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.
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