircraft performs a descent. In the course of phase 2, the available sun radiation decreases until becoming zero. This means that there is a first part of phase 2 during which electrical power is still available from the sun radiation, yielding a powered descent. In the second part of phase 2, the sun radiation is zero so that the descent is performed in terms of a glide. Phase 2 ends when the aircraft approaches the lowest altitude. Thereafter, phase 3 begins. The flight in phase 3 takes place at a lower altitude limit which is given by a safety margin to the terrain. This phase is basically a steady-state cruise where the controls as well as the speed and the altitude take on practically constant values. After phase 3 has ended, a new day-night cycle begins showing the same properties of the state and control variables as the described one. According to the periodicity of the flight, the following periodic boundary conditions hold

)0()(

)0()(

)0()(

)0()(

hth

t

VtV

QtQ

cyc

cyc

cyc

BcycB

=

=

=

=

γγ (12)

The total time of the three phases depicted in Fig. 7 is the cycle time denoting the period of a complete day-night cycle after which a new one with the same properties follows. The cycle time cyct is not a fixed value equalling 24 h of a full day. Rather, it is the time after which the aircraft has reached a position in terms of )( cycts at which the sun radiation equals the value at

3768 G. Sachs, J. Lenz and F. Holzapfel the beginning of the flight path cycle at 0=t . The cycle time cyct is subject of the overall optimization. 5.2 Periodic Optimal Control Problem Formulation The periodic optimal control problem can be formulated as to find a state function

5],0[:),,,,( RtshVQx cyc →= γr (13)

a control function

3],0[:),,( RtCqu cycLPB →= δr (14)

and the cycle time cyct which minimize the cost function

( ) Min)(max →= cycB tQJ (15) subject to a set of differential equations

))(),(()( tutxftx =& (16) given by Eqs. (1) and (8) describing the battery charging and the vehicle dynamics as well as to state constraints

max

maxmin

maxmin

0 qqnnnhhh

≤≤≤≤≤≤

(17)

to control constraints

3,2,1,max,min, =≤≤ iuuu iii (18) and to periodic boundary conditions given in Eq. (12). For solving the described optimal control problem, efficient optimization methods and computational techniques are required which are capable of coping with complex functional relationships including various kinds of constraints. The mathematical procedure used to solve the range maximization problem is a direct optimization method. The numerical investigation was performed using the parameterization optimization technique ALTOS with the graphical environment GESOP [22, 23]. The whole modelling part has been completed in MATLAB/SIMULINK. This concerns the right hand side and the cost function. Furthermore, the constraints and the model initialization

Periodic optimal flight of solar aircraft 3769 are specified in MATLAB/SIMULINK. In a second step, the model is compiled into C-code using the MATLAB Real-Time Workshop. The compiled code is then edited using the graphical user interface in GESOP to be handed over to the optimization method PROMIS. PROMIS is able to combine control parameterization with multiple shooting and features automatic function scaling. The optimization itself is performed by SNOPT, software for large scale nonlinear programming [24]. Finally, the optimization results are exported to MATLAB for plotting purposes. For dealing with the optimization problem, the time of a complete day-night cycle,

cycttt <<= 00 , is subdivided into intervals (Fig. 8), yielding a state grid given by

cycn tsm=<<<= τττ ...0 10 (19)

In order to increase stability and ease initial guesses, the integration process is restarted at each node of the state grid, leading to defects of the state vector at the nodes. The state grid comprises smn intervals where, apart from the last node, at each node initial conditions for the states are given. The components of the control vector

))(),(),(()( tCttqt LPB δ=u (20) are parameterized, yielding

),()( tUtu ii p= (21) where ),( tU i p is an estimation of the i-th control using a piecewise linear function or a higher order spline (Fig. 8). Equality and inequality path constraints are checked at the nodes of an additional path constraint grid. A solution of the optimal control problem is obtained as soon as the change of the cost function is below a specified value and no defects are present at the state grid nodes. Furthermore, all boundary conditions and path constraints have to be matched. The stopping criterion (optimization tolerance or convergence criterion) for the optimization process is the Karush-Kuhn-Tucker (KKT) condition. The major optimality tolerance is set to a specified value, with reference to the cost function normalized to one. The constraint tolerance or constraint violation is also user defined. For this purpose, appropriate values are specified, with the related variables normalized to the magnitude of one. These values are checked at the control nodes.

3770 G. Sachs, J. Lenz and F. Holzapfel 6 Periodic Optimal Trajectory Results Characteristic features of the optimized trajectory for a complete day-night cycle can be identified when examining the altitude profile. The altitude profile of a periodic optimal flight yielding the minimum of the battery weight penalty is presented in Fig. 9. A result of the optimization is the optimal cycle time for a complete day-night cycle, amounting to h22.23=cyct . Thus, the optimal cycle time is less than 24 h of a full day. This is due to the eastward movement of the aircraft and the rotation of the Earth which is accounted for in the trajectory optimization. In Fig. 9, the three characteristic phases described in the preceding section are visible, marked by Nos. 1 to 3. When sufficient sun radiation is available in the morning, phase 1 begins during which the aircraft performs a climb to the maximum altitude (admissible or possible). After having reached the maximum altitude, phase 2 of the optimized trajectory takes place. It is a descent to approach the lowest altitude. In the first part, it is a powered descent until the sun radiation vanishes. This is followed by a glide, without engine thrust. A third characteristic feature of the optimized trajectory identified as phase 3 is that a flight portion in the night takes place at the lowest altitude admissible. Since there is no sun radiation, the energy required for the propulsion system is taken from the batteries. The constancy in the altitude in phase 3 suggests that this is basically a steady-state flight. The optimal time history of the speed which is presented in Fig. 10 shows a behaviour which corresponds with that of the altitude. This is relating to the phases 1 and 2 where the speed is increasing and decreasing, respectively, as well as to phase 3 showing a practically constant value. The constancy of the speed in phase 3 confirms the steady-state nature of this flight portion. Results on the optimal behaviour of the controls are graphically presented in Figs. 11 and 12. In Fig. 11, the battery charging rate and the motor control setting are shown in terms of the respective electrical power quantities. The optimal time history of the lift coefficient is depicted in Fig. 12. Fig. 11 reveals interesting features of the optimal power management for supplying the motor with electrical power and for charging the batteries during the availability of the sun radiation. First, there is an optimal distribution of the used power between the motor for propelling the vehicle (including all other aircraft consumers requiring electrical power) and the batteries. This is a result of the overall optimization. Second, there is more solar power available as needed by the electrical power consumers of the aircraft and for charging the batteries. Thus there is a certain amount of solar power that is not used. A characteristic feature of the optimized lift coefficient control (Fig. 12) is that it shows only small changes throughout the whole cycle. In particular, there is a practically constant part which is related to phases 2 and 3. The constancy in the lift coefficient in phase 3 is in

Periodic optimal flight of solar aircraft 3771 accordance with the behaviour of the motion variables and the steady-state nature of this phase as described above. 7 Conclusions A solar-powered aircraft is dealt with which shows an unlimited endurance capability. This is possible due to storing solar energy in batteries such that the flight can be maintained during the whole night until the following day when the sun radiation is again available. Thus, an unlimited endurance performance can be achieved. During the day, the energy from sun radiation is used to propel the aircraft and to charge the batteries. The described endurance performance goal is treated in terms of a trajectory optimization problem. The trajectory of a complete day-night cycle is optimized with the goal to minimize the required battery capacity and, thus, the related weight penalty. Realistic mathematical models of the energy management system (solar cells, batteries, electrical motor, etc.) and the dynamics of the aircraft are applied, and an efficient optimization method is used to generate results. It turns out that there are characteristic features of the optimal trajectory. During daytime, a climb to the highest altitude is performed. Thereafter, a descent follows to reach the lowest altitude. At the lowest altitude, a practically steady-state flight is conducted. This takes place during a portion of the flight in the night, before the next day-night cycle begins. References [1] R. Voit-Nitschmann: Solar- und Elektroflugzeuge – Geschichte und Zukunft.

WechselWirkungen, Jahrbuch aus Lehre und Forschung der Universität Stuttgart 2001, pp. 88-99, 2001.

[2] A. Noth: History of Solar flight. Autonomous Systems Lab, Swiss Federal Institute of Technology, Zürich, October 2006.

[3] W. Scholz, R. Voit-Nitschmann and M. Rehmet: icaré – Solatflugzeug der universität Stuttgart. Idaflieg-Berichtsheft XXI, pp. 199-210, 1995.

[4] M. Rehmet, R. Voit-Nitschmann and B. Kröplin: Eine Methode zur Auslegung von Solarflugzeugen. Idaflieg-Berichtsheft XXIII, pp. 1-35, 1997.

[5] A. Noth: Design of Solar Powered Airplanes for Continuous Flight. PhD Thesis, Autonomous Systems Lab, ETH Zürich, Switzerland, 2008.

[6] G. Romeo, G. Frulla, E. Cestino, and G. Corsino. Heliplat : Design, Aerodynamic, Structural Analysis of Long-Endurance Solar-Powered Stratospheric Platform. Journal of Aircraft, 41(6):1505–1520, 2004.

[7] E. Rizzo and A. Frediani. A Model for Solar Powered Aircraft Preliminary Design. In Proc. of International Conference on Computational & Experimental Engineering & Sciences 04, volume 1, pages 39–54, Madeira, Portugal, July 2004.

[8] H. Ross: Solarangetriebene Flugzeuge = The True All Electric Aircraft – Eine Übersicht. DGLR-2006-150, 2006.

[9] R.J. Boucher: Sunrise, the World's First Solar-Powered Airplane. Journal of Aircraft, Vol. 22, No. 10, pp. 840-846, 1985.

[10] R.J. Boucher: History of Solar Flight. SAE, and ASME, 20th Joint Propulsion Conference, Cincinnati, OH, June 11-13, AIAA-1984-1429, 1984.

3772 G. Sachs, J. Lenz and F. Holzapfel [11] P. B. MacCready, P. B. S. Lissaman, W. R. Morgan, and J. D. Burke. Sun-Powered

Aircraft Designs. Journal of Aircraft, Vol. 20, No.6, pp.487–493, 1983. [12] Icare at Uni Stuttgart. http://www.ifb.uni-stuttgart.de/ icare/Englisch/ icare2eng.htm. [13] Solar-Power Research at NASA Dryden. http://trc.dfrc.nasa.gov/Newsroom/FactSheets/

PDF/FS-054-DFRC.pdf. [14] Noth, A., Siegwart, R. and Engel, W. Autonomous Solar UAV for Sustainable Flight. In:

Advances in Unmanned Aerial Vehicles, State of the Art and the Road to Autonomy, Ed. Kimon P. Valavanis, Springer Verlag, pp. 377-405, 2007.

[15] B. Keidel: Auslegung und Simulation von hochfliegenden, dauerhaft stationierbaren Solardrohnen. Dissertation, Technische Universität München, 2000.

[16] B. Keidel and G. Sachs: Development of a Solar Powered Aircraft for High Altitude Long Endurance Flight. UNCONF '97 „Unconventional Flight Analysis“, Book 1, Selected Paper of the First International Conference on Unconventional Flight, Published by the Department of Aircraft at Technical University of Budapest and Ships and R-Group Ltd, Budapest, ISBN 963 420 699 9, pp. 55-65, 1999.

[17] H. Ross: “Fly around the World with a Solar Powered Airplane”. ICAS 2008-10.11.1 / AIAA 2008-8954, 2008.

[18] G. Sachs, J. Lenz, O. da Costa und H. Ross: Solar Aircraft Trajectory Optimization for Performance Improvement. Jahrbuch 2007 der Deutschen Gesellschaft für Luft- und Raumfahrt, Vol. VI, ISSN 0070-4083, pp. 3589-3596, 2008.

[19] C.-J. Winter, R.L. Sizman and L.L. Vant-Hull: Solar Power Plants. New York: Springer-Verlag, 1991.

[20] F. Kasten and A.T. Young: Revised Optical Air Mass Tables and Approximation Formula. Applied Optics, 28 (1989) 4735-4738.

[21] J.A. Duffie and W.A. Beckmann: Solar Engineering of Thermal Processes. John Wiley & Sons, 1980.

[22] N.N., ALTOS – Software User Manual, Institut für Flugmechanik und Regelung, University of Stuttgart, August 1996.

[23] N.N., GESOP (Graphical Environment for Simulation and Optimization), Softwaresystem für Bahnoptimierung, Institut für Robotik und Systemdynamik, DLR, Oberpfaffenhofen, 1993.

[24] Gill, P.E., Murray, W., Saunders, A.: User's Guide for SNOPT 7.1: a Fortran Package for Large-Scale Nonlinear Programming. Report NA 05-2, Department of Mathematics, University of California, San Diego, 2005.

Periodic optimal flight of solar aircraft 3773

MPPTη≈0.95

On-Board Consumption750W

Battery withBattery Manager

η≈0.96Converterη≈0.985Engineη≈0.93

η≈0.2

Solar Cell

η≈0.77

Propeller

Fig. 1 Energy management and propulsion systems

hornv

SunnvSunZ

Sunθ

Fig. 2 Solar zenith-angle, SunΖ , and elevation angle, Sunθ ,

3774 G. Sachs, J. Lenz and F. Holzapfel

Fig. 3 Geocentric frame

Fig. 4 Effects of daytime and altitude on solar radiation (Oberpfaffenhofen airport, Germany, 155th day of the year)

hornv

Sun nv

μ

0=λ[west] λ

ω

δ

x1

z1

y1

Sun Z

Periodic optimal flight of solar aircraft 3775

Fig. 5 Effects of daytime and latitude on sun radiation (155th day of the year)

Fig. 6 Forces at solar aircraft

Fig. 7 Scheme of periodic optimal trajectory for complete day-night cycle

3776 G. Sachs, J. Lenz and F. Holzapfel

Fig. 8 Grids used in optimization computation

Fig. 9 Optimal altitude time history for minimizing required battery capacity (curve begin: 7.45 h), optimal cycle time: h22.23=cyct

Periodic optimal flight of solar aircraft 3777

Fig. 10 Optimal speed time history for minimizing required battery capacity

Fig. 11 Optimal power management during availability of sun radiation for minimizing required battery capacity

3778 G. Sachs, J. Lenz and F. Holzapfel

Fig. 12 Optimal lift coefficient time history for minimizing required battery capacity

Received: June, 2008