optimal scheduling of fuel-minimal approach …...technische universität münchen institute of...

Technische Universität München

Institute of

Flight System Dynamics

Optimal Scheduling of Fuel-Minimal

Approach Trajectories

3rd International Air Transport and Operations Symposium

18 – 20 June 2012

Delft, the Netherlands

Florian Fisch, Matthias Bittner, Prof. Florian Holzapfel Institute of Flight System Dynamics, Technische Universität München, Garching, Germany

Institute of

Flight System Dynamics Optimal Scheduling of Fuel-Minimal Approach Trajectories 2

Outline

1. Introduction

2. Aircraft Simulation Model

3. Multi-Aircraft Optimization Problem

4. Results

5. Summary & Outlook

Institute of


Outline

1. Introduction

Institute of


Introduction

Computation of fuel minimal and noise minimal approach trajectories:

So far:

⇨ Optimization of stand-alone approach trajectories

⇨ Limitations due to other aircraft in the vicinity of an airport are

not taken into account

⇨ Optimization results can not be put into practice due to the limitations

arising from the remaining air traffic and the daily airport business

Here:

Simultaneous optimization of the approach trajectories of multiple

aircraft present in the vicinity of an airport

Landing sequence is not pre-determined and has to be found

by the optimization procedure

More realistic results

Institute of


Outline

2. Aircraft Simulation Model

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Aircraft Simulation Model

Point-Mass Simulation Model:

Position Equations of Motion (NED-Frame):

Translation Equations of Motion:

O

G

K

G

K

G

K

G

K

G

K

G

K

G

K

G

K

E

OV

V

V

z

y

x

sin

cossin

coscos

K

G

K

G

K

G

K

G

K

EO

K

G

K

G

K

G

K

m

V

V

V

F

1

1

cos

1

1

Institute of



Total sum of external forces:

Thrust modeling:

Aerodynamic coefficients:

O

G

GKOK

G

PK

G

AK

GFMFFF

2

20 LDDD CCCC

CMDALLL CCC ,0

maxTT T

4,

2

3

2

1max,

,5max,max

1

1

TcISAeffISA

Tc

Tc

TcISA

effISATcISA

CTT

hCC

hCT

TCTT

Institute of



Aerodynamic Forces:

Dynamic pressure:

Force vector:

2

20 LDDD CCCSqCSqD

ALLL CCSqCSqL 0

0 AQQ CSqCSqQ

25.0 AVq

AAA

A

G

PA

G

AA

G

L

DTT

L

D

0

0

00FFF

Institute of



Noise model:

Sound pressure level:

Sound exposure level:

Number of Awakenings:

drcrbTarTLA 2

lglg,

2

1

10

0

101

lg10

t

t

tL

AE dtt

L A

i

iiAEAW pLn79.1

, 300087.0

Institute of



Atmospheric model (DIN ISO 2533):

Fuel consumption:

hr

hrH

E

EG

Geometric

Altitude [km]

100

90

80

70

60

50

40

30

20

10TROPOSPHERE

STRATOSPHERE

MESOSPHERE

Mesopause

Stratopause

Ozone

Zirrus

Cumulonimbus

Temperature [K]

180 200 220 240 260 280 300

Pressure [mbar]

0.0001

0.001

0.01

0.1

1

10

100

1000

Tropopausemax

2

1max, 1 TC

VCm

f

Affuel

1

1

Tr

S

R

g

G

S

TrS H

T

1Tr

S

R

g

G

S

TrS H

Tpp

idlefuelfuelTidlefuelfuel mmmm ,max,,

4

3, 1f

fidlefuelC

hCm

Institute of


Outline

3. Multi-Aircraft Optimization Problem

Institute of


Multi-Aircraft Optimization Problem

Determine the optimal control histories

and the corresponding optimal state trajectories

that minimize the Bolza cost functional

subject to

the state dynamics

the initial boundary conditions

the final boundary conditions

the interior point conditions

the equality constraints

and the inequality constraints

m

iopti t ,u

n

iopti t ,x

iiiiiiii tttt ,,uxfx

iq

iiiii tt ,0,0,0,0 , ψ0xψ

ip

ifififiif tt ,,,, , ψ0xψ

ik

iiii tt r0xr ,

is

iineqiiiiiiineq ttt ,, ,, C0uxC

ir

ieqiiiiiieq ttt ,, ,, C0uxC

N

i

t

t

iiiiiiiififii

if

i

dttttLtteJ1

,,

,

,0

,,, uxx

Ni ,...,1

Institute of


Initial boundary conditions:

Defined by the entry position into the considered air space

Final boundary conditions:

Assure that the aircraft are finally located on the ILS glide path

Final approach fix: located at the origin of the Local Fixed Frame 𝑁

at an altitude of ℎ𝐹𝐴𝐹

ILS glide path: directed parallel to the 𝑥-axis of the Local Fixed Frame 𝑁,

into the direction of the positive 𝑥-axis


Institute of


Final boundary conditions:

Northward position:

Eastward position:

Altitude:

Glide-path angle:

Heading angle:

Kinematic velocity:

xxtx FAFf

FAFf yty

fILSKFAFf txhth ,tan

ILSKfK t ,

ILSKfK t ,

ILSKfK VtV ,


Institute of


Inequality path constraints:

Load factor:

Kinematic velocity:

Angle of attack:

Bank angle:

Thrust lever:

Eastward position:

Altitude:

Aircraft distance:

UBZZLBZ ntnn ,, 15.185.0

UBKKLBK Vh

kmtV

h

kmV ,, 1000200

UBCMDACMDALBCMDA ,,,,, 05.2073.5

UBCMDKCMDKLBCMDK ,,,,, 4545

UBCMDTCMDTLBCMDT ,,,,, 0.10.0

tytyty UBLB

ththth UBLB

0min dtdij


Institute of



Path constraints are formulated such that the aircraft have to follow

the ILS glide path once the FAF has been passed

Eastward position:

Altitude:

Kinematic velocity:

0.100tanh15.0 LBLB ytxaty

0.100tanh15.0 UBUB ytxaty

txhhtxath ILSFAFLBLB tan0.100tanh15.0

txhhtxath ILSFAFUBUB tan0.100tanh15.0

0.100.10tanh15.0 ,,,, ILSKILSKLBKLBK VVVtxatV

0.100.10tanh15.0 ,,,, ILSKILSKUBKUBK VVVtxatV


Institute of



Path constraints are formulated such that the aircraft have to follow

the ILS glide path once the FAF has been passed

Permitted airspace Path constraints w.r.t. kinematic velocity 𝑽𝑲


Institute of



A certain separation distance between the aircraft has to be maintained

Aircraft distances:

Normalization of flight times w.r.t. final flight times:

Introduction of one single parameter for all flight times:

The time elapsed is the same for all aircraft

Constraints w.r.t. the minimum distances can be checked directly

(because of the direct correlation of the time elapsed)

NijNitztztytytxtxtd jijijiij ,...,1,,...,1 ,2 2 2

Nit

t

if

ii ,...,1,

,

Nffff tttt ,2,1, ...


Institute of


Cost function:

Fuel-minimal approaches: maximize aircraft masses at the final times

Noise-minimal approaches: minimize maximum sound pressure level

or number of awakenings

Integral cost functions:

The same flight time for all aircraft is enforced

Aircraft are located on different positions on the ILS glide path

(i.e. they have covered different distances)

Equal weighting of the aircraft has to be achieved

Portion of the integral cost function originating from flight along ILS glide path

is not incorporated into the integral cost function:

N

i

ifi tmJ1

,

fuelfuelefffuel xtxatmtm tanh15.0,


Institute of


Full-Discretization Method – Forward (explicit) Euler:

⇨ Time discretization (e.g. equidistant):

⇨ Discretization of controls and states at time discretization points:

⇨ Approximation of differential equations:

⇨ Additional equality constraints:

1,...,1 ,),,(=1 Nih iiii puxfxx

1,...,1 ,),,(1 Nih iiii 0puxfxx

Niii ,...,1 ,, ux

1= ,,1,= ,1)(=

0

0

N

tthNihit

f

i


Institute of


Discretized Optimal Control Problem (Euler):

Determine the optimal parameter vector

that minimizes the cost function

subject to

the inequality constraints

and the equality constraints

SNOPT (sequential quadratic programming SQP)

T],[ uxz

zJ

0zzxC ,ineq

0

zzxψ

zzxr

uxfxx

zzxC

zzxψ

C

,

,

),(

,

,

1

0

f

iiii

eq

h


Institute of


Solution Strategy:

(1) Optimization without distance path constraints

(2) Simultaneous optimization with distance path constraints,

using previous results as initial guess

Distance path constraints fulfilled by initial guess:

Initial guess = Optimal solution of constrained problem

Distance path constraints not fulfilled by initial guess:

Separation of aircraft until path constraints are met

Assumptions:

Optimal solution of unconstrained problem = Excellent initial guess of constrained

problem

Cost function of a specific optimization problem is always less than or

equal to the cost function of the same trajectory optimization problem with

additional constraints


Institute of


Outline

4. Results

Institute of


Results

Generic scenario:

The optimal landing sequence and the optimal approach trajectories

for four aircraft are sought

Initial conditions:

AC No. 1 2 3 4

𝑥𝑖,0 −100000 𝑚 −80000 𝑚 −40000 𝑚 −100000 𝑚

𝑦𝑖,0 100000 𝑚 50000 𝑚 −120000 𝑚 −70000 𝑚

ℎ𝑖,0 4000 𝑚 4000 𝑚 5000 𝑚 6000 𝑚

𝜒𝐾,𝑖,0 −60.0° −45.0° 95.0° 45.0°

𝛾𝐾,𝑖,0 0.0° 0.0° 0.0° 0.0°

𝑉𝐾,𝑖,0 450.0 𝑘𝑚/ℎ 360.0 𝑘𝑚/ℎ 540.0 𝑘𝑚/ℎ 600.0 𝑘𝑚/ℎ

Institute of


Results

Optimized approach trajectories

-100 -50 0 50

200

400

600

800

1000

Position x [km]

Ve

locity V

K [km

/h]

Bounds

Aircraft 1

Aircraft 2

Aircraft 3

Aircraft 4

Optimized aircraft velocities 𝑽𝑲

Institute of


Results

Optimized time histories of aircraft controls

Distances between aircraft

0 200 400 600 800 1000

0

10

20

Time [s]

A

,CM

D [°

]

0 200 400 600 800 1000

-40-20

02040

Time [s]

A

,CM

D [°

]

0 200 400 600 800 1000

0

0.5

1

Time [s]

T,C

MD

[-]

0 200 400 600 800 10000

50

100

150

200

250

Time t [s]

Dis

tan

ce

d [km

]

AC 1 - AC 2

AC 1 - AC 3

AC 1 - AC 4

AC 2 - AC 3

AC 2 - AC 4

AC 3 - AC 4

Lower bound

Institute of


Outline

5. Summary & Outlook

Institute of


Summary

The approach trajectories of multiple aircraft in the vicinity of an airport

have been optimized simultaneously

Path constraints have been introduced so that the aircraft are finally

located on the ILS glide path and keep a certain separation distance

The optimal landing sequence is determined by the optimization procedure

Outlook

Utilization of splines to describe the centerlines of the allowed flight path corridors

for the involved aircraft

Introduction of path constraints w.r.t. the maximum lateral and horizontal

deviation from the centerlines

More sophisticated distance path constraints between aircraft

Institute of


Thank you very much for your attention!

optimal scheduling of fuel-minimal approach …...technische universität münchen institute of...

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