Vertical Navigation Trajectory Optimization Algorithm ...

Vertical Navigation Trajectory Optimization Algorithm For A Commercial Aircraft Alejandro Murrieta Mendoza∗ and Ruxandra Botez † ´ University of Qu´ebec, Ecole de Technologie Sup´erieure Laboratory of Research in Active Controls, Avionics and Aeroservoelasticity LARCASE (www.larcase.etsmtl.ca) Montreal,Quebec,H3C-1K3, Canada

Flight trajectory optimization is an alternative to reduce flight costs and contaminant emissions generated by fuel consumption. The objective of this work is to develop an algorithm to find the most economical vertical navigation profile between two points. The global flight cost analyzed is a compromise between fuel burned and flight time. This compromise is achieved using a variable called cost index, which assigns a cost to flight time in terms of fuel consumption. The optimization is performed by calculating a candidate cruise trajectory profile using an aircraft performance database. This candidate cruise profile reduces the search space, as only those profiles around the optimal candidate one are analyzed in terms of their account climb and descent costs. During cruise, step climbs are evaluated at every hour of flight. The different profiles are compared and the most economical one is defined as the optimal vertical navigation. The algorithm was evaluated for a commercial aircraft using the same performance database as a currently operational Flight Management System. The algorithm was developed in MATLAB, and its validation was performed using a complete aerodynamic model in the software FlightSIMTM developed by PresagisTM and the profiles generated by the Part Task Trainer of a commercial Flight Management System.

Nomenclature AP U AT C CDA CI CO2 Di0 Di1 Di2 Di3 Di test FMS HC ISA IAS N Ox P DB RT A T AS ∗ Ph.D.

Auxiliar Power Unit Air Traffic Control Continuos Descent Approach Cost Index Carbon dioxyde Total Distance Horizontal distance needed to travel during 2,000 ft to 10,000 ft climb Horizontal distance needed to travel from 10,000 ft to desired cruise. Estimated descent distance. Distance to calculate the pre-optimal cruise profile Flight Management System Hydrocarbons International Standard Atmosphere Indicated Air speed. Nitrogen oxides. Performance Database Required Time of Arrival True Air Speed Student, University of Quebec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada. University of Quebec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada.

† Professor,

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T OC T OD LN AV V N AV W0 W1 W2 Wtest

Top of Climb Top of Descent Lateral Navigation Vertical Navigation Initial aircraft weight Fuel burned during 2,000 ft to 10,000 ft climb Fuel burned to reach desired cruise altitude from 10,000 ft. Weight needed to calculate calculate the pre-optimal cruise profile

I.

Introduction

he aerospace industry has been motivated to reduce fuel costs and the polluting emissions generated by T fuel consumption. Many companies and airlines are developing services and products that reduce fuel consumption. Multiple opportunities for fuel savings, and thus to reduce polluting emissions, have been identified in aircraft route planning. In 2011, according with the Air Transport Action Group1 fuel cost for commercial airlines was approximately 178 billion dollars – 26% of their total operating expenses. The amount of fuel consumed not only means less profit for the airlines, but is also proportional to the amount of pollution generated. Among the main emissions from the burned fuel are carbon dioxide (CO2 ), the combination of nitrogen oxide and nitrogen oxides (NOx), hydrocarbons (HC). CO2 is one of the major greenhouse effect gases and its release to the atmosphere is one of the principal causes of global warming. In 2011, 649 million tons of CO2 were released to the atmosphere by commercial airplanes. Almost 80% of this CO2 was released in flights longer than 1000 kilometres where there was no other practical way of travelling. Around 2% of all the CO2 released to the atmosphere is attributable to commercial air traffic. HC are known to contribute to the greenhouse effect. NOx was identified by Ravishankara et al.2 as components that contribute to depleting the ozone layer in the stratosphere. Water vapour at high altitudes can cause contrails at non-conventional altitudes; these contrails can act as a greenhouse gas.3 Calculating fuel consumption has been of interest during descent. Murrieta et al.4 and Dancila et al.5 developed models to calculate the fuel burned during a missed approach (or go-around) procedure using the emissions inventory guidebook from the European Environment agency. Descent has been of importance due to noise and fuel polluting emission released near urban centres. A technique known as Continuous Descent Approach (CDA) has been widely studied and implemented in some airports. The CDA proposes a constant slope descent reducing the engine utilization instead of the conventional stepped descent. Studies to determine the Top of Descent (TOD) location behavior with parameters such as weight, speed and winds have been conducted6 as well studies to measure the fuel burned and noise reduction on its implementation in airports such as Louville International7 in Kentucky. Airlines have implemented programs to reduce fuel consumption and the associated emissions. Air Transat∗ added improvements such as washing engines to improve their efficiency, changing the fleets tires to lighter ones, reducing the use of the auxiliary power unit (APU), taxiing with only one engine, reducing the weight of food related items, and varying the cost index (fuel to time ratio) during flight. The cockpit avionics equipment that helps a pilot to plan and maintain a route is the Flight Management System (FMS). The main tasks that a FMS performs are flight guidance, control of the lateral and vertical aircraft paths, monitoring of the flight envelope, computing the optimal speed for every phase of the flight and providing automatic control of the engine thrust.8 Lid´en9–11 was one of the first researchers to study the FMS trajectory optimisation; his work was mostly focussed on 4D trajectory guidance – to arrive at a destination at a given time with minimum fuel burn. In Ref 9, the effects of tail and headwind on the flight cost were studied in cruise by varying the speed, but no step climbs were performed. The no-wind effects on cost were also studied. A method was proposed to calculate the optimal cost index by adding a penalisation if the Required Time of Arrival (RTA) was not accomplished. This penalisation considers the cost of connection flights lost by the passengers. In Ref 10, the effects of step climb in cruise with and without meteorological condition were investigated. Methods to determine when it would be most advantageous to perform a step climb were studied. In Ref 11, the effects of the step climb and of winds were studied to identify the optimal cost index for a long flight. A procedure ∗ http://resp.transat.com/en/greenhouse-gas-reduction.html

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was established to eliminate discontinuities in the time versus cost index relationship. F´elix et al.12 implemented genetic algorithms in the lateral navigation profile to find the most economical route by taking advantage of tailwinds and avoiding headwinds. In this algorithm, the roulette method and mutations were used to avoid finding local optimal trajectories and thus only focus on global optimal trajectories. Using this methodology, savings of 0.5 % were reported when the optimal trajectories were compared with the geodesic route, or great circle (the shortest path between two points in a sphere). Dancila et al.13 proposed an algorithm to find the optimal altitude in cruise by measuring flight time and the fuel flow three different aircraft. In this algorithm, the cruise trajectories were not divided into sub-trajectories as done by a typical commercial FMS o, but a fuel cosumption estimator14 was used. Climb and descent phases were assumed to have no effect on the flight optimal cruise altitude. F´elix et al.15,16 proposed an algorithm where the optimal combination of speed schedule and cruise altitude was found using the Golden Section method for flight distances less than than 500 nm. For flight distances greater than 500 nm, this algorithm evaluated the possible step climbs in every waypoint defined in the route. A combined average optimization of 2.57% was attained for two commercial aircraft with respect to a commercial FMS algorithm. Nevertheless, this algorithm requires a complete analysis of all the available IAS/mach couples during climbs, as well as all the mach/IAS couples during descents. Calculating all climb and descent combinations makes the algorithm too time consuming even with a special interpolation method.17 Gagn´e et al.18 proposed an algorithm that found the optimal vertical flight profile by inspecting the majority of the combinations that compose a complete trajectory. All of the possible combinations of climb, cruise and descent were analyzed to identify the most economical one. In order to improve the cost reduction during cruise, the possibility of a step climb at each 25 nm was evaluated by measuring the aircraft fuel flow. A precise method using weather forecast information was developed to accurately estimate temperature and wind effects during a flight. An optimization of 1.92% was obtained for the commercial aircraft used in this paper. The high number of interpolations needed to calculate all of the trajectory combinations made the calculations too complicated. Sidibe19 solved the optimization problem by calculating the optimal profile by using dynamic programming. Fays et al.20 found the optimal trajectory for a Boeing 747-300 aircraft by modelling no fly zones (zones where flying is prohibited) as obstacles around the route. Metaheuristic optimization algorithms such as taboo were used to solve this problem. The algorithm proposed in this paper aims to find a better optimal vertical profile than that of the commercial FMS of reference, this is achieved by determining a pre-optimal cruise profile and evaluating the complete trajectories in the vicinity of the pre-optimal cruise profile. Climb, cruise and descent costs are considered in the optimization process. Special efforts were made to minimize the level and number of required calculations. Throughout this paper, the term optimal will refer to the most economical trajectory including fuel and the time-related costs.

II. A.

Methodology

Aircraft Model

The aircraft model information needed to perform the calculations is contained in the form of a Performance DataBase (PDB) used in the FMS of a current commercial aircraft. Data in the PDB are given in discrete form. The PDB can be divided into seven sub databases, which are described in Table 1. Each subdatabase requires specific inputs and provides output data of interest to calculate fuel consumption and flight time. In Table 1, IAS stands for Indicated Air Speed and ISA stands for International Standard Atmosphere.

B.

Route Calculations

The main phases of a flight are climb, cruise and descent. Using the PDB for the cost calculations of these phases, Lagrange lineal interpolations in ISA deviation temperature and weight are required. The Lagrange interpolation is shown in Eq. (1). Interpolations in speed and altitude are not performed, only those discrete available values in the PDB are used. Thus, the solution is to find the optimal speeds and altitudes that compose the most economical flight.

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Table 1. PDB Sub-databases Sub-database Climb IAS

Acceleration

Climb Mach

Cruise

Descent Mach

Deceleration

Descent IAS

Inputs IAS(knots) Gross Weight (kg) ISA deviation temperature ( ◦ C) Altitude (ft) Gross Weight (kg) Initial IAS (knots) Altitude when acceleration begins (ft) Delta speed to accelerate (knots) Mach Gross Weight(kg) ISA deviation temperature ( ◦ C) Altitude (ft) Mach Gross Weight (kg) ISA deviation temperature ( ◦ C) Altitude (ft) Mach Gross Weight (kg) ISA deviation temperature ( ◦ C) Altitude (ft) Gross Weight (kg) Initial IAS (knots) Altitude when deceleration begins (ft) Delta speed to decelerate Gross Weight (kg) IAS (knots) ISA deviation temperature ( ◦ C) Altitude (ft)

p1 (x) = 1.

Output Fuel Burn (kg) Horizontal distance traveled (nm)

Fuel burn (kg) Horizontal distance traveled (nm) Altitude needed (ft.) Fuel burn (kg) Horizontal distance traveled (nm)

Fuel flow (kg/hr)

Fuel burn (kg) Horizontal distance traveled (nm)

Fuel burn (kg) Horizontal distance traveled (nm) Altitude needed (ft) Fuel burn (kg) Horizontal distance traveled (nm)

x − x1 x − x0 f0 + f1 x0 − x1 x1 − x0

(1)

Climb Calculations

The complete climb can be divided into four smaller trajectories (sub-phases). Initial Climb: Due to regulations such as the 14 Codes of Federal Regulations section 91.117 in the U.S, speed restrictions exist within certain altitudes. Flights cannot go faster than 250 IAS at altitudes below 10,000 ft. The climb speed in the algorithm is considered to be 250 IAS from the initial altitude of 2,000 ft to the altitude of 10,000 ft. Acceleration: If the required climb speed is greater than 250 IAS, an accelerated flight starts from 250 IAS in order to attain the required IAS climb. Climb IAS : The aircraft climbs at a constant IAS until the crossover altitude is reached. The crossover altitude is the altitude where the true airspeed (TAS) of the IAS is equal to the TAS of the desired Mach. At higher altitudes than the crossover altitude the speed reference must be changed from IAS to mach. Climb Mach: The aircraft climbs at a constant mach until the Top of Climb (TOC) is reached. During the initial climb, IAS and Mach climb calculations of the needed fuel and of the horizontal distance traveled during a given climb are calculated by steps of 1,000 ft until the TOC. In every step, the fuel needed to climb the last 1,000 ft step is subtracted from the total aircraft’ weight to improve the accuracy of calculations.

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2.

Cruise

Cruise begins after the TOC and ends at the Top of Descent (TOD). During this phase, Mach is always constant. The fuel burned during cruise is subtracted from the aircraft total fuel weight every 25 nm. The route followed during cruise is always the geodesic one. Every hour of flight, the algorithm evaluates a step climb to verify if cruising at a higher altitude would reduce the flight cost. These higher altitudes are always 2,000 ft higher than the current altitude. The 2,000 ft climb was selected to conform to normal air traffic control (ATC) operational procedures.21 The hour of flight was selected under the hypothesis that it is not possible to request ATC to change of altitude at often. 3.

Descent Calculations

Similarly to climb calculations, descent is composed of four sub-phases, just as described during climb in Section II.B.1, but in the inverse sense. It begins with Mach descent, changes to IAS after the crossover altitude, followed by a deceleration and then the final descent from 10,000 ft to 2,000 ft at 250 IAS. Descent calculations are dependent upon cruise calculations. If the aircraft’s final location at 2,000 ft is not as expected, the distance needed to reach or that has surpassed the final point is either added to or reduced from the cruise, thus the TOD point location changes. The descent is then recalculated from the corrected TOD. This process is repeated as many times as needed until the aircraft position at 2,000 ft. satisfies the design conditions: The design conditions for this paper are that the aircraft cannot be located after the final coordinates, but can be located 500 m before the final coordinates. The 500 m distance was selected because less than 500 m does not represent an important change to the total cost, and many factors can influence this distance such as weather conditions and the pilot’s skills. All these flight phases are described in Fig. (1). A more detailed explanation of these phases can be found in the workd done by Murrieta.22

Step Climb

Top of descent

Top of climb MACH decent

MACH climb

Altitude (ft)

Crossover altitude

KIAS descent

KIAS climb

Acceleration 10,000 ft

Deceleration 10,000 ft

KIAS climb @ 250 KIAS

KIAS descent @ 250 KIAS

Distance (nm)

Figure 1. Complete Flight Phases.

C.

Global flight cost

Flights cost cannot be measured only by the quantity of fuel needed to fly a given distance. There are many factors that influence the flight cost, such as the salary of the crew, the aircraft maintenance costs, the cost of arriving too late or too early to a given gate, etc. To consider these costs, airlines and commercial FMSs use a parameter called the Cost Index (CI). The CI is a ratio that allows a compromise between fuel and time-related costs. A higher CI would give priority to a faster flight over reducing fuel consumption, while as a lower CI would give priority to the reduction of fuel consumption over flight time. For example, a CI of 0 gives no importance to flight time and focuses only on fuel reduction. The expression that defines the total cost of a flight is defined in Eq. (2), where T otal Cost and T otal F uel Consumed are expressed in kg, time (T ) is expressed in minutes and CI is expressed in kg/hr. 60 is a conversion to minutes; this value is used to obtain results to be compared with those generated by the commercial FMS of reference.

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T otal Cost = T otal F uelconsumed + CI · T · 60

(2)

The CI value is chosen by the airline and can change from one flight to another.While as it is possible to change the CI in flight, in this paper it is always kept constant. D.

Vertical navigation flight optimization

To find the optimal trajectory, flight trajectories with the many speed/altitude combinations available in the PDB have to be calculated and compared. If step climbs are recommended, trajectories condisering this altitude change have to be calculated incrementing the total number of available combinations. Finally, all trajectories are compared and the one with the lowest cost is selected as the optimal Vertical Navigation (VNAV) profile. To avoid performing all the calculations, a discriminating optimization method called Pre-cruise optimization is presented. 1.

Pre-cruise optimization method

The objective of the pre-cruise optimization method is to find a first guess of the optimal cruise altitude/mach profile. The first guess is defined as the pre-cruise optimal candidate. Climb, final cruise and descent explained above are only calculated for the pre-cruise optimal candidate and in its vicinity. The total amount of space search, thus computations is therefore reduced. The advantage of applying the method describe here for a commercial aircraft PDB is that when climb IAS will be computed, instead of computing the crossover altitude for 13 possible mach numbers, this crossover altitude will only need to be computed for five. It can be interpreted as a reduction of eight mach crossovers for each IAS climb (160 combinations for 20 IAS climbs available in the PDB), and eight mach climb calculations. It can be inferred that a reduction in the number of possible optimal solutions will be achieved. To be able to implement the pre-cruise optimization algorithm, a test cruise trajectory has to be determined. The procedure to compute the test distance and the pre-optimal cruise pair is the following:

1.- Calculate the distance (Di0 ) between airport A and airport B. 2.- Calculate the climb from 2,000 ft. to 10,000 ft at 250 IAS. The fuel burned during this climb (W1 ) as well as the horizontal distance traveled (Di1 ) variables are saved. 3.- Airplanes are designed to have an optimal Mach speed at a certain altitude. This speed and altitude have to be known or estimated a priori in order to choose their correct values for the estimated climb cost. For the aircraft of reference, 0.82 mach is the optimal cruise speed and 36,000 ft is a typical cruise altitude. With the maximum weight step in the PDB being 210,000 kg for the studied aircraft, and using the PDB in the climb Mach mode, the fuel consumption (W2 ) and the traveled distance (Di2) are fetched directly and saved. 4.- For the descent, the traveled distance data are found directly from the descent IAS PDB table. The IAS is selected arbitrarily; a descent from 36,000 ft at 300 IAS was the altitude information used to fetch data. The weight used was the maximum weight allowed for a descent, 170,000 kg for the aircraft of reference. The only data saved are the horizontal distances travelled during the descent (Di3 ). Only the weight at the beginning of the cruise is needed; the fuel remaining at the TOD is of no interest in determining the pre-optimal cruise profile. 5.- The saved distances (Di1 , Di2 , and Di3 ) are added together. This accumulated distance is then subtracted from the total distance (Di0 ) to obtain the test distance (Di test). 6.- The saved weights (W1 and W2 ) are subtracted from the total weight, thereby obtaining the test weight. 7.- Calculate the cruise cost for every Mach at every available altitude, as described in Section II.B.2, without evaluating the step climb and with only eight waypoints during cruise regardless of the flight distance. These eight points were selected to assure a rapid calculation and to allow weather data to be added.

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8.- Compare and identify the least expensive profile and declare it as the pre-optimal cruise mach/altitude profile. Equations (3) and (4) can be deduced from the steps above to determine the weight and the test distance. Di test is the test distance used to perform the cruise, W test is the weight required to perform the test at the beginning of the cruise, and W0 is the initial weight of the aircraft. Di test = Di0 − Di1 − Di2 − Di3

(3)

W test = W0 − W1 − W2

(4)

Figure (2) shows a graphical representation of the trajectory and the weight calculations. A

B

Complete trajectory 36,000 ft

W test 0.82 MACH

W1

W0

2,000 ft Di1

Di2

Di Test

Di3

Figure 2. Pre-optimal cruise weight and distances.

Figure (3) displays the results in terms of fuel burned variation in relation to speed and alditude (every asterisk represents a Mach) during a pre-cruise evaluation for a CI = 0. In this particular case the least expensive altitude is cruising at 36,000 ft at a Mach of 0.81. The pre-cruise optimal candidate is then 36,000 ft at 0.81 Mach. 4

1.2

x 10

Each star represent a MACH in cruise 1.18

Fuel burned (Kg)

1.16

1.14

1.12 Pre-optimal value 1.1

1.08

1.06 31

32

33

34

35 36 37 Altitude (x1000 ft)

38

39

40

Figure 3. Pre-optimal cruise selection graph.

Because the pre-cruise optimal candidate is obtained by focusing mainly on the cruise (and not on other phases), this does not give good results for short flights, where the climb has a significant influence on the flight’s total cost. During a long cruise of around 700 nm and longer, the climb cost does not significantly affect the optimal mach/altitude cruise.

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2.

Climb and descent IAS/mach selection

After the selection of the pre-cruise profile, the climb computations are performed for all the available IAS and for five mach values: the pre-cruise optimal candidate mach, the two discrete values in the PDB before the candidate mach and the two discrete values after the candidate mach. By observation, it was determined that two discrete values (+/- 0.1 mach) comprised an acceptable search space. The calculated maximal altitude is the discrete value available in the PDB after the candidate altitude (+/- 2,000 ft); this value was also defined by observation. As stated above, the number of IAS/mach/altitude combinations are reduced, and thus the search space is reduced. The results of these computations are tabulated. Tables containing the climb cost and the horizontal distance traveled are obtained and used within the algorithm. Using these two tables, the ratio of nautical miles traveled per cost kilogram is calculated. Notice the term cost kilogram; this is the cost defined by Eq. (2)where the CI and the flight time are considered. The ratio with the maximum value is selected as the best climb profile. For the descent phase, only the cruise mach is analyzed. All the IAS values are calculated, and once again the results are tabulated. The most economical mach/IAS descent is the profile selected as providing the best descent. 3.

Cruise calculations

To find the optimal profile, different combinations of altitudes and speeds are evaluated around the optimal pre - cruise candidate. The altitudes from the PDB to be evaluated are two discrete values: the one available after and the one before the the pre-optimal cruise candidate altitude; one discrete value represents 2,000 ft. The evaluated Mach are the two discrete Mach available after and the two discrete mach before the candidate speed; two discrete values represent 0.1 mach. They correspond to the altitudes and speeds calculated during the climb phases. Step climbs are evaluated every hour of flight in every combination. Every 25 nm of cruise, the aircraft weight is updated and the geographical coordinates are calculated. The cruise costs contain the descent costs. 4.

VNAV optimal selection

Once all the possible cruises with their step climbs (if available) are calculated, the algorithm has all the information it needs to determine the optimal trajectory. Along the calculations performed by the algorithm, cost tables were created for each flight stage, the complete trajectories costs are then constructed including the influence of each stage. The least expensive trajectory is selected as the optimal trajectory. The climb, cruise and descent profiles are then available.

III. A.

Results

Accuracy of results

In order to verify the validity of the trajectory calculations, two flights were performed on the full aerodynamic model for the studied aircraft with FlightSIMTM by PresagisTM . This aerodynamic model is assumed to be a good representation of the aircraft. The same two flights were performed using the algorithm presented here. The results of these tests can be visualized in Table 2. The flight distance used for these tests is 272 nm. Table 2. Calculation accuracy — comparisons

Flight 1 2

FlightSIMTM (kg) 4753.25 4853.52

Fuel Burned Algorithm (Kg) 4836.45 4925.72

Diff 83.20 72.21

% 1.75 1.49

FlightSIMTM (H) 0.6915 0.7105

Flight Time Algorithm(H) 0.6915 0.7105

Diff 0.0001 0.0068

% 0.02 0.94

It can be seen that the difference between results obtained with the algorithm and with FlightSIMTM is very small. For Flight 1 for example, the difference between the fuel burned by the algorithm comparing to the fuel burned by the FlightSIMTM model is 83.20 kg, this is only 1.75 % more than the FlightSIMTM

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model. For the flight time it can be seen that the difference is practically zero. Fuel burned error can be attributed to the fact that the PDB used by the algorithm does not have exactly the same model as the complete aerodynamic model given in FlightSIMTM , and to the error induced with the interpolations. Flight time is calculated based on the aircraft speed and the selected distance in cruise. B.

VNAV optimization comparison

The test procedure consisted of the selection of the initial and final coordinates of the route to be followed, the selection of the CI, and determining the initial total weight. The developed algorithm was executed and the flight profile, the fuel burned, the flight time and the total cost were saved. The same initial and final coordinates, weight and CI were introduced in the commercial FMS Part Task Trainer (PTT), the same information was saved. The final costs were compared. During these tests, the CI selected for the tests was always 0. This value was selected because the PTT available for tests only had the option of minimum fuel, which is equivalent to selecting a CI of 0. Vancouver was chosen as the maximal destination because the flights that the available FlightSIM is able to perform cannot exceed five hours. Table 3 presents the flights performed and the economization of the algorithm versus that of the PTT. Table 3. Optimization comparison between the algorithm and the commercial FMS Flight 1 2 3 4

Departure Montreal Edmonton Los Angeles Montreal

Arrival Winnipeg Chicago Minneapolis Vancouver

Distance(nm) 984 1233 1323 1988

Optimization(%) 2.99 1.56 1.27 1.20

From Table 3, it can be seen that in all four cases the optimization of the algorithm presented in this paper was superior to the optimization provided by the PTT profiles. This shows that the VNAV flight profile provided by the algorithm is better than the one provided by the PTT. The optimisation mean for this aircraft was 1.75%. This percentage is comparable to the 1.94% found by Gagn´e et al 18 and the 2.57% found by F´elix et al.15 It is important to observe that for the PDB of this aircraft, the speeds and altitudes calculated correspond to a search space reduction of 8 discrete mach and 9 discrete altitude values. This reduction of the search space has a non-negligible impact on the computation time. The computation time for flight 3 from Table 3 for the algorithm developed here was observed to be 12.97 s, whereas the computation time for Gagn´es algorithm in Ref. 18, was observed to be around 40 seconds. The computation time diminution for the same trajectory is a direct consequence of the search space reduction. These tests were performed using an AMD PhenomTM Quad-core processor at 2.29 GHz with 3.48 Gb of RAM.

IV.

Conclusion

The algorithm described in this work was successfully implemented for a commercial aircraft PDB. By taking the costs of climb, cruise and descent into consideration, the new algorithm not only calculates and optimizes the costs for one phase of flight, but for the complete flight. Following a comparison of the results provided by algorithm versus those provided by the PTT, it can be seen that the algorithm provided better profiles for all the evaluated cases. For the studied trajectories simulated by a commercial aircraft, the average optimisation was 1.75%. The algorithm’s flight computations were compared with the results from FlightSIMTM for accuracy and from the PTT for optimizations. The VNAV optimisation computation time for this algorithm was lower than the time required by other algorithms presented in previous works (Ref 16 and Ref 19). This improved optimization time was achieved by reducing the available combinations by implementing the pre-cruise optimisation method. Reducing the computation time is especially crucial to allow the implementation of the new algorithms on an FMS that requires reduced computation time.

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Acknowledgments The authors would like to thank Mr. Rex Haygate at CMC Electronics – Esterline, the Green Aviation Research & Development Network (GARDN), and Mr. Oscar Carranza at the The Research Laboratory in Active Controls, Avionics and Aeroservoelasticity (LARCASE). A. M. M would like to acknowledge the funding provided by the Consejo Nacional de Ciencia y Tecnologia (CONACYT).

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