SOLAR POWER PROFILE PREDICTION FOR LOW EARTH ORBIT SATELLITES

Jural Mekaikal Jue 2009, No. 28, 1-15 SOLAR POWER PROFILE PREDICTION FOR LOW EARTH ORBIT SATELLITES Chow Ki Paw, Reugath Varatharajoo* Departet of Aerospace Egieerig Uiversiti Putra Malaysia 43400 Serdag, Selagor ABSTRACT This paper presets the developet of algoriths usig MATLAB codes to predict the i-orbit satellite power profile. Satellite power requireet is a crucial paraeter for its i-orbit operatio. I this case, it is best to idetify the power profile which idicates the aout of power geerated over a tie frae of a orbit. However, the deteriatio of the satellite power profile requires substatial aout of efforts to copute ad this is largely due to the coplex uerical treatets. Orbital paraeters are deeed to affect the deteriatio of the satellite power profile. Therefore, a coputer progra has bee writte to solve all the goverig equatios leadig to the satellite power profile predictio for a orbit ad evetually for a year. The power profile validatio was doe aalytically usig the goverig equatios before the profile is geerated through the coputer codes. This work cotributes greatly towards the sall satellite (<100 kg) power sizig effort ad eliiates the eed of a costly coercial power sizig software. Keywords: Orbit, power profile, sall satellite desig. 1.0 INTRODUCTION Satellites are spacecraft that receive sigals ad sed the back to Earth. Satellite s ca be see operatig for global telecouicatio idustry which provides couicatio coverage o Earth [1]. To eable various issios, Low Earth Orbit (LEO) satellites are required to fulfill their requireets o power cosuptio to sustai their life i space. As typical oes use photovoltaic cells to geerate solar power, the presece of Su phase is iportat. Apart fro the issue of havig the Su to geerate power, a greater cocer shall also be placed o its eclipse duratio, whe geeratio of solar power is ipossible. *Correspodig author : reu99@gx.de 1

Jural Mekaikal, Jue 2009 Hece, the positio of a satellite i a orbit dictates its power profile. I ay tie withi its life, a solar powered satellite ay fall ito either the Su or eclipse regio [2]. Satellites positio ad their orietatio i space are well represeted usig positio vectors. A vector is represeted i for of a atrix. Spatial coputatios usig rotatio vectors ad atrices provide iforatio regardig orietatio of solar paels towards the Su [3]. The orietatio deduced as Su agle β therefore dictates the istataeous icidet power P over a orbit period [4]. Correctios are iposed to the value P accordigly wheever the oral vector of solar pael surface faces agaist the sulight (β > 90 o ). Also, the power profile is ot without soe reductio due to iefficiecies. The efficiecies for various types of solar paels were evaluated accordig to aterials ad are give i several refereces [5, 6, 7, 8]. Power profile aalysis ay be difficult whe it cosiders large duratio of orbits or eve orbits for a year. Hece, this paper will explai the developet of a coputer progra which perfors those aalyses [9]. This should overcoe the atheatical coplexity ad save cosiderable work tie ad effort. 2.0 MATHEMATICAL MODELS AND ALGORITHMS This sectio describes various atheatical odels ad forulas, which are ipleeted i the power predictio algoriths. The itroductio of rotatio vectors, atrices ad goverig equatios ito the algoriths is show towards the ed of this sectio. 2.1 Goverig Equatios A list of goverig equatios is give as follows: µ i. Satellite agular velocity; 3 e ω = (1) S RS 1 2 ii. Satellite orbit period; T o 2π = (2) ω S iii. Earth s agular chage about the Su per orbit;; To ϕ = 2 π Ty (3) T iv. Satellite s agular chage per iute; ϕ = sat 2 π To (4) v. Su agle; cos β i, = Np i Ssat, (5) 2

Jural Mekaikal, Deceber 2008 β i, a dot product of i-th ad -th eleets of vectors Np with S sat, respectively. The idex i is a iteger which rages fro 1 to p. β rages fro zero to π/2. vi. Istataeous icidet power; P = η y Ap S cos β (6) vii. Eclipse etry tie; T viii. Eclipse exit tie; T E, etry E, exit 2.2 Rotatio Vectors A list of rotatio vectors is give as follows: ω ste = TEpeak 2 (7) ω ste = TEpeak + 2 (8) sit i. Rotatio vector of Earth about the Su; E = cost (9) 0 T, the -th eleet of a array of Earth s rotatio about the Su ad rages betwee zero to 2π for every steps of ϕ. cosu ii. Rotatio vector of satellite about the Earth; B = siu (10) 0 U, the -th eleet of a array of satellite s rotatio about the Earth, rages fro zero to 2π for every steps of ϕ sat. iii. First rotatio about Z axis; 1 0 0 A Ω = 0 cos( Ω) si( Ω) (11) 0 si( Ω) cos( Ω) iv. Secod rotatio about X axis; A 1 = 0 0 0 cos( i ) si( i ) 0 si( i cos( i ) ) (12) 3

Jural Mekaikal, Jue 2009 v. Third rotatio about X axis; A i 1 = 0 0 0 cos( i) si( i) 0 si( i) cos( i) (13) vi. Rotatio of the Su about the satellite i SCO frae; S sat, siθ = 0 cosθ s, s, cosθ 0 siθ s, s, 0 1 Se 0 (14) S sat,, the -th eleet of S sat array. SCO frae axes are defied as follows: a. X-axis poits alog flight directio b. Y-axis is oral to flight directio c. Z-axis poits towards ceter of the Earth (adir) 2.3 Algoriths The GUI allows four ai optios to coece satellite power profile aalyses. Those optios are to copute the satellite solar power geeratio; preset the power profile per orbit, ad per year; as well as the power requireet per year. Upo activatio of the COMPUTE pushbutto, the progra will perfor the followig procedures i steps: Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: The progra calls for Algorith 1 (see Figure. 1) ad display the iforatio i MATLAB Workspace. The progra calls for Algorith 2 (see Figure. 2) ad display the iforatio i MATLAB Workspace. The progra calls for Algorith 3 (see Figure. 3) ad display the iforatio i MATLAB Workspace. The progra iposes Earth s rotatio about the Su for 10 to 360 with icreet of 10. For each icreet, the followig steps are the followed: i. The Su Phase is set fro 1 to its axiu cout (obtaied fro Algorith 3) with steps of 100 uit. ii. For each step, the progra calls for Algorith 4 (see Figure. 4) ad the Algorith 5 (see Figure. 5). Iforatio fro the both algoriths is displayed i MATLAB Workspace. The progra calls for Algorith 6 ad display the iforatio i MATLAB Workspace. The progra saves all iforatio i MATLAB Workspace ito.mat file. Upo activatio of the POWER PROFILE PER YEAR pushbutto, the progra will perfor the followig procedures i steps: Step 1: The progra attepts to load previous iforatio. Step 2: i. If iforatio is ot available, the error warig is displayed. 4

Jural Mekaikal, Deceber 2008 ii. Else, the progra produces a surface plot for the power profile per year. The figure is defied with the followig particulars: a. X-axis labeled Duratio of Orbit [i] b. Y-axis labeled Rotatio about the Su [ ] c. Z-axis labeled Power [W] d. Title labeled Geerated Solar Power for a Year Upo activatio of the POWER REQUIREMENT PER YEAR pushbutto, the progra will perfor the followig procedures i steps: Step 1: Progra will attept to load previous iforatio. Step 2: i. If iforatio is ot available, the error warig is displayed. ii. Else, the progra will produce a lie plot for iiu power requireet per year. iii. The figure is defied with the followig particulars: a. X-axis labeled Rotatio about the Su [ ] b. Y-axis labeled Power [W] c. Title labeled Power Requireet for a Year Upo activatio of the POWER PROFILE PER ORBIT pushbutto, the progra will perfor the followig procedures: Step 1: Progra will attept to load previous iforatio. Step 2: i. If iforatio is ot available, the error warig is displayed. ii. Else, the progra will request user for positio of the Earth rotatio about the Su. The it will produce a lie plot for the power profile per orbit. The figure will be defied with the followig particulars: a. X-axis labeled Duratio of Orbit [i] b. Y-axis labeled Power [W] c. Title labeled Satellite Power Profile Per Orbit All the algoriths for the i-orbit satellite power profile geeratio progra are show i Figures 1 to 6. h = str2double(iputdlg( Orbit altitude, Iput )) uit [k] i = str2double(iputdlg( Orbit icliatio, Iput )) uit [ ] η PV = str2double(iputdlg( PV efficiecy, Iput )) η T = str2double(iputdlg( Teperature effect, Iput )) η desig = str2double(iputdlg( Desig & Assebly, Iput )) η S = str2double(iputdlg( Shadowig factor, Iput )) η = η η η η total solar pael efficiecies y PV S T desig Call Algorith 7 Figure 1 : Algorith 1 requests user for iforatio regardig the satellite s orbit paraeters as well as additioal data. 5

Jural Mekaikal, Jue 2009 R = R + ( h 1000) S E radius of orbit Equatio (1) Equatio (2) satellite agular velocity satellite s orbit period Figure 2: Algorith 2 coputes orbit period of the satellite. Equatio (3) Earth s agular chage about the Su per orbit φ The value φ is used to create a arithetic sequece with eleets T : T = φ : φ : 2π T = T 1, T 2, T 3 T ax = φ, 2φ, 3φ 2π Equatio (9) i = -23.5 rotatig vectors for the Earth about the Su E Figure 3: Algorith 3 deteries rotatio of the Earth about the Su. icliatio of the Earth about the ecliptic plae: Equatio (4) satellite s agular chage about the Earth per iute φ sat The value φ sat is used to create a arithetic sequece with eleets U : U = φ sat : φ sat : 2π U = U 1, U 2, U 3 U ax = φ sat, 2φ sat, 3φ sat... 2π Each uber i this sequece represets the shift i satellite s positio as it rotate about the Earth for each oe iute. Equatio (10) rotatig vector for the satellite about the Earth B The referece plae for all rotatig vectors is trasfored to ecliptic plae by eployig trasforatio atrices show by Eqs. (11) to (13). S S p e = A A AΩ B B with referece to the ecliptic plae i i = A A AΩ E E with referece to the ecliptic plae Equatio (14) The rotatig vector for the satellite about the Su S sat i the SCO frae Figure 4: Algorith 4 deteries rotatio of the Su about the satellite with referece to ecliptic plae 6

Jural Mekaikal, Deceber 2008 Equatio (5) Equatio (6) Su agle cos β The istataeous icidet power for all paels. Calculated power geerated for all pael is the totaled up at a give icidece. Ceter of eclipse per orbit is deteried fro usig for ad if coditio. This is doe by deteriig the tie whereby the dot product at a icidece is at axiu ceter of eclipse = ax( E B) 2 π 1 RE α = si 1 2 half agle eclipse plae RS Total eclipse agle is twice the value of α. α T E = ( T o ) eclipse tie π T = T T tie i Su phase su o E Eqs. (7) ad (8) eclipse etry ad exit ties The istataeous icidet power profile is set to zero for a orbit wheever ceter of eclipse T. T E, etry E, exit This algorith registers P as the power geerated over a aout of Su phase per orbit ad also the power requireet. The power requireet per orbit is obtaied by perforig the followig operatio: su of eergy per orbit PwrRqd = T o Figure 5: Algorith 5 aalyses the satellite s power profile for each orbit. Recall Algorith 5 the power geerated over the aout of Su phase P is allocated ito the duratio of Su phase. I (:, aout Su phase[diesioless]) = I(:, Tsu[i]) Eclipse phase is icluded ito the duratio of orbit by extedig the duratio ito a full orbit T o ad zero the power geerated P after T su. Figure 6: Algorith 6 fuctios by coputig power profile for a year. 7

Jural Mekaikal, Jue 2009 The algoriths show i Figures 1 to 6 are operated i sequece as show i the block diagras i Figure 7. START GUI Press COMPUTE pushbutto Iitiate Power Profile Aalysis Call algoriths for iput paraeters Algorith 1 Propt for h ad Algorith 7 i fro user + Call algoriths, i the followig sequece Algorith 2 Algorith 3 Algorith 4 Algorith 5 Perfor for loop fuctio to correct Ω for 0 < Ω 360, with steps of 10 + + Execute algorith Algorith 6 MATLAB Workspace saved as.mat file RETURN TO GUI Figure 7 : Flow chart shows process path to copute geerated power by pressig COMPUTE pushbutto 8

Jural Mekaikal, Deceber 2008 The flow chart i Figure 7 is a brief descriptio of data flow ad processes udertake by the progra to deterie the satellite power profile for a orbit ad a year. The display of the satellite power profiles is achieved by processes show i Figure 8. START GUI Press POWER PROFILE PER ORBIT or POWER PROFILE PER YEAR or POWER REQUIREMENT PER YEAR Attept to load previous iforatio YES Is the file available? NO * Plot Power Profile TERMINATE PROGRAM, WITH ERROR WARNING RETURN TO GUI Display Power Profile Figure 8 : Flow chart shows process path to display either power profile per orbit; power profile per year; or power requireet per year To display the satellite power profile per-orbit, the progra shall propt the user for the Earth s positio about the Su. This additioal process is ebedded i a block arked with a asterisk (*) i Figure 8. 9

Jural Mekaikal, Jue 2009 3.0 RESULTS AND DISCUSSION This sectio presets ad discusses the geerated i-orbit satellite power profile per orbit ad per year. The followig power profiles were the outcoe fro usig differet iput paraeters as idicated alog with the figures. Note that the results cosider a cubic satellite body with solar pael arrageets as show i Figure 9. -Y -Z -X +X +Z +Y Figure 9 : The arrows idicate each with 0.048 2 area ad a solar pael Efficiecy factor of 0.15 The power profile validatio was doe aalytically (Figure. 10 (a)) usig the all goverig equatios before the power profile is geerated through the coputer codes (Figure. 10 (b)). The peak power i both cases is the sae; thus, the coputer codes are validated. (a) (b) Figure 10: Satellite power profile per orbit at h = 500 k; i = 23.5 10

Jural Mekaikal, Deceber 2008 3.1 Power Profile Per Year Figures 11 (a), (b), (c) ad (d) are geerated as the result fro the usig differet altitude ad icliatio paraeters. Satellite power profile geeratio for a year provides iforatio o the aout of solar power geerated over the duratio of orbit i a year. The solar power geerated is quatified i Watts whereas other paraeters such as the duratio of orbit ad steps i a year are give i iutes ad degrees agle, respectively. (a) (b) (c) (d) Figure 11 : 3D Satellite Power Profile Geeratio siulated for a year at; (a) h = 500 k with i = 53 ; (b) h = 1, 000 k with i = 53 ; (c) h = 500 k with i = 23.5 ; ad (d) h = 500 k with i = 82.5 3.2 Power Requireet Per Year Havig the 3D power profile per year, the power requireet for power sizig works ca be deteried as show i Figures. 12 (a), (b), (c), ad (d). Note that the satellite power sizig is doe by cosiderig the lowest value, as idicated by the dashed lies. 11

Jural Mekaikal, Jue 2009 Power requireet Power requireet a) h = 500 k with i = 53 ; b) h = 1, 000 k with i = 53 Power requireet Power requireet c) h = 500 k with i = 23.5 d) h = 500 k with i = 82.5 Figure 12: Satellite Power Profile Geeratio for i-orbit satellite siulated for power requireet i a year. 3.3 Power Profile Per Orbit After settig the power requireet, the satellite power profile for a orbit ca be viewed as well. Figures. 13 (a), (b), (c) ad (d) show the correspodig power profile per orbit. Note that the satellite power requireet liit is idicated by the dashed lies as show i Figures 13 (a), (b), (c), ad (d). 12

Jural Mekaikal, Deceber 2008 Power requireet Power requireet a) h = 500 k with i = 53 b) h = 1, 000 k with i = 53 Power requireet Power requireet c) h = 500 k with i = 23.5 d) h = 500 k with i = 82.5 Figure 13: 2D Satellite Power Profile Geeratio for i-orbit satellite siulated for per orbit. 3.4 Discussio The uerical ad siulatio results obtaied fro this progra are useful i the issio aalysis ad the sall satellite power sizig. Based o the orbit icliatio, altitude ad solar pael sizes, the coputer progra ca geerate the i-orbit power profile ad the issio power requireet. By usig this progra, the tie cosuig aalytical ethod ca be replaced by this uerical ethod for the solar power profile geeratio. A user friedly GUI progra eables the iput paraeters such as altitude, orbit icliatio ad photovoltaic cell efficiecies for the progra. The developed progra is validated aalytically usig the power profiles geerated per-orbit. 13

Jural Mekaikal, Jue 2009 4.0 CONCLUSION A coputer code was developed usig MATLAB to siulate the i-orbit sall satellite power profile. By defiig the issio paraeters such as the altitude, orbit icliatio ad photovoltaic cell efficiecies, the progra geerates the iorbit solar power profiles. Iitially, the i-orbit siulatio for a orbit was successfully ipleeted. The siulated result was validated usig the result obtaied aalytically. Apparetly, both results were foud to be siilar. Evetually, the geeratio of power profile for a orbit was successfully exteded to siulate a i-orbit 3D satellite power profile for oe coplete year orbit. As a coclusio, the developed progra successfully deteries the i-orbit power profiles as well as the power requireet for differet sall satellites (< 100 kg). NOMENCLATURES Ap Photovoltaic cell surface area [ 2 ] B Satellite s positio vectors i orbit plae [] E Earth s positio vector cetered at the Su [] ESA Europea Space Agecy GUI Graphical User Iterface GUIDE Graphical User Iterface Developet Eviroet h Altitude [] i Earth icliatio [deg] i Satellite orbit icliatio [deg] ϕ Agular step chage [rad] LEO Low Earth Orbit Np Noral vector of solar pael P Power [W] p Nuber of solar pael R S Radius of orbit fro ceter of the Earth [] R E Radius of the Earth, by default is 6, 378.140 k S Solar costat, by default is 1, 358 W/ 2 SCO Spacecraft Cetered Orbit S e Satellite positio vector (Earth cetered) [] T Step chage i tie [1 i = 60 s] T E Duratio of eclipse [s] T o Orbit Period of a satellite about the Earth [s] T su Duratio of Su phase [s] T y Orbit Period of the Earth about the Su, by default is 31,58 8,149.548 s η desig Structural ad desig efficiecy [%] η pv Photovoltaic cells efficiecy [%] η S Shadig factor [%] η T Theral efficiecy [%] η Solar pael efficiecy [%] Ω y Right ascesio of ascedig ode [deg] 14

Jural Mekaikal, Deceber 2008 ω S Agular velocity of satellite about Earth [rad/s] µ Earth gravitatioal costat, is 3.986005 10 14 3 /s 2 e α β Half-agle of eclipse regio [deg] Agle betwee the Su vector ad surface oral vector [deg] REFERENCES 1. Satellite Power Systes Solar Eergy Used i Space Techology Progras, 2003. Europea Space Agecy, ESA Publicatios Divisio, Netherlads, BR-202. 2. Kostolaskŷ E., 2002. O the Duratio of the Total Eclipse of a Satellite of a Body. IMA Joural of Applied Matheatics, 67 (4), p. 401-410. 3. Aigstei P.A. ad Peña R.S., 1998. Aalysis of Solar Pael Orietatio i Low Altitude Satellites. IEEE Trasactios o Aerospace ad Electroic Systes, Vol. 34, No. 2, IEEE. 4. Pitchaiai M., Gaesh T. S., Soa P. ad Shivakuar S.K., 2006. Strategies for Ehacig Power Geeratio i Satellites with Large Local Tie Agles, SpaceOps 2006 Coferece Paper, Aerica Istitute of Aeroautics ad Astroautics, pp. 1-7. 5. Larso, W.J. ad Wertz J.R., 1992. Space Missio Aalysis ad Desig, 2 d Editio. Microcos, Ic. Torrace, Califoria. 6. Fortescue, P.W., Stark J. ad Swierd G., 2003. Power Syste Desig. Spacecraft Syste Egieerig, 3 rd Editio, Wiley Itl. 7. Patel, M.R., 2005. Spacecraft Power Systes, CRC Press, Florida. 8. Pisacae, V.L., 2005. Fudaetals of Space Systes, 2 d Editio. Oxford Uiversity Press, NY. 9. MATLAB, 1984-2005. 7.1.0.246 (R14) Service Pack 3, The MathWorks, Ic. US 15