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Aircraft flight trajectories optimization through
genetic algorithms for a LNAV and VNAV
integrated path
Roberto Salvador Félix Patrón and Ruxandra Mihaela Botez
ETS, Laboratory of Research in Active Controls, Avionics and AeroServoElasticity
(www.larcase.etsmtl.ca), Montreal, Quebec, H3C-1K3, Canada
Abstract
For long flights, the cruise is the longest phase and where the largest amount of fuel is
consumed. An in-cruise optimization method has been implemented to calculate the optimal
trajectory that reduces the flight cost. A 3D grid has been created coupling LNAV and
VNAV profiles. With a dynamic analysis of the wind, the aircraft can perform a horizontal
deviation or change altitudes via step-climbs to reduce fuel consumption. As the number of
waypoints and possible step-climbs is increased, the number of flight trajectories increases
exponentially, thus a genetic algorithm has been implemented to reduce the total number of
calculated trajectories. The aircraft’s model has been obtained from a performance database,
which is currently used in the commercial flight management system studied in this paper. A
5% flight cost reduction has been obtained.
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Nomenclature
ATC = Air Traffic Control
CI = Cost index
FMS = Flight Management System
GDPS = Global Deterministic Forecast System
ISA = International Standard Atmosphere
LNAV = Lateral navigation
PDB = Performance database
RTA = Required Time to Arrival
SC = Step climb
TOC = Top of climb
TOD = Top of descent
UTC = Coordinated Universal Time
VNAV = Vertical navigation
WP = Waypoint
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I. Introduction
The global aviation industry produced 689 million tons of CO2 in 2012, which represents
around the 2% of the total anthropogenic emissions produced worldwide [1]. The CO2
emissions contribute to global warming, which is one of the biggest environmental problems
encountered today. At short term, aviation industry has committed to improve by 1.5% per
year the fuel efficiency. At long term the goal is to reduce by 50% its net carbon footprint
compared to 2005.
To achieve the reduction of polluting effects produced by aircraft, the aviation industry has
been actively funding different types of research projects where the objective is to create
green aircraft (improved aircraft performance). Multiple solutions to reduce aircraft
emissions have been put forward. These can be divided in three major categories: aircraft
technology improvement, use of alternative fuels, and improvements in air traffic
management and airline operations [2]. Each of these categories could increase aircraft
efficiency and thereby reduce fuel burn and emissions.
One of the research areas in the aircraft technology improvement category is focused on
increasing engine efficiency through lighter designs [3], increased compression rates [4] or
optimized aerodynamic patterns [5], to name a few. Airlines have been constantly reducing
aircraft weight by changing to lighter seats
1
. Techniques to install more efficient electrical
wiring have also been studied [6]. Design studies to reduce drag through wing elasticity
1
Air Transat “Air Transat’s new cabin” - www.airtransat.ca, 2014.
4
improvements [7] or by increasing the aircraft efficiency through the addition of winglets
[8] have also contributed to this area.
To reduce its impact on climate change, the aviation industry has been studying sustainable
biofuels to provide a cleaner source of fuel [9]. Today, the aviation sector uses petroleum-
derived liquid fuels, which is not only a limited fuel resource, but it also contributes to CO2
emissions. Hendricks, Bushnell and Shouse performed a study on biofuels in which they
conclude that there is not only a large productive capacity for biofuels, but also the
potential for carbon emission neutrality and reasonable costs [10]. Airline companies, such
as Porter, already used a 50:50 biofuel/Jet A1 fuel blend to perform a complete flight,
which shows that biofuels are an important option for a greener aviation sector
2
.
Air traffic management and airline operation improvement would also reduce aviation’s
environment footprint. Air traffic, however, has increased significantly. By 2030, an
estimated number of 5.9 billion passengers is expected, doubling the amount from 2010
[11]. Over the past few years, this growth has influenced many researchers to include
increasing levels of air traffic as a part of the trajectory optimization process. This has also
opened a research domain in conflict detection algorithms to increase air security [12-14].
Air Traffic Control (ATC) is in charge of assigning the trajectories to airlines; once in-
flight, authorization from ATC is required to perform a trajectory deviation. The flight
management system (FMS) is an in-flight device, and can be used to identify optimal
trajectories to propose to ATC.
2
Porter airlines “Porter Airlines Operates Bombardier Q400 Aircraft in Canada’s First Biofuel-Powered
Revenue Flight” - www.flyporter.ca, 2012.
5
An improved method of communication between the FMS and the ATC must be established.
Mayer studied the benefits of an integrated aviation modeling and evaluation platform, in
which ATC and the FMS could be coupled to obtain better flight path planning [15]. Studies
on including aircraft traffic control as one of the FMS functions have been analyzed [16].
For both the Next Generation Air Transportation System (NGATS) in the USA, and the
Single European Sky ATM Research (SESAR) in Europe, the implementation of Required
Time of Arrival (RTA) as a part of the FMS and ATC was an important step towards better
air traffic control. De Smedt and Berz studied the characteristics of different FMS’
performance to determinate the accuracy of their RTA and the influence it could have on
ATC [17]. Friberg’s study showed that promising results in terms of the environment could
be achieved by establishing communication between the FMS’ RTA function and ATC
[18]. Margellos and Lygeros presented a study to increase ATC’s predictability, efficiency
and safety, using the Monte Carlo method in 4D [19]. Air traffic conditions have also been
identified as the cause of missed approaches [20]. Dancila created an analysis tool to
estimate the fuel cost and the emissions produced by aircraft during a missed approach
[21].
The creation of more efficient trajectories for aircraft would contribute to the reduction of
fuel burn, and therefore to the reduction of CO2 emissions to the atmosphere. For long haul
flights, the cruise is the phase where the most significant reduction can be obtained. In fact,
80% of the CO2 emissions produced by aviation come from long flights (more than 1,500
km or 810 nm), and the cruise is where most of the fuel is consumed [11]. To improve the
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VNAV (vertical navigation) profile, Lidén studied the variation of the optimal altitude as
fuel is burned during the flight [22]. Lovegren analyzed how the fuel burn could be reduced
during the cruise phase by choosing the appropriate cruise altitudes and speeds and
performing step climbs [23]. Jensen et al. presented in [24] a speed optimization method for
cruises with fixed lateral movement by analyzing radar information from the United States
Federal Aviation Administration’s (FAA) Enhanced Traffic Management System (ETMS)
[25]. Their results show that most flights in the United States do not fly at an optimal speed,
which increases their fuel consumption. Dancila et al. studied a new method to estimate the
fuel burn from aircraft to improve the precision in flight trajectory calculations [26, 27].
Miyazawa et al. developed a four dimensional algorithm using dynamic programming in
order to reduce fuel burn from aircraft in a congested airspace. They modeled the aircraft’s
performance using BADA (Base of Aircraft Data), which is an open-source database of
aircraft models, and the air traffic restrictions as obstacles to avoid during a trajectory, and
used real flight coordinates to perform their tests. The fuel consumption was minimized
while respecting arrival time constraints and the vertical distance safety separation from
other aircraft [28].
The influence of weather on aircraft flight has been considered as part of strategies to take
advantage of winds to reduce flight time and/or to avoid headwinds that could increase
global flight costs. Fays developed a 4D algorithm treating meteorological conditions or air
traffic restrictions in a specified air space, defining them as obstacles, to improve the
FMS’s trajectory-creation capabilities [29]. Murrieta presented an algorithm which
optimized the vertical and horizontal trajectories, taking into account the wind forces and
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patterns as well as the variation of the cost index [30]. Gagné performed an exhaustive
research of all possible speeds and altitudes to obtain the optimal trajectory and reduce fuel
burn [31]. Bonami et al. studied a trajectory optimization method capable of guiding
aircraft through different waypoints considering the wind factors and reducing fuel burn,
utilizing a multiphase mixed-integer control [32].
As an alternative trajectory optimization method, arranging aircraft in formation was
analyzed by Kent and Richards. Formation flights were used to reduce drag, thereby
reducing fuel burn. Kent and Richards used two different methods: an extension to the
Fermat-Torricelli problem allowing them to find optimal formations for many routes, and a
geometric method to be able to apply the influence of the wind [33]. Nangia and Palmer
reduced overall drag of the order of 15-20% for commercial aircraft flying in formation [34].
Calculation time constraints have to be considered when the objective is to implement these
trajectories’ optimization algorithms into a FMS. Genetic algorithms have been previously
used in aviation research as an optimization method. Turgut and Rosen used genetic
algorithms to obtain the optimal descent in terms of the fuel flow values and altitudes to
reduce the global descent cost [35]. Kanury and Song [36] used genetic algorithms to look
for the optimal trajectory under the presence of unknown obstacles, obtaining satisfactory
results in a short computing time; their algorithm obtained the optimal route and the
calculation time in their simulation was reduced. These algorithms are useful when
searching for a solution involving multiple imposed restrictions. Kouba [37] studied genetic
algorithms as a means to incorporate several constraints into a trajectory optimization
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problem, where the objective was to find the shortest route while considering different
restrictions.
More recently, various LNAV (lateral navigation) and VNAV profile optimization
algorithms were developed at the Research Laboratory in Active Controls, Avionics and
Aeroservoelasticity (LARCASE) [38-42].
The algorithm presented in this paper defines a new methodology to optimize flight
trajectories in the cruise phase, since as it was mentioned before, it is the phase where a
most significant flight cost reduction can be obtained for long haul flights. The performance
database-modeled aircraft used in this study represents an increased precision in terms of
fuel burn and flight calculation when compared with current equations-based aircrafts
models present in the literature. Even if this method could be used as a pre-flight
calculation algorithm, since it performs an extensive analysis of the winds and
temperatures, the structure of the performance database (PDB) and the weather database
allows an efficient implementation a genetic algorithm to reduce the total number of
calculations so it can reach its objective, which is to implemented in the commercial FMS
studied in this paper.
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II. Methodology
To reduce the flight cost, the proposed trajectories’ optimization algorithm analyzes
alternative trajectories which consider the influence of the wind, the outside air temperature
and the variation of the optimal altitudes as the fuel is burnt during flight.
Typically, flight trajectories are planned before each flight by large ground-based
computers, which consider the restrictions imposed by ATC. These trajectories incorporate
the current traffic, weather conditions, the aircraft’s weight and the airline’s operation
costs. However, due to changing weather, current traffic conditions and the variation of the
aircraft’s weight during the flight, these trajectories may not be optimal in terms of flight
costs. The optimization algorithm presented in this paper could be used as a pre-flight
calculation method, since it includes information on the weather and restrictions could be
manually imposed, however, it is intended to be implemented in a FMS, where the
processor capacities are usually lower than ground-based computers.
The algorithm presented in this paper has been developed using Matlab®. To reduce flight
costs, a 3D grid is created around the original flight trajectory, planned before the flight,
which allows the analysis of possible step climbs to reach the optimal altitude, as well as of
horizontal deviations to profit from the tailwinds or avoid the headwinds. Since this
algorithm is conceived in order to be implemented in a FMS, calculation time is an
important factor. To reduce the number of possible trajectories and thus the calculation
10
time, the aircraft’s speed will remain constant during the entire cruise, and a genetic
optimization algorithm is applied to calculate the optimal trajectory without calculating all
the possibilities within the grid.
The trajectories’ optimization algorithm analyzes the cruise once the aircraft is situated at
the Top of Climb (TOC), and the flight parameters are known, such as the aircraft’s weight,
speed, the initial cruise altitude and the flight time. Although the methodology and tests
presented in this paper include trajectories from TOC to Top of Descent (TOD), the starting
point of the algorithm could be redefined at any moment during the cruise.
The methodology is structured as follows: First, the model of the aircraft and the
calculation of each trajectory are defined. Next, a dynamic weather model is described, and
the grid where the possible alternative trajectories are analyzed is explained after. Finally,
the genetic algorithm applied to reduce the calculation time is defined.
A. Aircraft model – Performance Database
The aircraft’s model was obtained from a PDB for a commercial aircraft. This PDB includes
precise information on the main phases of the flight: climb, cruise and descent. The PDB, as
it considers real flight performance information obtained from actual tests, increases the
calculation precision compared to conventional methods that apply an aircraft’s equations of
motions.
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The trajectories’ optimization algorithm presented in this article analyzes the cruise phase
for long flights, with the possibility of performing step climbs to reduce flight cost. The
structure of the cruise and climb tables provided in the PDB is defined in this section. Table
1 presents the inputs and outputs of these tables.
Table 1 Inputs and outputs for a commercial aircraft’s PDB
Type of table
Inputs
Outputs
Climb
Speed (kt)
Gross weight (kg)
ISA deviation (°C)
Altitude (ft)
Fuel burn (kg)
Horizontal distance (nm)
Cruise
Speed (kt)
Gross weight (kg)
ISA deviation (°C)
Altitude (ft)
Fuel flow (kg/nm)
The cruise trajectory is divided into segments called waypoints. At each waypoint, the
algorithm analyzes the possibility of performing a step climb or a flight to an adjacent
horizontal waypoint. The flight cost for each possible segment is calculated.
To calculate the cost of each possible climb, the fuel burn and the horizontal values have to
be interpolated from the PDB using Matlab®. Figure 1 represents the interpolations
required to obtain these values for the climb phase.
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Figure 1 Interpolations to obtain the aircraft’s flight performance during climb from
the PDB
The speed remains constant and it is not interpolated. This means that only one speed is
calculated at a time, and only the speed values found in the PDB can be analyzed. The
Mach number ranges from 0.6 to 0.84. Variables gw_1 and gw_2 define the interval
boundaries of the aircraft’s gross weight in kg, and ISA_dev_1 and ISA_dev_2are the
interval boundaries on the ISA deviation input values in °C.
The altitude is not interpolated, and only the values found in the PDB are analyzed. The
cruise altitudes in the PDB are defined in steps of 1,000ft, and it varies from 20,000 to
40,000ft. The PDB sums the cost of each climb each 1,000ft; thus, in order to calculate the
cost in terms of Horizontal_distance and Fuel_burn, the interpolation results for altitude_1,
which represent the current aircraft altitude, have to be subtracted from the results obtained
for altitude_2, which refer to the desired climb altitude.
Horizontal_distance
Fuel_burn
speed
gw_1 ISA_dev_1 altitude_1
ISA_dev_2 altitude_1
gw_2 ISA_dev_1 altitude_1
ISA_dev_2 altitude_1
speed
gw_1 ISA_dev_1 altitude_2
ISA_dev_2 altitude_2
gw_2 ISA_dev_1 altitude_2
ISA_dev_2 altitude_2
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Possible step climbs of 2,000ft are analyzed. Even if real aircraft do sometimes perform
1,000ft step climbs, by convention, 2,000ft step climbs should be performed to avoid
aircraft flying in the opposite direction (eastbound and westbound flights) and to respect the
flight levels predefined by ATC [43].
An interpolation function has been created to obtain the Fuel_burn and Horizontal_distance.
The Fuel_burn is given in <kg>, while the Horizontal_distance is defined in <nm>.
The Lagrange linear interpolation function is applied to perform the interpolations, as in Eq.
1:
(1)
The interpolations required to obtain the fuel flow during cruise are presented in Fig. 2.
Figure 2 Interpolations to obtain the aircraft’s flight performance during the cruise
from the PDB
Fuel_flow speed
gw_1 ISA_dev_1 altitude
ISA_dev_2 altitude
gw_2 ISA_dev_1 altitude
ISA_dev_2 altitude
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The fuel flow is updated to calculate the fuel burn at each segment. The distance for each
segment is calculated using the coordinates of the grid and Vincenty’s method [44]. In the
PDB, the fuel flow is given in terms of kg of fuel burned per nautical mile. In order to
calculate the fuel burnt during the cruise, it suffices to multiply both values, as in Eq. 2.
Fuel_burn_cruise = Fuel_flow * Horizontal_distance
(2)
Where the Fuel_burn_cruise is given in <kg>, the Fuel_flow in <kg/nm> and the
Horizontal_distance in <nm>.
B. Dynamic wind model
The wind data used in this algorithm is extracted from Environment Canada. The
information is presented under a Global Deterministic Prediction System (GDPS) format.
The GDPS model provides a 601×301 latitude-longitude grid with a resolution of 0.6×0.6
degrees. At each point of this grid, information such as the wind direction, speed,
temperature, and the pressure can be obtained for different altitudes, in 3-hour time blocks.
This database is updated every 12 hours, and it is indicated in Coordinated Universal Time
(UTC).
The wind is defined as dynamic. As the aircraft advances in time, the weather is updated to
match the current position of the aircraft.
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Wind directly affects both the horizontal distance traveled with respect to ground level, and
indirectly affects the fuel consumption. The ground speed is calculated so that it can be
considered in the horizontal distance calculation. The speeds in Eq. 3 are expressed in knots
<kt>.
(3)
The airspeed is an aircraft’s speed relative to the air mass, and the wind is the horizontal
movement of this air mass relative to the ground. Here, the effective wind is the wind’s
component of the aircraft’s trajectory, and the crosswind is that component perpendicular to
the effective wind speed given in Eq. 4 [45]. These are illustrated in Fig. 3
(4)
Figure 3 Real wind, crosswind and effective wind
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As the aircraft flies on a straight path, the wind affects the aircraft’s speed. Depending on
the direction and speed of the wind, a distance factor is calculated. The distance traveled by
the aircraft will either be reduced or increased in a particular time segment. The horizontal
distance traveled at the ground level is the norm of the ground speed vector. Figure 4 shows
the influence of the wind of a mass moving from WPT(n) to WPT(n+1) [46, 47].
Figure 4 Wind factor calculation
The distance factor is calculated by the wind factor in the following way:
)cos(*
||||
||||
||||
||||)*sin(
arcsincos_
speed
speed
speed
speed
Air
Wind
Air
Wind
factorWind
(5)
The ground speed is obtained from the ratio between the true airspeed and the wind speed.
The ground distance can be calculated from the ground speed.
The wind data is interpolated in the optimization algorithm at each segment. For the
vertical interpolation, the wind vectors are analyzed according to the Earth’s Northern and
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Eastern axes (selected arbitrarily as a reference parameter) for two different altitudes.
Afterwards, an interpolation is made between these two axes at the required altitude to
obtain the wind vector (speed and direction). The horizontal interpolation is obtained
between consecutive waypoints. This process is sketched in Fig. 5.
Figure 5 Wind interpolation method
C. Flight cost
Airlines have different constraints when calculating their optimal trajectory. Business
trajectories could require to be optimized in flight time rather than fuel burn, while other
trajectories would permit to search for the maximal fuel reduction. The cost index (CI) is a
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parameter used by airlines to determine their flight cost. Each company has its own
parameters to define their CI. In this project, the CI is defined as in the commercial FMS
used for this study.
The CI varies from 0 to 99, and it is defined as the operation costs of the flight per unit of
time. Its units are defined in <$/hr>. The global cost of the flight is given by Eq. 6:
Global cost = ∑ Fuel burn * Fuel price + CI * Flight time
(6)
Where the Fuel burn is given in <kg>, the Fuel price in <$/kg>, the CI in <$/hr> and the
Flight time in <hr>.
The flight time is calculated using the aircraft’s ground speed.
It can be seen in Eq. 6, that when the CI is 0, the Flight time does not have an influence on
the Global cost. Minimal fuel optimization occurs when CI = 0. When the CI is set at its
maximum value of 99, the Flight time is most important. Minimal time optimization occurs
when CI=99.
Since the price of the fuel changes constantly, it would be simpler to show the results in
terms of kg of fuel. The Fuel price is considered in the operation costs of the flight. The Fuel
price could be removed from Eq. 6, and the CI can be represented in <kg/hr> since its
influence moves linearly independently of the Fuel price. Equation 7 represents the cost
definition used in this article to optimize the flight cost.
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Global cost = ∑ Fuel burn + CI * Flight time
(7)
To obtain the Global cost in terms of <$>, it suffices by multiplying the result by the Fuel
price of the specific flight day.
D. The grid
As mentioned before, the flight trajectories are planned before the flight by ground-based
computers, respecting the restrictions imposed by ATC. This proposed trajectories’
optimization algorithm, however, analyzes alternative trajectories once the aircraft is
already in the cruise phase, and finds those which, if approved by ATC, would reduce the
global flight cost.
A 3D grid is created around the original trajectory planned pre-flight. Horizontally, two
parallel trajectories are added to each side of the original trajectory. A total of five
trajectories on the horizontal profile are analyzed. The distance between horizontal
trajectories is variable, but predefined at 15nm for test purposes. The cruise, from the TOC
to the estimated TOD, is divided into an n number of waypoints. Vertically, the profile is
divided in m sections at each 1,000ft from the initial cruise altitude up to the maximal
cruise altitude (defined by the PDB for each aircraft). The size of the grid is 5 x n x m.
An example of this grid can be seen in Fig. 6, where a 3D grid has been designed around
the cruise horizontal path for a trajectory from Porto to Toronto.
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Original horizontal path
Figure 6 3D grid created around a real flight trajectory to calculate the optimal
trajectory
Within the grid, the aircraft can only fly to adjacent horizontal waypoints, and can perform
2,000ft step climbs.
Each waypoint in the grid is represented by a latitude, a longitude and an altitude. Each
waypoint is identified as a specific point in the grid. Each trajectory consists of n
waypoints.
Trajectories are randomly created to be introduced in the genetic algorithm explained in the
next section.
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E. The genetic algorithm
This trajectories’ optimization algorithm is used to reduce the number of calculations so it
can be implemented on board an actual FMS. The FMS does not have the same processing
capabilities as the ground-based computers that plan the trajectories before a flight. This
means that the calculation time has to be reduced as much as possible.
Within the grid, the number of possible alternative trajectories increases exponentially as n
increases. Calculating all the possible alternative trajectories is not only impractical, it is also
very time-consuming. Therefore, a genetic algorithm has been used to reduce calculation
time.
Genetic algorithms were selected because they have proved their ability to obtain optimal
solutions where nonlinear data is analyzed in a short calculation time [48, 49]. These
algorithms are based on Darwin’s evolution theory, where the fittest individuals in a
population survive to reproduce.
Genetic algorithms mimic the natural evolution process. Starting with an initial population, a
group of individuals selected by their fitness will reproduce, creating a second generation of
individuals. Once again, the fittest individuals will be selected to create a third, and so on.
This process is repeated for a predefined number of generations or until the optimal solution
is repeated for a predefined number of times.
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Genetic algorithms comprise the following steps: the definition of the individuals and the
creation of the initial population, the evaluation of individuals, the selection of the
individuals most-fitted to create the next generation, the reproduction and the process
termination conditions; each of which are explained in the following sections.
1. Individuals and initial population
Each individual is defined as a randomly-generated alternative trajectory. These trajectories
consist of a set of waypoints, defined by latitude, longitude and altitude. These alternative
trajectories are created respecting the previously defined grid, and all latitudes, longitudes
and altitudes must be within the grid. Table 2 describes the individuals created.
Table 2 Individuals parameters for the genetic algorithm
Waypoint
1
Waypoint
2
Waypoint
3
Waypoint
4
…
Waypoint
n
Individual
1
Lat1,1
Lon1,1
Alt1,1
Lat1,2
Lon1,2
Alt1,2
Lat1,3
Lon1,3
Alt1,3
Lat1,4
Lon1,4
Alt1,4
…
…
…
Lat1,n
Lon1,n
Alt1,n
Individual
2
Lat2,1
Lon2,1
Alt2,1
Lat2,2
Lon2,2
Alt2,2
Lat2,3
Lon2,3
Alt2,3
Lat2,4
Lon2,4
Alt2,4
…
…
…
Lat2,n
Lon2,n
Alt2,n
…
…
…
…
…
…
…
Individual
m
Latm,1
Lonm,1
Altm,1
Latm,2
Lonm,2
Altm,2
Latm,3
Lonm,3
Altm,3
Latm,4
Lonm,4
Altm,4
…
…
…
Latm,n
Lonm,n
Altm,n
Depending on the size of the grid, there could be thousands or millions of possible
trajectories. The initial population should represent a small percentage of all the possible
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solutions. The size of the initial population is defined depending on the size of the entire
number of possible solutions.
2. Evaluation
The evaluation process consists of calculating the flight cost of each trajectory using the
PDB.
After its evaluation, each individual is represented by the following:
Coordinates (latitudes and longitudes)
Altitudes at each waypoint (if step climbs were performed)
Aircraft speed (which remains constant throughout the entire cruise to save
calculation time)
Aircraft gross weight (updated dynamically as the aircraft advances)
Partial flight cost, fuel burnt and flight time (at each waypoint)
Wind speeds, directions and outside air temperature (calculated dynamically as
the aircraft advances)
Global flight cost, fuel burnt and flight time.
The fittest individual is defined as the flight trajectory that minimizes the global flight cost.
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3. Selection
According to Darwin’s theory of evolution, the best-fitted individuals are those that are more
likely to survive and have more chances to reproduce and preserve their genetic heritage.
This does not mean that less-fitted individuals do not have the right to be part of the
upcoming generations; they bring diversity to the population. This aspect makes it possible
to avoid local optimums more efficiently and reach the global optimum.
There are different methods for selecting the individuals to reproduce. Some examples are
uniform selection, rank selection, proportional selection (or the roulette wheel) and selection
by tournament.
In the uniform selection method, all the individuals are allowed to reproduce, independently
of their cost. This method is usually inefficient in terms of calculation time, since the
population is too diversified and the convergence to the optimal solution is slow.
The rank selection method has been applied to a VNAV optimization algorithm[49]. This
method sorts individuals according to their cost, and only the most-fitted individuals are
selected to reproduce. This method benefits from a quick convergence towards a solution;
however, depending on the complexity of the problem, it may lead to a quick convergence to
a suboptimal solution.
A selection by roulette wheel was implemented on an LNAV optimization algorithm in [48],
in which the non-linearity of the wind was added to the problem. This method consists of
25
assigning a piece of roulette to each individual depending on their cost. The more-fitted
individuals are represented by a bigger piece, while the less-fitted individuals are
represented by a proportionally smaller piece. The selection is performed randomly, as in a
roulette wheel. The more-fitted individuals thus have more chances of being selected than
the less-fitted individuals, who nonetheless still have a chance. This allows for a more
diversified population than the rank selection method, but one that is less diversified than
that created by uniform selection. This method is usually slower to converge than rank
selection, but the roulette wheel is more efficient at avoiding local optima.
In the genetic algorithm proposed in this paper, a selection by tournament was carried out.
This selection method makes the individuals compete against each other and preserves the
strongest one. Along with the roulette wheel selection, this method allows a diversified
population. However, the less-fitted individuals have a lower likelihood of reproducing,
allowing a quicker convergence towards the global optimal solution. After the first round of
the tournament, only half of the individuals survive to reproduce and create the next
generation.
4. Reproduction
After the tournament, the strongest individuals survived and half the population was
eliminated. The surviving individuals reproduce to create a new set of individuals; filling the
places made vacant after the first round. Since the strongest individuals are reproducing with
each other, more-fitted individuals are expected at each round or generation.
26
Each trajectory, as mentioned before, is represented by a set of waypoints defined by
latitude, longitude and altitude. A crossover method was used to create a new individual.
This method consists of taking one half of one individual, and combining it with a half from
another individual. The existing trajectories will be intersected at the middle waypoint, and
each part will be used to form a new individual, as in the example shown in Table 3.
Table 3 Reproduction by crossover
First half
Second half
Individual
1
Lat1,1
Lon1,1
Alt1,1
Lat1,2
Lon1,2
Alt1,2
Lat1,3
Lon1,3
Alt1,3
Lat1,4
Lon1,4
Alt1,4
Lat1,5
Lon1,5
Alt1,5
Lat1,6
Lon1,6
Alt1,6
Individual
2
Lat2,1
Lon2,1
Alt2,1
Lat2,2
Lon2,2
Alt2,2
Lat2,3
Lon2,3
Alt2,3
Lat2,4
Lon2,4
Alt2,4
Lat2,5
Lon2,5
Alt2,5
Lat2,6
Lon2,6
Alt2,6
New
individual
Lat1,1
Lon1,1
Alt1,1
Lat1,2
Lon1,2
Alt1,2
Lat1,3
Lon1,3
Alt1,3
Lat2,4
Lon2,4
Alt2,4
Lat2,5
Lon2,5
Alt2,5
Lat2,6
Lon2,6
Alt2,6
The order of the individuals to reproduce after the tournament is defined randomly.
After the new individuals are created, they will be evaluated to obtain their cost. A new
generation is thus obtained, consisting of old and new individuals in a 50/50 proportion.
The new generation will be sorted, and the most-fitted individual will be defined as the
optimal solution for that generation. In order to increase the diversity of the new generation,
the poorest-fit individuals will be eliminated automatically and replaced by a set of new
randomly-created individuals.
27
The process is repeated until a predefined number of generations are reached, or until the
optimal solution repeats itself for a predefined number of generations. The genetic algorithm
stops at a predefined stopping criterion, which is usually when the number of predefined
generations is reached. However, supplementary conditions could be added as stopping
criterion, such as to stop the algorithm when the optimal solution has been repeated a
predefined number of times.
The entire process is explained in the flow chart shown in Fig. 7.
START
INITIAL POPULATION
Creation of random
trajectories
EVALUATION
Flight cost calculation for
each trajectory
SELECTION
Tournament to define the
most fitted individuals REPRODUCTION
The winners at the
tournament reproduce to
create new trajectories
EVALUATION
Flight cost calculation for
the new trajectories
Poorest individuals are
eliminated
New random trajectories
are created to diversify
population
EVALUATION
Flight cost calculation for
the new trajectories
END
Lowest cost individual is
defined as the Optimal
trajectory
Is the optimal
solution repeated the
predefined number
of times?
Is the
predefined number
of generations
reached?
Yes
Yes
No
No
Figure 7 Genetic algorithm’s flow chart applied to optimize flight trajectories
28
III. Results
This section is divided into two parts. First, the performance of the genetic algorithm is
described in terms of calculation time and performance optimization. The second part covers
the proposed algorithm’s ability to reduce flight costs.
A. The genetic algorithm
A genetic algorithm has been implemented in order to reduce the calculation time. A
reduced percentage of the total number of possibilities is calculated. In order to define this
small percentage, first it is necessary to know the number of total possibilities.
The size of the grid is variable, since the number of waypoints can be modified to better
adapt it to the original trajectory, and the number of possible step climbs is defined by the
initial cruise altitude and the maximal climb altitude. The diagram presented in Fig. 8
represents how the total number of trajectories can be calculated in a 2D grid. The total
number of possible trajectories is defined by a sum of the trajectories arriving at each
waypoint from the precedent waypoints. As mentioned in the previous section, the aircraft
can only fly to adjacent waypoints.
29
+
+
++
+
+ +
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+
+
Figure 8 Dynamic diagram to calculate the total number of possibilities in a 2D grid
An example of the calculation in 2D (with no step climbs) for six waypoints, can be seen in
Fig. 9.
Start 1
1
1
3
2
1 3
+
+
++
+
+ +
+
6
+
+
+
2
+
+
1
+
7
+
+
+
6
+
+
+
3
+
+
+
+
+
16
+
+
+
19
+
+
+
16
+
+
+
51
+
+
+
Figure 9 Dynamic diagram’s example to calculate the total number of possibilities in a
2D grid for 6 waypoints
The diagram presented in Fig. 9 can calculate the total number of possibilities if no step
climbs are performed. In order to include step climbs in the calculation of total possible
30
trajectories, at each waypoint, the total sum of trajectories has to be multiplied by two, in
order to include the possibility of the aircraft remaining at the same altitude, or of
performing a step climb. Table 4 shows the total number of possible trajectories that can be
produced by varying the number of waypoints and the number of step climbs during the
entire cruise.
Table 4 Number of possible trajectories within the 3D grid varying the number of step
climbs (SC) and waypoints (WP)
SC
WP
0
1
2
3
4
5
Unlimited
6
51
102
204
408
816
1,632
1,632
7
139
278
556
1,112
2,224
4,448
8,896
8
379
758
1,516
3,032
6,064
12,128
48,512
9
1,035
2,070
4,140
8,280
16,560
33,120
264,960
10
2,827
5,654
11,308
22,616
45,232
90,464
1,447,424
11
7,723
15,446
30,892
61,784
123,568
247,136
7,908,352
12
21,099
42,198
84,396
168,792
337,584
675,168
43,210,752
Table 4 makes it clear that as the number of waypoints increases, the number of possible
trajectories grows exponentially. Adding a step climb to each trajectory doubles the number
of possibilities.
The results for trajectories divided into nine waypoints and allowing four step climbs during
the cruise are presented next.
To evaluate the performance of the genetic algorithm, a single flight was tested and repeated
100 times, thereby revealing the percentage of times the optimal trajectory was achieved.
31
The real flight information was taken from FlightAware
3
. This website provides real flight
information in the form of a database that contains flight coordinates, altitudes, time and
speeds of aircraft for each flight. These parameters are used as a reference, and the flight
trajectories optimization algorithm creates the 3D grid around the original flight trajectory
proposed by FlightAware. The costs of each flight, both the real and the optimal one, are
calculated using the PDB for a commercial aircraft.
The FlightAware website, however, does not provide information about an aircraft’s
weight. In order to perform the flight test, the aircraft’s standard weight is obtained from
the PDB, and the amount of fuel used is the minimal amount required to complete the
flight.
The selected trajectory is a flight from Lisbon to Toronto, which departed at 16:30 UTC, on
November 15th, 2013. The inputs were the following:
Aircraft type: Mid-range commercial aircraft
Aircraft weight: 100 ton
Aircraft fuel: 28 ton
Cost index: 0 (maximum fuel savings)
Maximal altitude: 40,000 ft
Altitude at TOC from the real flight: 31,000
Distance between alternative horizontal trajectories: 15nm
3
FlightAware. "FlightAware - Live Flight Tracking."
flightaware.com, 2014.
32
Aircraft speed: 0.70 Mach
For the genetic algorithm:
Individuals per generation: 50 individuals
Number of generations: 300
Maximal number of repetitions for the optimal solution: 50
The real flight is compared with the optimal flight in terms of latitudes, longitudes and
altitudes in Figure 10. A flight cost reduction of 5.71% of was obtained. The optimization
algorithm calculates the optimal trajectory at a constant speed in order to reduce calculation
time.
Figure 10 Example real flight trajectory compared with its optimal flight trajectory
-80 -70 -60 -50 -40 -30 -20 -10 38
40
42
44
3
3.2
3.4
3.6
3.8
x 104
Latitudes
3D Profile with grid around the original trajectory
Longitudes
Altitude (ft)
-80 -70 -60 -50 -40 -30 -20 -10
30
35
40
45
50
55
60
Longitudes
Latitudes
LNAV Profile
123456789
3.1
3.2
3.3
3.4
3.5
3.6
3.7
x 104
Waypoints
Altitude (ft)
VNAV Profile
Real trajectory
Optimal trajectory
TOD
TOC
TOD TOC
TOD
TOC
33
The results for the genetic algorithm can be seen in Table 5. Two different solutions are
defined here, obtained by the same optimization algorithm. Due to the randomness nature of
the genetic algorithms, the optimal solution is not always obtained. However, the optimal
solution represents a 5.71% flight cost reduction, and is obtained 45% of the time. The
suboptimal solutions averages represent a 5.60% flight cost reduction, and were obtained the
remaining 55%.
Table 5 Genetic algorithm optimization results
Original
flight cost
(kg)
Optimal
trajectory
cost (kg)
Average cost of
suboptimal
trajectories (kg)
Optimal trajectory
found by the
genetic algorithm
Average number
of trajectories
calculated by the
algorithm
26462
24950 (5.71%
cost reduction)
24979 (5.60%
cost reduction)
45%
23.2%
Table 5 represents the performance of the genetic algorithm. The optimal solution, which
reduces fuel consumption by 5.71%, was obtained 45% of the time, while calculating only
an average of 23.2% of the total possible trajectories. However, the suboptimal solutions still
reduce the fuel consumption by an average of 5.60%. If the number of individuals per
generation or the number of generations was increased, the optimal solution would be found
more often; however, that would substantially increase the number of calculations.
The purpose of this algorithm is to reduce the calculation time as much as possible so that
the proposed algorithm can be implemented in a FMS device.
34
B. Fuel cost reduction
A total of 20 real flights were compared by the trajectories optimization algorithm to reduce
the global flight cost. The results can be found in Table 6.
The inputs considered for the grid creation and the genetic algorithm are the following:
Number of waypoints: 9
Maximal number of step climbs: 4
Distance between alternative horizontal trajectories: 15nm
Individuals per generation: 50
Number of generations: 300
Maximal number of repetitions for the optimal solution: 50
All these flights were performed the same type of commercial aircraft.
35
Table 6 Genetic algorithm optimization results
Departure
Arrival
Date
Optimal
trajectory cost
Original
trajectory
cost
Flight cost
reduction
London
Toronto
23-Sep-13
26341.9
27112.6
2.84
Ponta Delgada
Boston
01-Oct-13
17650.7
17818.2
0.93
Paris
Toronto
04-Oct-13
26349.2
28039.4
6.02
Glasgow
Toronto
04-Oct-13
23899.7
25916.8
7.78
Lisbon
Toronto
04-Oct-13
25044.5
26692.4
6.17
London
Toronto
04-Oct-13
24896.4
29357.1
15.19
Paris
Montreal
11-Nov-13
24410.9
26107.3
6.49
Manchester
Toronto
11-Nov-13
24372.3
26657.2
8.57
London
Toronto
11-Nov-13
24726.6
26424.9
6.42
Lisbon
Cancun
11-Nov-13
36066.1
37974.5
5.02
Ponta Delgada
Boston
15-Nov-13
16684.2
17545.4
4.90
Paris
Montreal
15-Nov-13
25204.7
26424.4
4.61
Lisbon
Toronto
15-Nov-13
24950.7
26462.9
5.71
Paris
Montreal
18-Nov-13
24015.2
24621.0
2.46
London
Toronto
18-Nov-13
24509.6
25775.6
4.91
Montreal
Paris
21-Nov-13
24624.6
24953.7
1.31
Glasgow
Toronto
21-Nov-13
23876.7
24337.9
1.89
Paris
Montreal
21-Nov-13
25280.1
26024.1
2.85
Paris
Montreal
25-Nov-13
24446.1
25816.1
5.30
Glasgow
Toronto
06-Feb-14
24749.1
24886.8
0.55
Average flight cost reduction
5.00%
It can be seen in Table 6 that the trajectories optimization algorithm reduces the global flight
cost by 5%. These reductions were obtained by a better selection of the flight altitudes and a
better choice of the horizontal trajectory, which improves the fuel consumption by avoiding
headwinds or taking advantage of tailwinds.
The proposed algorithm analyzed the entire flight by using the mode of the real speed
vector, which represents the Mach number used for most of the flight. This speed remained
constant for the entire analysis of the optimal flight trajectory in order to reduce calculation
36
time. Varying the speed would result in a significant increase of the optimization algorithm’s
calculation time.
As future work, the trajectories’ optimization algorithm could consider varying the Mach
number during a flight to increase the cost savings. Varying the speed would also allow the
implementation of RTA restrictions at each waypoint, and thus, a 4D flight trajectory
analysis.
37
IV. Conclusion
The flight cost analysis was performed using a PDB for a commercial aircraft. This allows
a better precision and lower calculation time than conventional methods using an aircraft’s
equations of motion. By applying a genetic algorithm to reduce calculation time, the flight
trajectories optimization algorithm reduces the flight cost by 5%. This was accomplished
by analyzing real flight trajectories from FlightAware, which are trajectories that real
commercial airlines are flying today. It can be seen that actual trajectories are not optimal.
These airlines might be flying those trajectories because of ATC’s restrictions, by their own
flight policies or due to a bad analysis of the optimal performance of the aircraft and the
winds. Although the proposed trajectories optimization algorithm does not consider ATC
restrictions, it would serve as an in-flight method of calculating an alternative trajectory
which could be requested to ATC in order to reduce flight cost, through the FMS device.
This algorithm could also serve as a pre-flight analysis. However, pre-flight analysis
usually are performed by powerful ground based computers, in which case, the number of
calculations could be increased, and a more complete flight analysis could be performed.
The main objective of the present research is, however, to be implemented in a small
processing device such as the FMS, and therefore, parameters as the aircraft speed
remained constant to reduce calculation time. This algorithm analyses the behavior of the
winds, and uses them as a way to reduce fuel burn. At the same time, step climbs are
analyzed at each waypoint to improve the performance of the aircraft as the fuel is
consumed and the total aircraft weight reduced. The 5% obtained, even if ATC restrictions
38
are not considered, could be an important way to reduce aircraft’s fuel consumption and
pollutant emissions from aviation.
Acknowledgments
The authors would like to thank the Green Aviation Research & Development Network
(GARDN), CMC Electronics – Esterline and CONACYT for their financial support. Special
thanks are due to Mr. Rex Hygate, Mr. Dominique Labour and Mr. Claude Provencal for
their valuable assistance. Software support for the use of the FlightSIM product was
provided by Presagis.
39
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