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MULTI-OBJECTIVE DEPARTURE TRAJECTORY OPTIMISATION OF COMMERCIAL
AIRCRAFT ON ENVIRONMENTAL IMPACTS
Mengying Zhang
(1)
, Antonio Filippone
(2)
, Nicholas Bojdo
(3)
(1)
The University of Manchester, United Kingdom, M13 9PL, mengying.zhang@manchester.ac.uk
(2)
The University of Manchester, United Kingdom, M13 9PL, a.filippone@manchester.ac.uk
(3)
The University of Manchester, United Kingdom, M13 9PL, nicholas.bojdo@manchester.ac.uk
KEYWORDS: trajectory optimisation, noise,
emissions, genetic algorithm
ABSTRACT:
This research systematically evaluates the impacts
on the environment regarding aircraft noise and
exhaust emissions, aiming at identifying the
potential optimal aircraft flight trajectories with
respect to different objectives.
A trajectory
optimisation method based on genetic algorithms
is developed to cover a wider range of formations
of optimisation objectives without strict
requirements of problem formulation. To solve the
low computational efficiency problems caused by a
large number of free parameters, a new
parameterization approach is applied to discretize
the dynamics equations. Furthermore, a dynamic
bound method is proposed to define the upper and
lower bounds of free parameters to avoid
threatening potential search space. By applying
the proposed method, numerical simulation is
conducted addressing different optimisation tasks
with required operational constraints. Results
reveal that the proposed method is applicable,
reliable and flexible to solve multi-objective
optimisation problems.
1. INTRODUCTION
Air transportation is one of the most significant
traffic services in the word. While making a
balance between maximising the utilisation of
aviation growth and minimising the environmental
impact of aircraft operations might be drastic and
critical. With the rapid growth of commercial
aviation, there is an increasing public concern on
the environmental effects of air travel, such as
noise and air pollution. Currently, commercial
aircraft follow prescribed procedures during
departure and approach, such as Standard
Instrumental Departures (SIDs) and Standard
Terminal Arrival Routes (STARs)[1], [2]. Moreover,
in order to reduce noise impact for departing
aircraft, the ICAO proposed two general families of
procedures: the Noise Abatement Departure
Procedure (NADP) 1 to reduce noise in zones
close to the airport and NADP 2 to reduce noise in
zones far away from the airport [1]. However, such
standard procedures are designed for a bunch of
different types of aircraft under different flight
conditions, which means they need to be adapted
to specific aircraft and airport, otherwise, they
might not always be optimal in terms of
environmental impacts optimisation.
The possibility of using trajectory optimisation
methods to minimise noise and pollutant emissions
has generated wide interest during the past few
years. Visser et al. [3], [4] developed a trajectory
optimisation tool named NOISHHH, integrating a
noise model, a geographic information system and
a dynamics trajectory optimisation algorithm. The
core of the trajectory optimisation tool is the direct
optimal control problem solver which implements
the collocation method [5] to convert the
continuous optimal control problem into a finite-
dimensional nonlinear programming problem. A
similar algorithm has also been adopted in the
optimisation methodology developed by Hartjes et
al. [6], developed for multi-event aircraft
trajectories. Moreover, Prats et al. [7], [8] applied a
multi-criteria optimisation strategy with the
lexicographic-egalitarian technique to minimise the
noise annoyance impact in noise sensitive areas.
In the research of Khardi et al. [9], the direct
method was considered to be more adapted for
solving trajectory optimisation problem minimising
aircraft noise at reception points around airports
compared with the indirect method.
All the methods mentioned above fall into the
category of either gradient-based or derivative-
based methods. However, these kinds of
numerical algorithms have their own limitations
when dealing with optimisation problems for
discontinuous models. This means that their
objective function and constraints need to be
differentiable. With the increasing complexity and
integration of current optimisation problem
formulation, not all integrated problems can be
2
constructed with continuous models and functions
with continuous derivatives. This has led to a boom
of heuristic algorithms that are generally not
computationally competitive at present, but do not
need gradients of functions, making them more
suitable and flexible to search global optimal
solutions to the specific types of optimisation
problems described above. Torres et al. [10]
studied the implementation of a multi-objective
mesh adaptive direct search (multi-MADS) method
to minimise noise and NO
x
emissions for departing
aircraft. Similarly aimed at minimising noise impact
for arrival trajectories, Yu et al. [11], [12] conducted
state parameterization with Bernstein polynomials
to transform the infinite-dimensional optimal control
problem into a finite-dimensional parametric
optimisation problem. In his research, a genetic
algorithm was employed to optimise the
parameters within the search space to obtain
optimal trajectory both in vertical and lateral plane.
Similarly, Hartjes and Visser [13] applied the
genetic optimisation algorithms to design departure
flight path focusing on noise abatement and NO
x
reduction. However, genetic algorithms usually
have the disadvantage of being computationally
expensive, due to the need for a large number of
free parameters for the problem evaluation.
In this study, a trajectory optimisation method
based on genetic algorithms is developed to cover
a wider range of formations of optimisation
objectives without the strict requirements of
problem formulation. The aircraft flight mechanics,
aerodynamics, propulsion and acoustics models
used in this work are based on the configuration of
Airbus A320-211 with CFM56 engines. To solve
the low computational efficiency problems caused
by a large number of free parameters, a new
parameterization approach is applied to discretize
the dynamics equations on the vertical and lateral
motion planes. Furthermore, a dynamic bound
method is proposed to define the upper and lower
bounds of the free parameters to replace their
prescribed fixed values, for the reason of not
threatening the potential search space and
satisfying the majority of the path constraints.
The rest of this paper will be structured as follows.
First of all, the problem statement including the
trajectory optimisation problem formulation and the
objective model will be presented. Next, the
procedures of trajectory decoupling, segmentation
and state parameterization are described, followed
by the dynamic bound technique. Subsequently, a
numerical example and its results are presented.
Finally, the paper is concluded with the discussion
and future plan.
2. METHODOLOGY
2.1 Aircraft Dynamics Model
In order to minimize the environmental impact, a
constrained optimal control problem has been
developed to define the trajectory optimisation
process. Components including the aircraft flight
dynamics model, constraints of flight configuration
and safety issues, cost functions of different
concerns are introduced in the following sections.
In general, an aircraft can be modelled as a rigid
body with varying mass, aerodynamic, propulsive
and gravitational forces. Several assumptions are
made in order to simplify the problem: (1) the earth
is considered to be flat and non-rotational, (2) no
wind is presented in this work, (3) all force acting
on the aircraft go through its centre of gravity, (4)
the angle between the engine thrust
N
F
and the
longitudinal axis of the aircraft is assumed to be
equal to zero, (5) the angle of attack
α
is small.
Then a 3DOF flight dynamics model with a set of
differential algebraic equations associated with a
variable mass
m
is as following.
sin
cos cos
sin
cos
cos sin
cos cos
sin
N
F mg D
Vm
L mg
mV
L
mV
x V
y V
h V
m f
γ
µ γ
γ
µ
χγ
γ χ
γ χ
γ
− −
=
−
=
=
=
=
=
= −
&
&
&
&
&
&
&
(1)
where
V
is the true airspeed,
γ
is the flight path
angle,
χ
is the heading angle,
µ
is the bank
angle,
x
,
y
,
h
are the state variables to describe
the location of the aircraft in the three dimensional
space,
f
is the fuel flow rate. Based on the
comprehensive flight mechanics software
FLIGHT[14]–[16], the chosen model to calculate
the engine thrust
N
F
and fuel flow rate
f
are
expressed as
3
1
( , , )
N N
F F h Ma N
=
(2)
1
( , , )
f f h Ma N
=
(3)
where
Ma
is the Mach number,
1
[70%,103%]
N
∈
is the engine rpm.
The aerodynamic lift
L
and drag
D
are given by
2
1
2
L
L V C S
ρ
=
(4)
2
1
2
D
D V C S
ρ
=
(5)
where
ρ
is the atmospheric density,
S
is
reference area,
L
C
and
D
C
are the lift coefficient
and drag coefficient respectively both of which can
be obtained from the FLIGHT with required aircraft
configuration and motion parameters as well.
2.2 Parameterization
This section will introduce the method adopted to
discrete the trajectory in the time interval. Several
assumptions are made first for decoupling the
motions between horizontal and vertical planes.
Firstly, the force normal to the flight path is
assumed in equilibrium. This assumption
0
γ
=
&
is
made so that the lift and the portion of the weight
normal to the flight path is balanced during each
time step of the climb procedure, which leads to
cos cos
0
L mg
mV
µ γ
γ
−
= =
&
(6)
Then, lift coefficient is calculated by the Eq.7.
2
1
2
cos
cos
L
mg
CV S
γ
ρ µ
=
(7)
After assuming a parabolic drag polar, the drag
force could be derived by the Eq. 8.
2 2
1
02
( ( , ) ( , ) )
D L
D C h V k h V C V S
ρ
= + ⋅
(8)
where
0
D
C
and
k
are the parabolic drag polar
coefficients. Similarly, small value as it is, the
angle of attack
α
can be derived from the linear
equation
0
( , )
( , )
L L
L
C C h V
C h V
α
α
−
=
(9)
where
0
L
C
and
L
C
α
are the zero lift coefficient
and lift curve slope respectively. Then
α
will no
longer be a control variable.
However, since
0
γ
=
&
is assumed within each time
step, the value of flight path angle
γ
in each time
step should be defined as an input parameter,
which makes
γ
a control variable. Thus we have a
state variable vector
[ , , , , , ]
T
V x y h m
χ
=x
, and a
new control vector
1
[ , , ]
T
N
γ µ
=u
.Then the Eq. 1
can be decoupled in the vertical plane and
horizontal plane respectively in Eqs. 10-11.
tan
Horizontal:
cos sin
cos cos
g
V
x V
y V
µ
χ
γ χ
γ χ
=
=
=
&
&
&
(10)
sin
Vertical:
sin
N
F mg D
Vm
h V
m f
γ
γ
− −
=
=
= −
&
&
&
(11)
2.2.1 Horizontal Parameterization
Most departing commercial aircraft will follow
departure routes that have been based on the
modern navigation technology. Requirements such
as to make sufficient use of runway capacity, to
maintain safe separation between departures, to
avoid the increase in traffic control workload and to
mitigate new population exposed to noise have
been taken into consideration, giving rise to a
concentration of pre-programmed departure
procedure, especially for the lateral tracks. Among
those horizontal track parameterization methods,
waypoints are usually used to define the departure
routes. The lateral tracks can be expressed by a
series of waypoints enable to construct the flight
path between each two waypoints. Trajectories
models defined by spline interpolation have been
developed and introduced[17]. From the easiest
straight line segments to the more complex
piecewise cubic interpolation, different types of
splines have been applied either to connect each
4
waypoint or to construct the flight path along the
direction guided by the series of waypoints.
In this study, lateral track is expressed by a
sequence of waypoints and legs. Two different
types of legs: track-to a-fix (TF) legs and radius-to-
a-fix (RF) legs - are preferred. This means the
lateral trajectories can be constructed with straight
legs and constant radius turns[13], [18], [19]. The
arc one is commonly used for connecting two
straight legs which have different heading angles
(see Fig. 1). Assume the ground track is divided by
n
segments, then to define a constant radius turn,
the required parameters are the radius
i
R
and the
angle of fly-by-turn
i
θ
,
1,2,...,
i n
=
. Note that the
angle of fly-by-turn is equal to the absolute value of
the change of the heading angle
i i
θ χ
= ∆
.
Figure 1. Example Ground tracks with straight legs
and constant radius turns with
5n=
With a given radius
R
and the local true airspeed
V
, the control variable bank angle
µ
are able to
be derived from Eqs. 12-13.
2
tan
V
m mg
R
µ
=
(12)
2
1
tan ( )
V
gR
µ
−
= ±
(13)
To define a straight leg, only the length of the leg
l
is needed because the heading angle stays
constant with the one at the last waypoint, which
lead to a zero bank angle during the straight
segment.
0
µ
=
(14)
Thus the coordinate of the waypoints
( , ), 1
i i
x y i n≤ −
, can be expressed by the ground
track parameters. For straight segment,
1 1
1 1
sin
cos
i i i i
i i i i
x x l
y y l
χ
χ
− −
− −
= −
= +
(15)
and for arc segment
1 1
1 1
2 sin sin( )
2 2
2 sin cos( )
2 2
i i
i i i i
i i
i i i i
x x R
y y R
χ χ χ
χ χ χ
− −
− −
∆ ∆
= − +
∆ ∆
= + +
(16)
For a lateral track with fixed terminal point, only
one more freedom need to be constrained to
define the last two segments. To sum up, to
construct a ground track with
n
segments,
( 1) / 2
n+
straight lines and
( 1) / 2
n−
arcs are
needed to compose this ground track if the last
segment is straight. Parameters to define this kind
of ground track are shown in Tab. 1. The total
number of the parameters required is given in Eq.
17.
1 1 3 5
( 1) 2( 1) 1
2 2 2
n n n
N+ − −
= − + − + =
(17)
Table 1.Ground track parameters
segment Parameter needed
1
st
1
l
th
( 2,..., 2)
i i n= −
( , )
i i
R
χ
∆
for the arcs
i
l
for the straight lines
th
1n−
1n
R
−
th
n
none
2.2.2 Vertical Parameterization
2.2.2.1 Dynamic Variable Boundary
Before entering into the parameterization of the
vertical flight path, a technique dealing with
variables value range in this study will be
presented first. In solving trajectory optimization
problem, the application of constraints usually has
more requirements to be considered. For both
runway
5
derivative-based algorithms and non-based
algorithms, state variables or free parameters are
limited in some prescribed interval as is shown in
Eq. 18.
min max
x x x
≤ ≤
(18)
Although the constraints are made for the aircraft
to fly safely and reliably, to decide the lower and
upper bounds of the variable depends much on
transcendentally prescribed value, which may
manually narrow or enlarge the search space of
the original problem. Computational efficiency
might also be affected when the range of search
space is changed. As a consequence, an effective
and efficient way to define the boundary of the
variables and free parameters is of high
importance, especially for the non-derivative
algorithms (e.g. genetic algorithm). Unlike the
direct or indirect methods to solve this optimisation
problem, there are no dynamic constraints in the
process of GA solving. As a consequence, solution
space depends much on the search space made
from prescribed boundary values of free
parameters.
When applied in this study, the boundary values
for control variables are no longer constants but
change correspondently with real-time flight
condition in each time step.
1min, 1 1
1max,
( )
103%
i i
i
N N D
N
−
=
=
(19)
For level flight segment,
o o
min, max,
0 , 0
i i
γ γ
= =
(20)
For climb segment,
o
min,
1max 1
max, 1
1
0
1
sin
i
N i
iii
F D dV
m g g dt
γ
γ
−−
−−
=
−
= −
(21)
where
max 1max
( )
N N
F F n
=
at specific altitude and
velocity,
1 1 1
1
Ni i i
i
dV
F m g D
dt
− − −
−
= − −
, subscript
i
denotes the
th
i
time step. Before each time step, a
judgement will be made to assure the value of the
control variables
1
n
and
γ
are within their updated
boundaries. Otherwise the variables need to be
assigned reasonable values
1, 1min, 1max, min,
rand(1) ( )
i i i i
N N N N
= + × −
(22)
min, max, min,
rand(1) ( )
i i i i
γ γ γ γ
= + × −
(23)
2.2.2.2 Vertical Segmentation and
Parameterization
The departure procedure can be separated into
several segments of climb and acceleration. One
possible schedule is shown in Fig. 2 with example
parameter values. According to this flight operation
sequence, two accelerations and two constant
speed climb phases are taken into account. Those
parameters that describe the climb profile and their
effect on the noise perceived at the noise sensitive
points and emission pollution are the major
concerns we need to investigate.
Figure 2. Take-off and climb-out[14]
A. Phase 1 : A-B
In segment AB, the aircraft is assumed to take off
at maximum thrust until it reaches the point B
where the speed
B
V
and altitude
B
h
are
prescribed. As is shown in Fig. 2, during this phase,
several operations of high lift device and the
landing gear are usually taken into consideration.
Yet these operations do exert little impact on the
aircraft noise during the departure procedure. As a
result we assume the aircraft operates under a
clean configuration.
According to Eq. 6; the designed parameter in the
AB phase are
A
γ
and
B
V
, where
A
γ
is the initial
flight path angle,
B
V
is airspeed to be achieved at
the end point B. A sequence is adopted to prevent
the aircraft overfly the target terminal attitude and
airspeed. If the terminal attitude
B
h
is achieved
first, let
o
min
0
γ γ
= =
and accelerate until the
6
aircraft reaches the prescribed airspeed
B
V
. From
the other aspect, if
B
V
is reached first, let
0
V
=
&
,
which leads to a constant speed climb to the target
altitude .
Note that in reality, there are other parameters to
define this segment. For example, according to the
ICAO B procedure, the target altitude
B
h
could be
any value between 800 feet to 1,500 feet, which
makes it a free variable as well.
B. Phase 2: B-C
In this phase, the aircraft is assumed to climb at a
constant speed. The terminal condition is to reach
a final altitude
3,000 feet
C
h
=
. During this constant
speed climb phase, the only free parameter comes
to be the flight path angle
B
γ
.
The procedure to solve this flight phase is listed
below.
(1) Climb procedure starts from
i B
γ γ
=
(2) Calculate the lift coefficient
,
L i
C
and drag
i
D
according to Eqs. 7-8.
(3)Since
0
V
=
&
, then the thrust can be obtained by
, , 1
sin ( , , )
N i i i i D i i B
F m g D C h V
γ
−
= +
(24)
(4) Induce
i
γ
,
,
L i
C
,
i
D
and
,
N i
F
into the dynamics
equation to obtain
i
h
and
i
V
in this time step.
Then go to the next step (1) and repeat the
procedure until the final altitude
C
h
is reached.
C. Phase 3: C-D
In this phase, the aircraft is expected to do a level
acceleration until it reaches a target speed
D
V
for
the next climb. Therefore, the acceleration
becomes the variable that decides the profile of
this phase which could be control by the thrust or
the engine rpm
1
CD
N
. Assume the aircraft fly at a
constant
1
N
in this phase, a similar process is
applied until the aircraft reaches its target airspeed.
D. Phase 4: D-E
As is shown in Fig. 2, after the acceleration in
segment CD, one more climb segment has to be
added to the vertical profile in order to achieve the
terminal ground projection point
( , )
f f
x y
, which
needs one parameter
D
γ
to define the climb rate
of this segment. Procedure to constrain the flight
path angle and engine rpm is the same as what
has applied in the past segments. Constraints on
final attitude according to the air traffic control
requirements also need to be considered
max
D
h h
≤
,
where
max
h
is the SID’s upper limit (usually is
5,000 feet). Before reaching the ground target
point or the altitude limit, the aircraft keeps climb
with a constant speed. The problem of overpassing
the altitude limit at the end of the trajectory can be
solved by setting
o
min
0
γ γ
= =
once it achieves
5,000 feet. Another problem that is somehow more
difficult is that the constraints for thrust will be
violated if the thrust derived from Eq. 24 is smaller
than its lower bound
min
N
F
. To deal with this
specific problem, the thrust setting is changed to
be
min
N N
F F
=
. When
min
N N
F F
<
, the aircraft will
accelerate under the new power setting until it
reaches the terminal position. To sum up, seven
free parameters are needed to decide a vertical
climb profile
vertical 1
[ , , , , , , ]
T
A B B B CD D D
V h N V
γ γ γ
=p
(25)
2.2.3 Coupling of Horizontal and Vertical
Profiles
Previous sections present the trajectory
parameterization of the horizontal and vertical
profile respectively. From the Eqs.10-11, we
understand that the departure trajectory can be
parameterized through two sets of free parameters:
horizontal
p
to describe the lateral track on the ground
and
vertical
p
to define vertical motion. Fig. 3 depicts
the computational procedure of the 3D flight
dynamics model. As is shown in Fig. 3, three
control variables, namely the engine rpm
1
N
, the
flight path angle
γ
and the bank angle
µ
,
determine the motions on both horizontal and
vertical planes.
2.4 Constraints
In this section, constraints that are taken into
account are discussed. Firstly, path constraints for
state variables are
stall
250kt
V V
< ≤
(26)
0 max
h h h
≤ ≤
(27)
where
stall
V
is the stall speed,
0
h
is the initial
altitude. For a departure aircraft, neither
descending nor deceleration should be permitted.
7
Thus, a couple of path constraints are introduced
as
0
γ
≥
(28)
0
V
≥
&
(29)
vertical
p
horizontal
p
1
[ , , ]
T
i i
N
γ µ
=u
tan
cos sin
cos cos
g
V
x V
y V
µ
χ
γ χ
γ χ
=
=
=
&
&
&
sin
sin
N
F mg D
Vm
h V
m f
γ
γ
− −
=
=
= −
&
&
&
1,
,
i i
N
γ
2
1
tan ( ),
i
i
i
V
gR
µ
−
= ±
1
i i
= +
1 1
[ , , , , , ]
T
i i
V x y h m
χ
+ +
=x
Figure 3. Computation procedure of 3D flight
dynamics model
Secondly, for the control variables
1
[ , , ]
T
N
γ µ
=u
,
constraints on
1
N
and
γ
could be obtained from
Eqs. 24-26. However, as is discussed in Eqs. 13-
14, the bank angle
µ
is an implicit rather than an
explicit parameter in defining the ground track. As
a result, the maximum bank angle of the aircraft
could be deduced from the normal force equation
1 2
max
tan 1
n
µ
−
≤ −
(30)
where
n L mg
=
is the load factor.
Thirdly, the values of state and control variables of
the dynamics system at the initial and final time are
defined by the boundary conditions.
0 0 0 0 0 0
( ) , ( ) , ( )
h t h x t x y t y
= = =
(31)
max
( ) , ( ) , ( )
f f f f f
h t h x t x y t y
≤ = =
(32)
where
( , )
f f
x y
is usually chosen from the
coordinates of the waypoints where the aircraft
leaves controlled airspace of that typical airport.
Finally, other constraints on ground track include
the length of the straight segment
l
, radius of the
turn
R
, and the fly-by angle of each turn
χ
∆
. In
general, the definition of
horizontal
p
should be set in
coordinate with specific scenario. Yet, some
similarities are shared in common. For example, in
the first segment after take-off, usually no turns are
permitted below a warning altitude
warning
h
[20]
1 warning
cot
A
l h
γ
≥
(33)
For the constraints of the radius, no sharp turns
are allowed in the case of safety issue, therefore
the constraint can be given by Eq. 34.
2 2
2
max
tan
1
V V
Rgg n
µ
= ≥
−
(34)
2.5 Cost Function
Usually, different objectives such as flight duration,
fuel burnt, noise, emissions and so on can be set
as the optimisation objectives for the trajectory of
an aircraft. For the assessment of environmental
impacts, FLIGHT provides modules to evaluate
different indexes. Usual aircraft environmental
indexes, such as the EPNL (Effective Perceived
Noise Level), SEL (Sound Exposure Level),
max
A
L
(Maximum noise level), fuel consumption,
2
CO
(carbon dioxide),
NO
x
(oxides of nitrogen) and so
on are able to be calculated from this software.
Apart from the pure physical measurement
quantifying environmental impacts, an integrated
index to evaluate the financial cost of fuel
8
consumption and gaseous emissions is given in Eq.
36. Instead of summing up different objectives with
the weighted method, this model transforms the
fuel consumption , total amount of CO
2
emissions
and NO
x
emission under 3,000 feet[21] into a
common metric from the aspect of the expenses to
control the pollutants and energy consumed.
2 x
fuel CO 2 NO x
EC UC +UC CO +UC NO
F
= ⋅ ⋅ ⋅
(35)
where
EC
denotes the environmental cost for fuel
and emissions,
UC
is the unit cost: for the fuel
consumption, it is the unit jet fuel price; for the
gaseous emissions, environment-related taxes are
adopted.
3. NUMERICAL EXAMPLE
To demonstrate the capability of the method
proposed, we consider one Airbus 320-211 aircraft
departing toward the west off Runway 23R in
Manchester Airport[21]. The closest community
near the end of the Runway 23R/05L is Knutsford
which will be annoyed by the aviation noise most.
Thus, trajectory optimisation on departure for
minimizing noise impact on Knutsford as well as
local emission impact is the objectives.
1 2
EPNL, EC
J J
= =
(36)
With the origin point located in the coordinate of
Runway end 23R, the aircraft departures with initial
conditions that takes off at
0
916.7118m
x
= −
,
0
399.5492m
y
= −
,
0
3m
h
=
,
0
=75m/s
V
with landing
gear retracted and departure flaps selected. The
finial conditions are selected from an existing SID,
with final ground location fixed
18119.1951m
f
x= −
,
16708.6863m
f
y= −
,
1524m
f
h≤
. Lateral track to
be optimised can be segmented into five segments
with three straight legs and two constant radius
turns. Free parameters
horizontal 1 2 2 3 4
=[ , , , , ]
T
l R l R
χ
∆p
and their bounds to define the ground track are
listed in Tab. 2.
Table 2 Free parameters of ground track
segment
number Parameter
1
1
[4000m,6000m]
l
∈
2
2
[2000m,3000m]
R
∈
,
2
o o
[40 ,90 ]
χ
∆
∈
3
3
[2000m,4000m]
l
∈
4
4
[3000m,9000m]
R
∈
5 none
Similarly, the free parameters for vertical profile
vertical 1
[ , , , , , , ]
T
A B B B CD D D
V h N V
γ γ γ
=p
and their
initial value range are shown in Tab. 3. Note that a
reference value (i.e.
o
2.8
) according to the
existing SID is adopted for
D
γ
in order to decrease
dimension of the search space. Then we have six
free parameters to construct the vertical profile.
Therefore, a 3D departure trajectory is fully
described with 11 free parameters. Due to the
discrete formulation of this optimal control problem,
functions of objectives and constraints are not
derivable, which leads to a preference to use
algorithms that do not need any gradient
information. For this reason, a fast and elitist multi-
objective genetic algorithm (NSGA-II)[22] is
adopted as the optimiser.
Table 3. Vertical profile parameters
Parameters Lower bound Upper bound
A
γ
o
4
o
12
B
V
80m/s
100m/s
B
γ
o
4
o
12
B
h
243.84m
457.2m
1
CD
n
86%
103%
D
V
100m/s
128.6m/s
D
γ
o
2.8
o
2.8
4. RESULTS
The example presents the departure trajectory
optimisation aiming at minimizing Effective
Perceived Noise Level (EPNL) and environmental
cost for fuel consumptions and emissions (EC) in
the 3D space. Before proceeding to the result, a
reference trajectory is randomly picked as the
baseline for the purpose of performance
comparison.
The comparison result of the four representative
cases, namely the reference one (case1),, the
optimal solution (case2), EPNL-optimal case
(case3), and EC-optimal case (case4), are
summarized in Tab. 4. . Although time and
distance
d
were not considered as the
optimisation objectives in this example, they are
included in Tab. 4 as the performance indicator for
9
the four cases. Fig.4 illustrates the comparison of
their ground tracks.
Table 4. Comparison for the optimised trajectories
and the baseline
Case Time(s)
d
(km)
∆
EPNL(%)
∆
EC(%)
1 360.3607 35.1932 / /
2 309.3725 30.0413 -3.6900 -9.1161
3 280.3972 30.1672 -7.8633 -2.8634
4 271.0264 27.8373 +8.1540 -12.0977
Figure 4. Departure ground track comparison
With the comparison between the values of bio-
objectives, the multi-objectives nature of the
problem is demonstrated. The extreme points of
the Pareto solutions, case 3 and case 4, exhibit
significant difference on the environmental indexes.
Regarding the EPNL objective, both case 2 and
case 3 have a progress in reduce noise impact (-
3.6900% and -7.8633% respectively). By
comparing the ground tracks of these two cases, it
is clear that in both cases, a relatively early turn
with larger radius is preferred when the aircraft
circumvents the Knutsford area, which increases
the receiver-to-source-distance and reduces the
sound energy received. Case 4, in the contrary,
flies much closer to the noise sensitive area
resulting in a higher noise level compared with
other three cases. However, it is not always true
that the further aircraft moves away from the
receiver the less annoyance it will cause. It is
shown in the baseline trajectory that though the
ground track is the furthest among others, its
EPNL is not the lowest.
Then as for the second
objective EC, case 2 and
case 4 have a reduction with 9.1161% and
12.0977% respectively. Considering their
difference in EPNL, case 4 spends less on
emission and fuel consumption at the price of
bringing higher noise level for the reason that the
fuel consumption is less with shorter flight
distance(see Tab 4). Yet this leads to a
consequence of flying over an area that is
sensitive to noise. Therefore, case 2 shows a
capability of optimising multiple environmental
impacts (e.g. noise, fuel consumption and gaseous
emissions) compared with the reference trajectory.
Figure 5. Altitude profile comparison
Figure 6. Airspeed profile comparison
The vertical departure procedure described in
2.2.2.2 is followed by the four cases, which can be
seen in Fig. 5-6. It was founded that optimised
-20 -15 -10 -5 0
-15
-10
-5
0
East(km)
North(km)
Knutsford
Case1
Case2
Case3
Case4
0 10 20 30 40
0
500
1000
1500
2000
Ground Distance(km)
Altitude(m)
Case1
Case2
Case3
Case4
0 10 20 30 40
70
80
90
100
110
120
130
Ground Distance(km)
Airspeed(m/s)
Case1
Case2
Case3
Case4
10
solutions are similar in altitude profile yet differ
from each other in the airspeed profile. From the
Fig 6, it follows that there is a preference towards a
lower airspeed
B
V
during the initial climb phase
before reaching 3,000 feet, which exerts a quieter
operation. After passing 3,000 feet, it can be found
in Fig. 6 that the influence of airspeed on the noise
level is weakened due to a long distance between
the receiver and the noise source.
From the analysis above, distance
d
is the most
influential factor for the EC index in a terminal point
fixed departure. Yet after comparing the airspeed
profiles of case 2 and case 3 (both of which have
similar ground distance (see Tab. 4)), it is indicated
that lower final target speed
D
V
leads to lower EC.
The main reason for this is that EC is directly
linked to the fuel consumption. Especially for the
last part of climb, lower speed keeps the engine
maintains at a less intensive working condition,
which brings in less fuel burn and emissions in the
end.
5. CONCLUSIONS
This paper presented the optimisation method
based on genetic algorithm for departure trajectory
of commercial aircraft on multiple environmental
impacts on local communities around the airport. A
parameterization method with dynamic bounds
added for constraints is applied to discretize the
dynamics models on both the vertical and lateral
planes. The proposed method is tested in the
departure scenario with noise, fuel consumption
and emission impact considered.
In the numerical case, departure trajectory of a
commercial aircraft is optimised with receptive to
multiple objectives. Results show that below 3,000
feet, lower climb target speed for the initial
acceleration and constant speed climb is preferred
to obtain a quieter departure. Beyond 3,000 feet,
lower airspeed has less impact on noise level yet
is potential to reduce the cost of fuel burn and
emissions. It is also concluded that detour is an
effective option to avoid causing high noise level
but only with the flying range limited as well.
Because longer range means increased noise
exposure time and more sound energy received.
This also has a negative effect on emission cost
criterion since the flight duration has a direct link to
a growing amount of jet fuel consumption.
Future work will attempt to assess the capability of
extending the proposed method to other flight
phases with comprehensive operations, for
example, arrival trajectory in real scenarios.
6. ACKNOWLEDGMENTS
The authors gratefully thank the financial support
from the China Scholarship Council (CSC). The
development of the noise prediction code was
partly supported by the CleanSky Agreement
255750 (project Flight-Noise). Further information
of the FLIGHT computer program is available
online: www.flight.mace.manchester.ac.uk.
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