Analysis of Wind-Shear Effects on Optimal Aircraft Cruise

Analysis o Wind-Shea E ec s on Op imal Ai c a C uise Al onso Valenzuela, and Damián Ri as Depa men o Ae ospace Enginee ing Escuela Técnica Supe io de Ingenie ía, Uni e sidad de Se illa Se ille, Spain a [email protected], d i [email protected] Abs ac —An analysis o he e ec s o wind shea on op imal ai c a c uise ajec o ies is p esen ed. The p ocedu e conside ed is c uise a cons an Mach numbe and cons an al i ude, which is commonly lown by ai lines, ollowing ai - a ic-con ol ules. The op imal ajec o ies co espond o he case o minimum di ec -ope a ing-cos c uise wi h gi en ange, and a e ob ained using pa ame ic op imiza ion heo y. The main objec i e o he pape is o analyze he in luence o he wind shea on he op imum al i ude and speed. The esul s show ha , o a gi en cos index, he e ec o he wind shea on he op imum al i ude is qui e la ge. The e ec o he cos index on he op imum esul s is also analyzed. Resul s a e p esen ed o a model o a Boeing 767-300ER. Keywo ds-ai c a ajec o y op imiza ion; op imal c uise; wind shea I. I NTRODUCTION T ajec o y op imiza ion is an impo an subjec in ai a ic managemen , which aims a de ining op imal ligh p ocedu es ha lead o ene gy-e icien ligh s. In p ac ice, he ai lines conside a cos index (CI) and de ine he di ec ope a ing cos (DOC) as he combined cos o uel consumed and ligh ime, weigh ed by he CI. Thei goal is o minimize he DOC. This p oblem has been ea ed ex ensi ely in he li e a u e. Among many o he s, o example, minimum-DOC ajec o ies ha e been s udied by Ba man and E zbe ge [1], E zbe ge and Lee [2] and Bu ows [3] who analyze he minimum-DOC p oblem o global ajec o ies (climb-c uise-descen ); hey conside s eady c uise, and ake he ai c a mass as cons an . Bu ows [4] also analyzes he minimum-DOC p oblem o global ajec o ies, wi hou he assump ion o cons an mass, bu wi h he assump ion ha he c uise segmen akes place in he s a osphe e. Bilimo ia e al. [5] and Chak a a y [6] analyze he minimum-DOC, s eady c uise as he ou e solu ion o a singula pe u ba ion app oach, whe e he ai c a mass is aken as cons an . Lidén [7] p oposes compu ing algo i hms o be used in ligh managemen sys ems which op imize he c uise p o ile. Howe e , hese wo ks ei he do no conside wind-shea e ec s, as in [2-5], o only conside one pa icula wind p o ile, as in [1,6,7], no analyzing he e ec s o changing he wind p o ile. In his pape we add ess he p oblem o analyzing he e ec s o wind shea on minimum-DOC ai c a c uise ajec o ies. The p ocedu e conside ed is c uise a cons an Mach numbe and cons an al i ude, which is commonly lown by ai lines, ollowing ai - a ic-con ol (ATC) ules; he c uise ange is ixed. The c uise is uns eady, wi h a iable mass, subjec o a ho izon al, al i ude-dependen wind p o ile. The main objec i e o his wo k, wi h espec o he published li e a u e, is o p o ide an unde s anding o he in luence o he wind shea on he op imum c uise al i ude and c uise speed, aking in o accoun he cos index alue ixed by he ope a o . Fo comple eness, he in luence o he a e age wind speed (headwinds and ailwinds) is also analyzed. To op imize he c uise p ocedu e, a pa ame ic op imiza ion app oach is p esen ed. Pa ame ic ajec o y op imiza ion has been also ea ed ex ensi ely. Fo example, Be s and C ame [8] apply he di ec ansc ip ion echnique, which combines nonlinea op imiza ion wi h a disc e iza ion o he ajec o y dynamics, o he op imal design o ajec o ies ( o se e al pe o mance indices) subjec o ealis ic cons ain s ha ep esen he ajec o y phases o a mission p o ile. Sole e al. [9] elax some o he cons ain s imposed in [8] o gi e mo e oom o planning mo e e icien ajec o ies, o mula ing a single op imal con ol p oblem which is sol ed as a nonlinea op imiza ion p oblem. Menon e al. [10] op imize ligh s a egies o con lic esolu ion pa ame e izing he ajec o ies in e ms o ou -dimensional waypoin s, and app oxima ing he ajec o ies by piecewise-linea pa hs. Wu and Zhao [11] op imize he ajec o y om li o o ouchdown and quan i y he de ia ion om ac ual ajec o ies due o modeling e o s and/o ligh condi ions, de ining he ajec o y by a se ies o ligh segmen s speci ied by a se o ligh objec i es, such as speeds, al i udes o h o le se ings. To es e al. [12] o mula e a mul i-objec i e op imiza ion p oblem ha minimizes noise and pollu an s emissions o he depa u e p ocedu es, pa ame e izing he ajec o y h ough wo se s o a iables ha desc ibe he e olu ion o he ai c a speed and h us . Valenzuela e al. [13] op imize disc e e c uise p ocedu es cons ained o ha e Mach numbe s mul iple o 0.01, and al i udes de ined by ligh le els. In his pape he c uise p ocedu e is de ined by a ajec o y pa e n o med by i e segmen s commonly lown by ai lines, which is in ac a ligh in en ha de ines unambiguously how he ai c a is o ly. The segmen s a e as ollows: 1) s a ing a he ini ial al i ude i h , a ansi ion segmen a he ini ial Mach i M (descen /climb wi h idle/maximum c uise engine a ing) ending a he c uise al i ude c h , 2) a ansi ion segmen a cons an al i ude c h (decele a ion/accele a ion wi h idle/maxi- mum c uise engine a ing) ending a he c uise Mach c M , 3) he main c uise segmen a cons an Mach c M and cons an al i ude c h , ending when a dis ance c is lown, 4) a ansi ion segmen a cons an al i ude c h (decele a ion/accele a ion wi h idle/maximum c uise engine a ing) ending a he inal Mach M , and 5) a ansi ion segmen a cons an Mach M (descen /climb wi h idle/maximum c uise engine a ing) ending a he inal al i ude h . In his wo k, he ini ial and inal condi ions ( i h , i M , h , M ) a e gi en, so ha he c uise al i ude c h , he c uise Mach numbe c M and he dis ance c a e ee a iables, on which he op imiza ion is pe o med. Resul s a e p esen ed o a model o a Boeing 767-300ER, wi h comp essible ae odynamics and gene al speci ic uel consump ion and h us models, which is desc ibed in [14]. The esul s o linea wind p o iles show ha he e ec o he wind shea on he op imum al i ude is qui e la ge, depending s ongly on he cos index; on he con a y, he e ec o he a e age wind speed is much smalle . In pa icula , i is ound ha depending on he alue o he wind shea he op imal c uise akes place ei he in he oposphe e o in he s a o- sphe e, wi h opposi e beha io s as a unc ion o he cos index, namely, he op imum al i ude dec eases wi h he cos index in he oposphe e, whe eas i inc eases in he s a osphe e. II. P ROBLEM F ORMULATION A. Equa ions o Mo ion In his wo k, c uise ligh in a e ical plane is conside ed. The model adop ed o desc ibe he ai c a mo ion is ha o a poin mass wi h h ee deg ees o eedom, commonly used o ajec o y p edic ion (see Sla e y and Zhao [15]); he equa ions hen desc ibe he mo ion o he ai c a cen e o mass, conside ed as a mass- a ying body. The case o al i ude- dependen ho izon al winds con ained in he ligh plane is conside ed. The equa ions o mo ion o symme ic ligh wi h h us pa allel o he ai c a ae odynamic eloci y a e he ollowing (see Jackson e al. [16]): d d ( , , ) , d d , d( , ) , d d( ), d d, d V w m T D V h L mg mV h L mg mc V h T V w h hV γ γ γ = − − − = = − = + = (1) whe e he ollowing simpli ying assump ions ha e been made: 1 γ ≪ and / 0 V g γ ≈ ɺ . In he p e ious equa ions, V and γ a e he ae odynamic eloci y modulus and he ae odynamic pa h angle; m he ai c a mass; and h he ho izon al dis ance and he al i ude; w he wind speed; g he g a i y accele a ion; he ime; T , L , and D he h us , he li , and he ae odynamic d ag; and c he speci ic uel consump ion. Each ligh segmen is de ined by wo ligh cons ain s ( o example, o ly a cons an al i ude and cons an speed), which oge he wi h (1) o m a sys em o di e en ial algeb aic equa ions (DAE). The esolu ion o he DAE sys ems o he di e en ligh segmen s is based on he educ ion o he sys em o equa ions o a sys em o o dina y di e en ial equa ions (ODE) h ough he explici u iliza ion o he ligh cons ain s. The ODE sys ems a e hen sol ed using MATLAB's ode45 [17] (based on an explici Runge Ku a o mula). The compu a ion o each ligh segmen s a s wi h he co esponding ini ial condi ions and ends when he app op ia e s opping condi ion is eached ( o ins ance, eaching a gi en al i ude o a gi en Mach numbe ). Addi ionally, o compu e he ligh segmen s, some supplemen a y models a e needed: Ea h, ae odynamic and p opulsion models. In his pape , he Ea h has cons an g a i y, he a mosphe ic model is ISA, and ealis ic ae odynamic and p opulsion models a e conside ed, which a e desc ibed in [14]. The ai c a model p o ides he ollowing unc ions: comp essible d ag pola ( , ) D L C M C , speci ic uel consump ion ( , ) c M h , and a ailable h us ( , ) MCRZ T M h o maximum-c uise engine a ing and ( , ) IDLE T M h o idle engine a ing. The a mosphe e model p o ides he densi y ( ) h ρ and he wind speed p o ile ( ) w h . The li and d ag coe icien s a e de ined by 2 / 2 L L V SC ρ = and 2 / 2 D D V SC ρ =, whe e S is he e e ence wing su ace. B. T ajec o y Pa e n To model he c uise ligh in a e ical plane, he ajec o y pa e n shown in Fig. 1 is conside ed; he a ows in he igu e indica e he s opping c i e ion o each ligh segmen , and ER s ands o ixed engine a ing. The pa e n s a s om he ini ial al i ude i h and ini ial Mach numbe i M , and ends a he inal condi ions h and M .The pa e n is o med by i e ligh segmen s. The i s one is a ansi ion segmen , a descen /climb a cons an Mach i M and wi h idle/maximum c uise engine a ing ending a he c uise al i ude c h . The second one is also a ansi ion segmen , a decele a ion/accele a ion a cons an al i ude c h and wi h idle/maximum c uise engine a ing, ending a he c uise Mach numbe c M . The hi d one is a segmen a cons an Mach c M and cons an al i ude c h ending when a dis ance c is lown. Finally, because he c uise ligh has o end a he inal condi ions h and M , wo mo e ansi ion segmen s, as hose jus desc ibed, comple e he pa e n; ha is, a ansi ion segmen a cons an al i ude c h (decele a ion/accele a ion wi h idle/maximum c uise engine a ing) ending a he inal Mach M , and a ansi ion segmen a cons an Mach M (descen /climb wi h idle/maximum c uise engine a ing) ending a he inal al i ude h . Figu e 1. T ajec o y pa e n. Hence, one can see ha h ee di e en ypes o ligh segmen s a e conside ed, which comply wi h usual ATC ules, namely, segmen s wi h cons an Mach and cons an al i ude, ansi ion segmen s wi h ixed engine a ing and cons an Mach ( o descen /climb segmen s), and ansi ion segmen s wi h ixed engine a ing and cons an al i ude ( o decele a ing/accele a ing segmen s). The p ocedu e is de ined by se en pa ame e s: al i ude c h , Mach numbe c M , dis ance c , ini ial and inal al i udes, i h and h , and ini ial and inal Mach numbe s, i M and M . Depending on he applica ion, some o hese pa ame e s can be ixed, whe eas he es a e ee and used as a iables in he op imiza ion p oblem. In his pape , he ini ial and inal alues o al i ude and Mach numbe a e ixed. C. Pa ame ic Op imiza ion Once he ee pa ame e s o he ajec o y a e de ined, hey a e collec ed in a ec o x . The op imiza ion p oblem is o mula ed as a nonlinea p og amming (NLP) p oblem: minimize ( ) subjec o ( ) , ( ) , , J X = ≤ ∈ x x 0 g x 0 x (2) whe e X is he easible egion o he a iables. In his o mula ion, he op imali y c i e ion de ining he cos unc ion can be he minimiza ion o any p ope y o combina ion o p ope ies o he ajec o y ha can be de i ed om he compu a ion o he ajec o y. The equali y and inequali y cons ain s and he easible egion depend on he applica ion. Di e en echniques can be used o sol e NLP p oblems [18]. In his wo k, MATLAB's mincon is used, a sequen ial quad a ic p og amming (SQP) me hod, which is p oposed by Schi kowsky [19] as he mos e icien o sol e nonlinea p og amming p oblems. I mus be no ed ha SQP me hods, as g adien -based me hods, a e only able o ind one local minimum wi hin he easible egion; in case ha se e al local minima exis , he global minimum can be ob ained by subdi iding he easible egion in o app op ia e sub egions, sol ing he op imiza ion p oblem on each sub egion, and inally aking he bes local minimum ound. D. Minimum-DOC C uise The objec i e is o minimize he di ec ope a ing cos in c uise ligh wi h ixed ange. The DOC is a combina ion o uel and ime cos s, F DOC m CI = + (measu ed in kg), whe e F m is he uel consump ion, he ligh ime, and CI he cos index which measu es he ela i e impo ance o bo h cos s ( he case 0 CI = co esponds o minimum uel). No e ha al hough ai lines de ine he CI in uni s o $/hou di ided by cen s/lb, in his pape in e na ional uni s o measu e a e used, hence, he CI is measu ed in kg/s. Rep esen a i e alues o he CI a e in he ange 0 o 3 kg/s, which is conside ed in he nume ical simula ions. The ini ial and inal al i udes and speeds a e ixed, 30000 i h h= = and 0.79 i M M= = . The same ini ial and inal condi ions a e chosen, o be able o compa e all cases conside ed in he analysis; i M and M co espond o ypical climb and descen Mach alues and i h h = is a ypical c uise al i ude. The ange o be lown is 2000 A = km. The cos unc ion can be w i en as ( ) ( ) ( ), F J m CI = + x x x (3) whe e he uel consump ion and he ligh ime depend on he ee pa ame e s x . Because he o al lown dis ance is a unc ion o he ee pa ame e s, he gi en ange A is imposed by he equali y cons ain ( ) 0. A − = x (4) The inequali y cons ain s educe o equi ing ha he speeds and al i udes be wi hin he ai c a ope a ional en elope. Because he ini ial and inal condi ions a e gi en, he o al numbe o ee pa ame e s is h ee: he Mach numbe c M , he al i ude c h , and he dis ance lown du ing he hi d pa e n segmen c . The easible egion is gi en by [0.60,0.86] c M∈ , [20000,43000] c h∈ , and [0,2000] c ∈ km. Conside ing he di e en beha io o he a mosphe e in he oposphe e and in he s a osphe e, one could expec he objec i e unc ion o be non con ex in he easible egion, possibly ha ing a local minimum in each laye . Fo his eason, i has been decided o subdi ide he easible egion in o wo sub egions: [20000,36089] c h∈ in he oposphe e and [36089,43000] c h∈ in he s a osphe e. Fo he cases analyzed in Sec ion III, i has been obse ed ha he objec i e unc ion is con ex in each sub egion. The op imiza ion p oblem is hen sol ed in bo h sub egions and he bes local minimum ound is aken as he global minimum. E. Wind P o ile Fo he wind model, linea p o iles a e conside ed, wi h he absolu e alue o he wind speed inc easing wi h al i ude −40 −20 0 20 40 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5 x 104 ¯w[k ] h∗ c[ ] CI −40 −20 0 20 40 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 ¯w[k ] M∗ c[-] CI (see [6,20,21]). The p o iles, be ween wo gi en al i udes 1 h and 2 1 h h > , a e de ined as ollows ( ) , h h w h w w h h − = + ∆ − (5) whe e w is he a e age wind, w ∆ he wind-shea pa ame e and 1 2 ( ) / 2 h h h= + he a e age al i ude. Fo gi en alues o 1 h and 2 h , w ∆ de ines he wind shea d / d w h , and, in pa icula , 0 w ∆ = de ines a uni o m wind p o ile. No e ha he a e age wind speed w is gi en by 2 1 2 1 1 ( )d , h h w w h h h h =− ∫ (6) and, also, since he wind p o iles a e linea , w is he wind speed a he a e age al i ude, ha is, ( ) w w h = . In he ollowing, bo h ailwinds (TW) and headwinds (HW) a e conside ed, wi h he linea p o iles de ined as ollows: o TW one has 0 w > and 0 w ∆ ≥ , and o HW 0 w < and 0 w ∆ ≤ . To de ine he wind p o ile, he ollowing al i udes a e conside ed: 1 10000 h= , 2 33000 h= ; he a e age al i ude is 21500 h= . The a e age wind anges om 40 − k o 40 k , and he absolu e alue o he wind-shea pa ame e anges om 0 o 40 k . III. R ESULTS In his sec ion op imiza ion esul s a e p esen ed o 1200 W= kN. No e ha ins ead o ixing he ini ial ai c a weigh , which would depend on he alue o he CI, he inal weigh is ixed, he same o all he simula ions. Hence, he in eg a ion o he equa ions o mo ion is pe o med backwa ds. The e ec s o he a e age wind and o he wind shea on he op imal p ocedu es a e analyzed in Sec ions III.A and III.B, espec i ely. A. E ec o he A e age Wind The e olu ion o he op imum al i ude * c h wi h he a e age wind is shown in Fig. 2 o di e en alues o he CI and o a wind-shea pa ame e 0 w ∆ = . I can be seen ha he al i udes dec ease as he CI inc eases; his same beha io was ound, o example, by Ba man and E zbe ge [1] o sho -haul ai c a wi h cons an mass and no wind. Fo a ixed alue o he CI, he al i ude sligh ly inc eases as he a e age wind inc eases; he smalle he CI, he weake he al i ude inc ease. Fo ins ance, o 0 CI = , one has ha he di e ence in he al i ude ( o he ange o a e age wind conside ed) is 129 , and o 3 CI = kg/s is 727 . The op imum Mach numbe * c M is shown in Fig. 3. As expec ed, i inc eases as he CI inc eases. I can be seen ha he Mach numbe sligh ly dec eases as he a e age wind inc eases; he la ge he CI, he weake he Mach dec ease. Fo ins ance, o 3 CI = kg/s, one has ha he di e ence in he Mach numbe is jus 0.005, and o 0 CI = is 0.011. In summa y, i can be said ha he e ec o he a e age wind on he op imal p ocedu es is small. Figu e 2. Op imum al i ude s a e age wind o CI = 0,0.5,1,1.5,2,2.5, 3 kg/s, and 0 w ∆ = . Figu e 3. Op imum Mach numbe s a e age wind o CI = 0,0.5,1,1.5,2, 2.5,3 kg/s, and 0 w ∆ = . The global p ope ies F m , , and DOC a e shown in Figs. 4, 5, and 6, espec i ely. These h ee global p ope ies signi ican ly dec ease as w inc eases: hey a e, as expec ed, la ge o headwinds han o ailwinds. As an example, o 1.5 CI = kg/s one has he ollowing di e ences ( o he ange o a e age wind conside ed): 1756 kg in F m , 1347 s in , and 3777 kg in DOC . As he CI inc eases, F m inc eases and dec eases as expec ed, and DOC inc eases, because bo h F m and he p oduc CI inc eases. −40 −20 0 20 40 8000 8500 9000 9500 10000 10500 11000 ¯w[k ] mF[kg] CI −40 −20 0 20 40 7500 8000 8500 9000 9500 10000 ¯w[k ] [s] CI −40 −20 0 20 40 0.5 1 1.5 2 2.5 3 3.5 4x 104 ¯w[k ] DOC [kg] CI −40 −20 0 20 40 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 x 104 ∆w[k ] h∗ c[ ] CI CI CI Figu e 4. Fuel consump ion s a e age wind o CI = 0,0.5,1,1.5,2,2.5, 3 kg/s, and 0 w ∆ = . Figu e 5. Fligh ime s a e age wind o CI = 0,0.5,1,1.5,2,2.5,3 kg/s, and 0 w ∆ = . Figu e 6. DOC s a e age wind o CI = 0,0.5,1,1.5,2,2.5,3 kg/s, and 0 w ∆ = . B. E ec o he Wind Shea Now, he e ec o he wind shea is analyzed. In his analysis one has 30 w = − k and 0 w ∆ < o HW, and 30 w = k and 0 w ∆ > o TW. The e olu ion o he op imum al i ude wi h he wind-shea pa ame e is shown in Fig. 7. I can be seen ha he al i ude inc eases as he wind shea inc eases. This a ia ion is qui e la ge and s ongly depends on he CI. Fo ins ance, o 0 CI = , one has ha he di e ence in he al i ude ( o he ange o wind shea conside ed) is 5791 , and o 3 CI = kg/s is as la ge as 15977 . The jumps a 0 w ∆ = co espond o he di e en a e age winds conside ed. Figu e 7. Op imum al i ude s wind shea o CI = 0,0.5,1,1.5,2,2.5,3 kg/s; 30 w = − k o HW, 30 w = k o TW. Fo he ange o CI conside ed, he op imum al i ude is ound o be in he oposphe e o nega i e and small posi i e alues o w ∆ and in he s a osphe e o la ge alues o w ∆ . The way in which he ansi ion om one egion o he o he is pe o med depends on he alue o he CI. Fo ins ance, o 0 CI = , he op imum al i ude coincides wi h he opopause be ween 25 w ∆ = and 29 k , and o 1 CI = kg/s he al i ude jumps om 35650 o 36530 a app oxima ely 17.5 w ∆ = k . A mo e de ailed iew o his ansi ion is shown in Fig. 8 o CI anging om 0 o 0.5 kg/s. The a ia ion o he al i ude wi h he CI depends on he wind-shea pa ame e (see Fig. 7). Fo nega i e and small posi i e alues o w ∆ , one has ha he op imum al i ude dec eases as he CI inc eases; bu o la ge posi i e alues o w ∆ , when he op imum al i ude is loca ed in he s a osphe e, he beha io is e e sed: he op imum al i ude inc eases as he CI inc eases. The wo di e en beha io s a e be e obse ed in Fig. 9 whe e he e olu ion o he op imum al i ude wi h he CI is ep esen ed o posi i e alues o he wind-shea pa ame e : o small alues o w ∆ , * c h dec eases, and o la ge alues o w ∆ , * c h inc eases. As one can also see, o in e media e alues o w ∆ he op imum al i ude can be in he oposphe e o in he s a osphe e, depending on he alue o he CI. Fo ins ance, o 10 15 20 25 30 35 3.5 3.55 3.6 3.65 3.7 3.75 x 104 ∆w[k ] h∗[ ] CI opopause 0 0.5 1 1.5 2 2.5 3 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 x 104 CI [kg/s] h∗ c[ ] ∆w −40 −20 0 20 40 0.7 0.8 0.9 1 1.1 1.2 1.3 x 104 ∆w[k ] mF[kg] CI CI −40 −20 0 20 40 0.75 0.76 0.77 0.78 0.79 0.8 0.81 0.82 0.83 ∆w[k ] M∗ c[-] CI CI 17.5 w ∆ = k he op imum al i ude is in he s a osphe e o [0.89,1.77] CI ∈ kg/s and in he oposphe e in any o he case. These esul s show ha he wind shea does ha e an impo an e ec on he op imum c uise al i ude. Figu e 8. Op imum al i ude s wind shea o CI = 0,0.1,0.2,0.3,0.4, 0.5 kg/s, and 30 w = k . Figu e 9. Op imum al i ude s cos index o w ∆ = 0,5,10,15,17.5,20,25,30, 35,40 k , and 30 w = k . The e olu ion o he op imum Mach numbe is shown in Fig. 10. One can see ha , in gene al, i sligh ly inc eases o 0 w ∆ < and dec eases o 0 w ∆ > . The li le jumps abou 17.5 w ∆ = k ma ch wi h he al i ude jumps jus men ioned. The a ia ion o he Mach numbe wi h he wind shea is small and compa able o he a ia ion wi h he a e age wind. Fo ins ance, o 0 CI = , one has ha he di e ence be ween he maximum and minimum alue o he Mach numbe is jus 0.011, and o 3 CI = kg/s is 0.016. Figu e 10. Op imum Mach numbe s wind shea o CI = 0,0.5,1,1.5,2,2.5, 3 kg/s; 30 w = − k o HW, 30 w = k o TW. The global p ope ies F m , , and DOC a e shown in Figs. 11, 12, and 13, espec i ely. Fo a gi en CI, hey a e conside ably la ge o 0 w ∆ < , because in ha case one has headwinds, as opposed o he ailwinds one has o 0 w ∆ > . As be o e, F m and DOC inc ease, and dec eases as he CI inc eases. Fo ailwinds, he pe o mance index DOC dec eases as w ∆ inc eases; he la ge he wind shea , he be e . Fo headwinds, howe e , DOC inc eases as w ∆ inc eases; he la ge he wind shea (in modulus), he wo se. As an example, o 1.5 CI = kg/s one has he ollowing esul s: o ailwinds, when w ∆ inc eases om 0 o 40 k , he dec ease in DOC is 1933 kg; and o headwinds, when w ∆ inc eases om 0 o 40 k , he inc ease in DOC is 1773 kg. Figu e 11. Fuel consump ion s wind shea o CI = 0,0.5,1,1.5,2,2.5,3 kg/s; 30 w = − k o HW, 30 w = k o TW. −40 −20 0 20 40 7000 7500 8000 8500 9000 9500 10000 10500 ∆w[k ] [s] CI CI −40 −20 0 20 40 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 104 ∆w[k ] DOC [kg] CI CI Figu e 12. Fligh ime s wind shea o CI = 0,0.5,1,1.5,2,2.5,3 kg/s; 30 w = − k o HW, 30 w = k o TW. Figu e 13. DOC s wind shea o CI = 0,0.5,1,1.5,2,2.5,3 kg/s; 30 w = − k o HW, 30 w = k o TW. IV. C ONCLUSIONS An analysis o he e ec s o a e age wind and wind shea on op imal ai c a c uise ajec o ies has been p esen ed, which is based on he use o a p ede ined ajec o y pa e n and pa ame ic op imiza ion. The ajec o y pa e n is o med by i e segmen s commonly lown by ai lines ollowing ATC ules. The case o linea wind p o iles has been conside ed. The e ec o he a e age wind on he op imal p ocedu e (al i ude and Mach numbe ) has been shown o be small, and much la ge i s e ec on he global p ope ies ( uel consump ion, ligh ime, and cos ), as expec ed o he di e ence be ween ailwinds and headwinds. The esul s ha e shown ha he wind shea has a s ong e ec on he op imum c uise al i ude. In pa icula , i has been ound ha , depending on he alue o he wind shea , he op imal c uise can ake place ei he in he oposphe e o in he s a osphe e; in gene al, o ailwinds, he la ge he wind shea , he highe he op imal c uise, and, o headwinds, he la ge he wind shea (in modulus), he lowe he op imal c uise. I has also been shown ha he beha io o he op imum al i ude as a unc ion o he cos index is opposi e in he oposphe e and in he s a osphe e, namely, he op imum al i ude dec eases wi h he cos index in he oposphe e, whe eas i inc eases in he s a osphe e. On he con a y, he e ec o he wind shea on he op imum c uise Mach has been shown o be qui e small. Finally, he e ec o he wind shea on he op imal pe o mance has been also shown o be impo an : o ailwinds, he la ge he wind shea , he be e he pe o mance; on he con a y, o headwinds, he la ge he wind shea (in modulus), he wo se he pe o mance. The analysis o c uise ligh wi h la ge ange, wi h se e al c uise s eps (s epped climb c uise), is le o u u e wo k. I is expec ed ha o la ge, posi i e alues o he wind shea ( o ailwinds) he whole op imal c uise ake place in he s a osphe e; o lowe alues, one may ha e he op imal c uise s a ing in he oposphe e and ending in he s a osphe e; and o nega i e alues ( o headwinds) one may expec ha he whole op imal c uise ake place in he oposphe e. The analysis o o he ypes o wind p o iles and o cases ha include wind a ia ion along he ajec o y a e also le o u u e wo k. R EFERENCES [1] J. F. Ba man, and H. E zbe ge , “Fixed- ange op imum ajec o ies o sho -haul ai c a ,” Jou nal o Ai c a , ol. 13, no. 10, 1976, pp. 748- 754. [2] H. E zbe ge , and H. Lee, “Cons ained op imum ajec o ies wi h speci ied ange,” Jou nal o Guidance, Con ol, and Dynamics, ol. 3, no. 1, 1980, pp. 78-85. [3] J. W. Bu ows, “Fuel-op imal ai c a ajec o ies wi h ixed a i al imes,” Jou nal o Guidance, Con ol, and Dynamics, ol. 6, no. 1, 1983, pp. 14-19. [4] J. W. Bu ows, “Fuel op imal ajec o y compu a ion,” Jou nal o Ai c a , ol. 19, no. 4, 1982, pp. 324-329. [5] K. D. Bilimo ia, E. M. Cli , and H. J. Kelley, “Classical and neo- classical c uise-dash op imiza ion,” Jou nal o Ai c a , ol. 22, no. 7, 1985, pp. 555-560. [6] A. Chak a a y, “Fou -dimensional uel-op imal guidance in he p esence o winds,” Jou nal o Guidance, Con ol, and Dynamics, ol. 8, no. 1, 1985, pp. 16-22. [7] S. Lidén, “Op imum c uise p o iles in he p esence o winds,” IEEE-0- 7803-0820-4, IEEE/AIAA 11 h Digi al A ionics Sys ems Con e ence, Sea le, WA (USA), 05-08 Oc obe , 1992, pp. 254-261. [8] J. T. Be s, and E. J. C ame , “Applica ion o di ec ansc ip ion o comme cial ai c a ajec o y op imiza ion,” Jou nal o Guidance, Con ol, and Dynamics, ol. 18, no. 1, 1995, pp. 151-159. [9] M. Sole , A. Oli a es, and E. S a e i, “Hyb id op imal con ol app oach o comme cial ai c a ajec o y planning,” Jou nal o Guidance, Con ol, and Dynamics, ol. 33, no. 3, 2010, pp. 985-991. [10] P. K. Menon, G. D. Swe iduk, and B. S idha , “Op imal s a egies o ee- ligh ai a ic con lic esolu ion,” Jou nal o Guidance, Con ol, and Dynamics, ol. 22, no. 2, 1999, pp. 202-211. [11] D. Wu, and Y. J. Zhao, “Pe o mances and sensi i i ies o op imal ajec o y gene a ion o ai a ic con ol au oma ion,” AIAA-2009- 6167, AIAA GNC Con e ence, Chicago, IL (USA), 10-13 Augus , 2009. [12] R. To es, J. Chap al, C. Bès, and J.-B. Hi ia -U u y, “Op imal, en i onmen ally iendly depa u e p ocedu es o ci il ai c a ,” Jou nal o Ai c a , ol. 48, no. 1, 2011, pp. 11-22. [13] A. Valenzuela, D. Ri as, and A. F anco, “C uise op imiza ion using ajec o y pa e ns,” AIAA-2010-9140, 10 h AIAA ATIO Con e ence, Fo Wo h, TX (USA), 13-15 Sep embe , 2010. [14] A. F anco, D. Ri as, and A. Valenzuela, “Minimum- uel c uise a cons an al i ude wi h ixed a i al ime,” Jou nal o Guidance, Con ol, and Dynamics, ol. 33, no. 1, 2010, pp. 280-285. [15] R. Sla e y, and Y. Zhao, “T ajec o y syn hesis o ai a ic au oma ion,” Jou nal o Guidance, Con ol, and Dynamics, ol. 20, no. 2, 1997, pp. 232-238. [16] M. R. Jackson, Y. Zhao, and R. A. Sla e y, “Sensi i i y o ajec o y p edic ion in ai a ic managemen ,” Jou nal o Guidance, Con ol, and Dynamics, ol. 22, no. 2, 1999, pp. 219-228. [17] L. F. Shampine, M. W. and Reichel , “The MATLAB ODE sui e,” SIAM Jou nal on Scien i ic Compu ing, ol. 18, no. 1, 1997, pp. 1-22. [18] R. Fle che , P ac ical me hods o op imiza ion, John Wiley & Sons, 1987, pp. 331-336. [19] K. Schi kowski, “NLPQL: a Fo an sub ou ine sol ing cons ained nonlinea p og amming p oblems,” Annals o Ope a ions Resea ch, ol. 5, 1985, pp. 485-500. [20] R. M. Osegue a, and D. H. Williams, “Fligh e alua ion o he CTAS descen ad iso ajec o y p edic ion,” P oceedings o he Ame ican Con ol Con e ence, Sea le, Washing on, June 1995, pp. 3435-3439. [21] S. Lidén, “P ac ical conside a ions in op imal ligh managemen compu a ions,” Jou nal o Guidance, Con ol, and Dynamics, ol. 9, no. 4, 1986, pp. 427-432.