Resolution of an Antenna-Satellite assignment problem by means of Integer Linear Programming

Resolu ion o an An enna-Sa elli e assignmen p oblem by means o In ege Linea P og amming Ra ael V´azquez1, Fede ico Pe ea2, Jo ge Gal´an Vioque3 1Depa amen o de Ingenie ´ıa Ae oespacial y Mec´anica de Fluidos. Uni e sidad de Se illa (Spain). [email p o ec ed]. 2Depa amen o de Es ad´ıs ica e In es igaci´on Ope a i a Aplicadas y Calidad. Uni e si a Poli `ecnica de Val`encia (Spain). pe e[email p o ec ed] 3Depa amen o de Ma em´a ica Aplicada II e Ins i u o de Ma em´a icas de la Uni e sidad de Se illa (Spain). [email p o ec ed] Abs ac G ound s a ion ope a o s ha e o assign di e en an ennas in hei g ound s a ions ne wo k o passes o sa elli es om cus ome s ha ha e eques ed he use o he ne wo k. Howe e , o ope a o s ha suppo a high numbe o sa elli es, in many cases hese eques s yield con lic s (which appea when mo e han one sa elli e eques s he same ime slo on he same an enna). I he e a e many con lic s, he p ocess o decon lic ing (i.e., mo ing passes o o he an ennas o si es o cancelling hem so ha con lic s a e a oided) is no e y e icien when done manually, due o he la ge numbe o in e ac ing eques s. Thus, he e is a need o an au oma ic ool ha is able o man- age he An enna-Sa elli e assignmen p oblem o a la ge numbe o passes, by conside ing he p oblem globally o a gi en ime- ame ( o ins ance, a week). In his pape we p opose o add ess he decon lic ion p ocess by means o In ege Linea P og amming. Models ha ake in o accoun he basic decon lic ing ope a ions (mo ing an enna, mo ing si e, sho ening, o cancelling), a e p oposed and es ed on eal da a p o ided by he company ha posed his p oblem, yielding be e esul s han he solu ions ob ained by hei p e ious sys em. Keywo ds: Decon lic ion, In ege P og amming, An enna-Sa elli e Alloca ion. P ep in submi ed o Ae ospace Science and Technology May 14, 2015 1. In oduc ion Sa elli e owne s need he suppo o g ound ne wo ks o be able o up- load commands o download ga he ed da a. Inc easing da a equi emen s a e esul ing in a conside able g ow h o eques s o ime alloca ion o sa el- li e passes o g ound an ennas, pa icula ly hose loca ed on s a egic geo- g aphical loca ions— o ins ance, g ound s a ions loca ed close o he poles, which a e mo e equen ly accessible o sa elli e in sun-synch onous o bi s (which includes he majo i y o Ea h Obse a ion Sa elli es, Ba e and Cu is (1992)), and hus a e expe iencing a conside able inc ease o ime alloca ion eques s om cus ome s. While g ound ne wo ks con inue o ex- pand, and build si es in di e se loca ions h oughou he wo ld, he numbe o sa elli e cus ome s con inues o inc ease. Also, he appea ance o dis ibu ed ne wo ks o small sa elli es (Schilling (2009)) could push he capabili ies o he ne wo ks o he limi . The numbe o eques s is leading o e y complex p oblems o an enna-sa elli e alloca ion ope a ions wi h se e al peculia i ies, which make he assignmen planning a cumbe some ask i done by hand. The sa elli e-an enna assignmen p oblem is o en called “Sa elli e Range Scheduling” (SRS) p oblem, o which some s a egies ha e al eady been p oposed in he li e a u e. Ba bulescu e al. (2002) apply he SRS o he US Ai Fo ce Sa elli e Con ol Ne wo k (AFSCN) employing 100 sa elli es, 16 an ennas, 9 s a ions, 500 eques s pe day, p oducing se e al schedules pe day. The conside ed objec i e is he educ ion o he numbe o con lic s ( ypically 120). The au ho s obse ed ha gene ic algo i hms pe o med be e . Ba bulescu e al. (2004) analyze he SRS bo h empi ically and o - mally, p o ing ha he p oblem is NP-comple e. They also p o ide new algo i hms imp o ing hose in Ba bulescu e al. (2002). La e , Ba bulescu e al. (2006a) p esen he e olu ion o he p oblem in he las 10 yea s in he AFSCN. They analyze possible al e na i es o he cos unc ion such as minimizing he sum o o e laps. The same g oup o au ho s s udy mo e heu is ics o he SRS in Ba bulescu e al. (2006b), combining se e al algo- i hms. Chien e al. (2012) s udies he concep o imeline and e iews he gene ic sys ems ha ha e been de eloped. Clemen and Johns on (2005) desc ibe he SRS o he Deep Space Ne wo k (DSN) wi h 16 an ennas, 20 spacec a s, ou -mon h ime- ames, 1650 passes pe week. They gene a e and epai schedules, desc ibing he p oblem and heu is ics o sol ing i wi h emphasis in he e-scheduling p oblem. Co ao e al. (2012) in eg a e Ge- ne ic Algo i hms, G aph Theo y and Linea P og amming in o de o build 2 con lic - ee plans, and apply hei app oach o a case s udy ob ained om a sa elli e se ice company. Lee e al. (2008) s udy he scheduling o a single geos a iona y sa elli e. Ma inelli e al. (2011) o mula es he p oblem as an ILP model, which is decla ed as in easible. The p oblem is sol ed by means o a Lag angian elaxa ion. As a case s udy hey apply hei app oach o he s a ion Galileo. Schmid and Schilling (2009) conside academic p ob- lems consis ing o sa elli es and s a ions dis ibu ed all o e he Ea h. They maximize edundancy in o de o sol e possible ailu es in communica ion. They sol e a oy example wi h 6 sa elli es and 4 s a ions yielding 51 con ac windows. Xha a e al. (2013) use S uggle Gene ic Algo i hms and STK. Zhang e al. (2014) p opose an -colony algo i hms, sol ing examples wi h 17 sa elli es, 11 o 13 an ennas yielding a ound 400 passes. Zu e ey e al. (2008) apply g aph colo ing algo i hms o e a se o 500 ealis ic ins ances. Howe e , due in pa o adi ion, and in pa o he complexi ies o he p oblem, manual elabo a ion o schedules a e s ill usual p ocedu e o g ound ne wo ks manage s. This p oblem has also a isen in he con ex o g ound s a ion ne wo ks (Schmid e al. (2008); Bes e (2009)) p o ided by esea ch ins i u ions o small sa elli es used o academic p ojec s, which usually ha e some speci ic needs such as edundancy and lexibili y, and he e o e equi e speci ic algo i hms. When planning, i mus be aken in o accoun ha a gi en sa elli e can be suppo ed jus by a subse o he a ailable an ennas, and du ing speci ic ime in e als which mus be compu ed by using knowledge o he sa elli e o bi and he an enna geog aphical loca ion and allowable posi ions o Azimu h- Ele a ion o he an enna (i.e. which egion o he sky is accessible o he an enna). Each pass has a de aul an enna, which is he one eques ed by he use ; his would be conside ed as he mos p e e ed an enna. Since di e en use s make eques s independen ly o one ano he , p e e ed alloca ions may cause con lic s, i.e., ime in e als whe e eques s om di e en passes o e - lap on he same an enna. Such con lic s can be add essed by a numbe o al e na i es. Fi s , he p e e ed op ion would be ealloca ing some passes o compa ible an ennas loca ed in he same si e; o he op ions would be chang- ing he pass o ano he si e (which could imply a conside able change in he p e iously alloca ed ime o access i he new si e is a away om he o igi- nal si e), sho ening he pass (up o a minimum du a ion) o accommoda e se e al passes, o , i no o he op ions a e a ailable, canceling he pass. These ope a ions migh be a ailable only o a subse o passes i he e is a numbe o al eady alloca ed passes ha mus be hono ed ( o ins ance o p e e ed 3 clien s o p e ious commi men s). I also mus be aken in o accoun ha some sa elli es o cus ome s could be o highe p io i ies han o he s. In his pape we p opose he use o In ege Linea P og amming (ILP) models o sol e he p oblem o decon lic ion posed by a g ound ne wo k ope a o ha manages se e al si es wi h dozens o an ennas, om now on called “ he company”. While he scheduling p oblem has been epo ed NP comple e (Ba bulescu e al. (2004)), using ou models we ha e been able o sol e in a easonable ime (less han a minu e) eal-wo ld ins ances o he p oblem o ealis ic dimensions ( housands o passes o e dozens o an ennas) and ime ames (a week) by using an open sou ce ILP sol e . The use o In ege Linea P og amming modelling has p o en ui ul o o he space mission op imiza ion p oblems, such as he p oblem o swa h acquisi ion planning o mul iple Ea h Obse a ion Sa elli es, see Gal´an- Vioque e al. (2011). The emainde o his pape is s uc u ed as ollows. In Sec ion 2, he p oblem is o mally s a ed and he no a ion used h oughou he pape is in oduced. The di e en decon lic ion objec i es, and he esul ing models a e desc ibed in Sec ion 3, o mula ed as In ege Linea P og amming p ob- lems. Compu a ional esul s, aken om eal da a, a e analyzed in Sec ion 4. We inish wi h some concluding ema ks in Sec ion 5. 2. Inpu da a We nex show which da a a e needed on sa elli es, an ennas and passes and which compu a ions a e equi ed o o mula e he An enna-Sa elli e as- signmen p oblem. 2.1. Inpu da a The inpu da a o he p oblem a e: •The ime- ame o he planning p oblem is an in e al [T0, T ] gi en by he ini ial and inal imes T0and T . A imes we migh e e o his in e al as T(in ou case, usually a week). •A se So sa elli es. The o bi o he sa elli es can be gi en in any con en ional o ma , o ins ance gi en as Two-Line Elemen s (TLEs) o a ce ain epoch (which should be close o he ime- ame o be able o p ecisely de e mine he passes). 4 •A se A={A1, ..., Ana}o an ennas, gi en by hei geog aphical lo- ca ions. An ennas which a e geog aphically close o each o he a e conside ed o be in he same si e, whe eas an ennas loca ed a away om each o he a e in di e en si es. Fo each an enna, we also assume ha we know i s admissible ange o Azimu h-Ele a ion, which would depend on obs acles and local geog aphy ( o ins ance moun ains) and he equi ed minimum ele a ion abo e he ho izon o a oid a mosphe ic e ec s. This is ma hema ically o mula ed as he se Ωa={(Az,El)} o accessible poin s in he sky gi en by hei azimu hs and ele a ions. •O he ele an inpu da a include he minimum du a ion o a pass o be conside ed alid minsa (which can depend on he sa elli e and an enna) and he se o compa ible an enna-sa elli e pai s C ⊂ A × S. 2.2. Compu a ion o passes The nex s ep o o mula e he p oblem is o calcula e he se o possible passes o all sa elli es Sand an ennas A. Fo each e olu ion o sa elli e s∈ S o e he Ea h we ob ain a pass Pwhen he e a e ime in e als o he o m [ 0, 1]⊂Tdu ing which a sa elli e is accessible o one o mo e an ennas a∈ A, gi en ha he du a ion o he accesses, 1− 0, is g ea e o equal han he minimum du a ion minsa and he an enna is compa ible wi h he sa elli e equi emen s, i.e. (a, s)∈ C. We assume ha he e is an an enna o which he pass is o iginally assigned; he possible an ennas o which he pass can be assigned (di e en om he o iginal one) a e called al e na i e an ennas. To compu e he passes, he i s s ep is o p opaga e he o bi al elemen s o he sa elli es du ing he mission ime- ame. This can be done using any o he many possible me hods a ailable in he li e a u e, which inco po a e mo e o less accu a e models o o bi pe u ba ions (see o ins ance Vallado and McClain (2007), and e e ences he ein). Once he elemen s a e known a all imes ∈T, he ec o posi ion ~ s( ) in he geog aphical e e ence ame ( ha o a es wi h he Ea h) can be compu ed (Cu is (2009)), o all s∈ S. Then, using he an enna geog aphical coo dina es he ec o posi ion o he an ennas ~ a o all a∈ A can be also compu ed. Then, by p ojec ing he ela i e posi ion o he sa elli e wi h espec o he an enna, ~ as( ) = ~ s( )−~ aon he opocen ic ame cen e ed in he espec i e an enna, one can compu e he azimu h and ele a ion o each compa ible an enna- sa elli e pai , (Azas( ), has( )) o (a, s)∈ C. Each o he ime in e als 5 in which (Azas( ), has( )) ∈Ωa o a leas he minimum du a ion minsa cons i u es an access which will gi e an al e na i e o he pass. Sa elli es will gene a e a pass only each ime he g ound ack passes close o a gi en an enna ( o mos loca ions once o wice a day). No ice ha o en imes sa elli es will ha e sun-synch onous o bi s since hese a e he mos equen ly used o bi o Ea h Obse a ion Sa elli es (due o cons an ligh ing p ope ies), which ansmi la ge amoun s o da a and he e o e cons i u e a la ge amoun o he sa elli es eques ing passes. Gi en ha hese sa elli es ha e almos -pola low o bi s, i is ad an ageous o loca e bases close o he poles o he Ea h, since hen one would ob ain passes on mos o bi e olu ions (a ound 13 passes each day). 2.3. Addi ional inpu da a o he passes Once he passes ha e been compu ed, we ha e se s P={P1, . . . , Pnp},A= {A1, ..., Ana}consis ing o nppasses and naan ennas, espec i ely. The ime in e al ha each o hese passes Pico e s in a gi en an enna Akis gi en by he in e als [αi k, βi k]. Passes a e classi ied as accep ed o ee. In he i s case, he eques ed an enna and ime slo a e conside ed o be ixed, while o ee passes, one is allowed o change he eques ed condi ions (an- enna eques ed, pass leng h), o e en o cancel hem in o de o imp o e he con lic s s a us. The ollowing pa ame e s a e addi ional inpu da a o ou p oblem: •F⊂ {1, . . . , np}is he se o ee passes, i.e., passes which can be modi ied wi h espec o he o iginal eques . •pik : p io i y o pass Piin an enna Ak.pik < pi0kmeans ha Piis mo e p e e ed han Pi0 o an enna Ak. •aik : minimum leng h o ime in which Pimus be ac i e i an enna Ak is o ge i s da a (Such leng h o ime includes p e and pos -p ocessing imes, which depend on he sa elli e-an enna pai .) •The bina y pa ame e eik akes he alue 1 i pass Piis o iginally e- ques ed o be assigned o an enna Ak. We assume ha Pna k=1 eik = 1 o e e y pass Pi, ha is, o iginally pass Piis assigned o one and only one an enna. •Ciis he se o an ennas which ha e access and a e compa ible wi h pass Pi.The bina y pa ame e cik akes alue 1 i pass Pihas access 6 and is compa ible wi h an enna Ak, and 0 o he wise. In o he wo ds, cik = 1 i and only i k∈Ci. •(αi k, βi k) is he pe iod o ime in which pass Pihas access o an enna Ak, k ∈Ci. 2.4. Compu a ion o ime in e als Once all he passes ha e been compu ed and he inpu da a on he passes has been ga he ed, he o mula ion o he An enna-Sa elli e assignmen p ob- lem equi es he analysis o he di e en ime in e als in which, o a gi en an enna, possible passes can o e lap. •Fo each an enna Ak, we conside he in e sec ions o all possible in- e als o ime (αi k, βi k) o compa ible passes. The esul is ns(ns≤ 2np−1) in e als I1k, . . . , Insk,wi h leng hs l1k, . . . , lnsk. The in e als a e so ed in such a way ha he beginning o in e al Ij k is equal o he end o in e al Ij−1,k, o la ge i he e is a “gap” du ing which no compa ible passes exis o he an enna. •Sik ⊂ {1,2, . . . , ns}is he se o indices o he so ed in e als {Ijk, j = 1, ..., ns},in which Pican be ac i e in an enna Ak. Taking in o accoun he accep ed ( ixed) passes, in some in e als, he an enna will al eady be occupied by hese accep ed passes, and hus, by cons uc ion, o such in e als Ijk we ha e j6∈ Sik. We show a simpli ied example o such a imeline in Figu e 1, which con- side s h ee passes (p1,p2and p3) and wo an ennas (A1and A2). Fo he sake o simplici y he passes could be loca ed in ei he an enna wi h he same s a and end imes. F om he igu e we see ha he beginning o p1 in ei he an enna is α11 =α12 = 0, he ending o p1is β11 =β12 = 2, and simila ly o p2we ha e α21 =α22 = 1and β21 =β22 = 4, and o p3we ha e α31 =α32 = 3and β21 =β22 = 5. The esul ing in e als o bo h an ennas a e I11 =I12 = [ 0, 1], I21 =I22 = [ 1, 2], I31 =I32 = [ 2, 3], I41 =I42 = [ 3, 4] and I51 =I52 = [ 4, 5]. Assuming no passes a e ixed we ha e S11 =S12 ={1,2},S21 =S22 ={2,3,4}and S31 =S32 ={4,5}, which means ha he i s pass spans (in ei he an enna) he ime in e als 1 and 2, he second pass he ime in e als 2, 3 and 4, and hi d pass he ime in- e als 4 and 5. I Figu e 1 ep esen s he o iginally p oposed schedule hen we see he e is a con lic in an enna 1 be ween passes 1 and 2, which can 7 be i ially esol ed ei he by mo ing pass 1 o an enna 2 (solu ion 1) o by swi ching an enna be ween passes 2 and 3 (solu ion 2), as show in Figu e 2. I5 A1 A2 con lic p1 p3 p2 0 1 2 3 4 5 I1 I2 I3 I4 Figu e 1: Simple example o cons uc ion o ime in e als and con lic . 3 p2 p1 p3 solu ion 1 A2 A1 A2 A1 solu ion 2 p1 p3 p2 4 5 0 1 2 3 4 5 I1 I2 I3 I4 I5 I1 I2 I3 I4 I5 0 1 2 3 p2 p1 p3 solu ion 1 A2 A1 A2 A1 solu ion 2 p1 p3 p2 4 5 0 1 2 3 4 5 I1 I2 I3 I4 I5 I1 I2 I3 I4 I5 0 1 2 Figu e 2: Two possible solu ions o he example. 3. ILP Models The models de eloped he e aim a sol ing con lic s. A con lic is p oduced i he e is an o e lap, i.e i o e he same ime in e al, wo passes a e assigned o he same an enna. Using ou no a ion, his occu s when he e exis an an enna Akand an in e al Ijk such ha X i∈P:j∈Sik eik >1. When con lic s a e ound, he company equi es ha one o he ollowing h ee decon lic ing ope a ions is done: 8 1. Mo ing passes o a di e en an enna (see Sec ion 3.1) which could be in he same si e o in ano he si e. 2. Sho ening a pass ime alloca ion on he an enna (Sec ion 3.2). 3. Cancella ion o passes (Sec ion 3.3). These ope a ions a e w i en in o de o p e e ence, i.e., i s , i possible, con lic s should be add essed by mo ing passes o an ennas di e en o he de aul ones (and i possible wi hin he same si e). Only i con lic s exis a e his ope a ion, some passes should be sho ened ( aking in o accoun he minimum du a ion o a pass). S ill, i some con lic s pe sis , some passes (beginning wi h hose wi h lowe p io i y) should be canceled un il a easible solu ion is ound. In wha ollows we o mula e an in ege linea p og amming (ILP) model ha includes all he possible decon lic ing ope a ions and a he same ime op imizing a pe o mance index di ec ly ela ed wi h he p io i y o he di - e en sa elli es. 3.1. Mo ing passes o a di e en an enna We add ess he p oblem o e-alloca ing ee passes o an ennas so ha con lic s disappea . P io i ies o he di e en passes a e aken in o accoun . Ou ILP model uses he ollowing bina y a iables: •Fo each i∈F, o each k∈Ci,i.e., o each ee pass and compa ible an enna, de ine he bina y a iable yik which akes he alue 1 i pass Piis assigned o an enna Akand 0 o he wise. The cons ain s o he model would be: •E e y ee pass has o be assigned o one and only one an enna. X k∈Ci yik = 1,∀i∈F. (1) •Fo a gi en an enna Akand a ime in e al Ij k a ailable o ee passes, i.e., wi h ∪i∈FSik 6=∅, he e should be no con lic among he nppasses. X i∈F:j∈Sik,k∈Ci yik ≤1,∀k, j :∪i∈FSik 6=∅.(2) 9 •To sol e he esul ing ILP p oblem, he ee-so wa e package Lpsol e sol e was used. We ha e used e sion 5.5.2.0 o windows 32 bi s( h p://lpsol e.sou ce o ge.ne /5.5/). •All expe imen s we e un on a lap op, In el Co e i7 2 GHz and 4 GB o RAM memo y, O.S. Windows 7 P o essional 32 bi s. The ollowing s aigh o wa d conclusions o hese esul s can be no ed : •On a e age, 2605.2 passes we e analyzed in each ins ance wi h 287.9 con lic s. The co esponding p oblems had 9154.7 a iables and 9013.7 cons ain s. The a e age compu a ional ime equi ed o sol e hese p oblems was 64.8 seconds. •Only 3.13% o passes we e canceled. •Only 0.02% o passes we e sho ened (no e howe e ha many o he passes did no allow o sho ening). •19.32% o passes we e mo ed o o he an ennas (0.38% o o he si e). F om he esul s cas in able 1 we can a i m ha he algo i hm p oposed in his p ojec p o ides a eal ime (a ound a minu e) op imal solu ion o p oblems o conside able size (up o 4000 passes) ha ypically co espond o a ull week o ope a ion. The compu a ion ime does no co ela e di ec ly wi h he size o he numbe o con lic s bu depends mo e on he complexi y o he con lic s. The ope a ion cons ain s and p io i ies ha e been e icien ly in eg a ed in he modeling and di e en adjus men s can be achie ed by uning he cos weigh s acco ding o he speci ica ions o he passes. 5. Conclusions In his pape we ha e in oduced In ege Linea P og amming models o e icien ly manage he scheduling o passes o a mul i-an enna, mul i-si e g ound ne wo k se ing nume ous cus ome s. A d ama ic g ow h in he num- be o eques s has ende ed manual scheduling planning i ually un easible. The aim o ou me hods is o sol e con lic s in he bes possible way while e- spec ing p e e ed assignmen s and p io i ies. A con lic appea s when, o e 16 he same ime in e al, wo o mo e passes a e scheduled o he same an- enna. Se e al possibili ies can be applied o sol e such con lic s: mo emen o passes o o he an ennas (possibly loca ed on o he si es), sho ening o acquisi ions o cancela ion o passes. We ha e modeled hese p oblems using a basic ool o ope a ions esea ch: In ege Linea P og amming. Ou models ha e been es ed o e a numbe o ealis ic ins ances p o ided by he g ound ne wo k ope a o , which was p e iously scheduling he passes manually. Con e sa ions wi h he company ep esen a i es le us know ha he pe o mance o ou p ocedu es exceeded he ope a o ’s expec a ions in e ms o speed and quali y o solu ions ( ew numbe o mo emen s, e en ewe numbe o cancela ions) wi h espec o hei p e ious manual sys em. Among u u e possible e inemen s, we could men ion he inclusion o ad- di ional objec i es, such as ai ness c i e ia (penalizing mul iple cancella ions o he same cus ome ) o he de elopmen o ad anced ools such as adap- i e online scheduling (which would imply an schedule unning online wi h capabili ies such as including las -minu e eques s o passes as hey come, o immedia ely adap ing o dynamic cons ain s, o ins ance, an enna ailu es). Acknowledgmen s The au ho s acknowledge he coope a ion o Tai us So wa e (www. ai usso wa e.com) and i s eam and in pa icula i s ounde and CEO (Felipe Ma ´ın C espo), which in oduced his p oblem o us and p o ided in eg a ion wi h i s o bi al mechanics isual so wa e SaVoi . 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Jou nal o Scheduling 11, 263–277. 19 Passes Con lic s Cancella ions Sho enings Mo emen s o al (o he si e) Va iables Cons ain s Time (s) 3356 537 116 1 839 (13) 8090 9322 64 3066 196 75 3 703 (5) 11219 12157 61 3356 231 94 0 517 (6) 12465 12245 73 3566 306 114 1 561 (11) 12360 12714 73 3470 253 100 0 557 (17) 12016 12788 80 3408 196 91 0 478 (5) 12289 12025 71 1573 68 0 0 487 (7) 7143 4467 64 384 33 21 0 101 (0) 1247 1075 22 1586 6 2 0 479 (2) 7892 4964 63 2287 1053 250 4 312 (35) 6820 8380 77 Table 1: Decon lic ing esul s on 10 eal scheduling p oblem ins ances. 20