Branch & Bound Based Algorithm For Aircraft VNAV ...

Branch & Bound Based Algorithm For Aircraft VNAV Profile Reference Trajectory Optimization Alejandro Murrieta Mendoza∗ and Bruce Beuze † and Laurane Ternisien ‡ 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 Computing the vertical navigation reference trajectory is investigated as a way to reduce fuel consumption. Future Air Traffic Management functions might be able to allow aircraft to fly at their most economical profiles allowing fuel consumption reduction. The vertical navigation reference trajectory solution is a combination of the possible Indicated Air Speed, Mach number and altitude of the different flight phases. This paper considers these speeds and altitudes as discrete values, which area available in a Performance Database. The possible combinations are modeled as a tree-like graph. The graph was browsed using a mixture of Best-First Search and Depth-First Search method. A Branch & Bound based algorithm was implemented to reduce the number of computations required to find the optimal combination. A bounding function to estimate the cost at each node was developed and a parameter defined as the Optimism Coefficient was introduced to vary the accuracy of the bounding function. Comparing the experimental results to an exhaustive search algorithm proved the optimal solution and the fuel reduction potential of this algorithm. This algorithm tries to calculate the least possible combinations making it a good choice in low processing power devices such as the Flight Management System.

Nomenclature Altitude AP U AT M BOU N DOU T CDA Copt CI ClimbIAS ClimbM ach CO2 DescentsIAS FMS ISA IAS ListIAS

Altitudes available in the PDB Power Unit Air Traffic Management Cost assigned to a given node Continuous Descent Approach Optimist Coefficient Cost Index Climb IAS available in the PDB Mach numbers available in the PDB Carbon dioxide Descents IAS available in the PDB Flight Management System International Standard Atmosphere Indicated Air speed. First Level IAS nodes waiting list.

∗ Ph.D.

Student, University of Quebec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada. Student, University of Quebec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada. ‡ Undergaduate Student, University of Quebec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada. § Professor, University of Quebec - LARCASE, 1100 Notre Dame West, Montreal, QC, H3C-1K3, Canada. † Undergaduate

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ListM ach LN AV mean N extGEN P DB P DBmin SESARS T OC T OD V N AV V N AVOP T

Second Level Mach Number nodes waiting list. Lateral Navigation Average of the values found in the PDB Next Generation Air Transportation System Performance Database Minimum value found in the PDB Single European Sky ATM Top of Climb Top of Descent Vertical Navigation Optimal Reference Vertical Navigation Trajectory

I.

Introduction

ith the ambitious goal of creating efficient aircraft, the aeronautical industry has successfully reduced W fuel consumption with the evolution and development of new technology. Fuel reduction is one of the main motivators for improving aircraft performance, since fuel cost is highly unpredictable, and by itself represents a high percentage of airline operation costs. Concerns about global warming also serve to push for improvements in aircraft performance. In addition to their cost, fossil fuels release contaminants to the atmosphere that negatively affect global warming. One of these contaminants is carbon dioxide (CO2 ), known for its contribution to the greenhouse effect and thus to global warming. The aeronautics industry intends to reduce its total CO2 contribution from all its activities to 50% of the values recorded in 2005 by the year 2050.1 One of the simplest ways to reduce CO2 is to reduce fuel consumption. To attain this goal, various technological implementations have been studied to reduce fuel consumption. Among the most well-known efforts are the introduction of biofuels2 , component weight reduction, improved aerodynamics by the use of new designs such as morphing wings,3 or by aircraft modifications such as the implementation of winglets4 , engine improvements,5 flight procedure improvements, and more recently, a change in the use of airspace by traffic management improvements. Continuous Descent Approach/Arrival, also known as CDA (i.e., descent in a continuous angled path with the engines in IDLE) is a procedure improvement for descent that has proved to save a significant amount of fuel as well as to reduce noise6, 7 Researchers have observed that the Top of Descent (TOD) location, the aircraft weight at the TOD, and winds each play an important role in this procedure. Different techniques have been developed to estimate these parameters and to better know the best aircraft trajectories.8–12 It is important to perform high-quality descents to avoid missed approaches (or go-around) procedures. Having to execute a go-around procedure has a clear negative impact on fuel consumption and CO2 released to the atmosphere.13, 14 Airspace improvements are intended to offer better traffic management by improving safe traffic separation while allowing, as much as possible, aircraft to fly in or close to their most economical trajectory. These airspace changes are being studied, managed and implemented by the Next Generation Air Transportation System (NextGen) in the United States and by the Single European Sky ATM research (SESAR) in Europe. Although the importance of flying the most optimal route has been desirable for airlines in order to reduce fuel consumption, studies performed by the Massachusetts Institute of Technology (MIT)15, 16 suggested the existence of an important opportunity for reference trajectory optimization to improve fuel consumption, since their studies showed that many flights in the United States did not fly at their optimal speed and/or altitude. These speeds and altitudes followed from point A to point B define the Vertical Navigation (VNAV) reference trajectory. Different authors have worked on the optimal trajectory problem. Normally the aircraft equations of motion (EOM) and an optimal control approach have been proposed.17–20 Other authors have used EOM or look-up tables as an aircraft model along with dynamic programming, or an adaptation to find the optimal trajectory considering factors such as winds and altitude changes.21–24 Some of these approaches could be used by ground teams before takeoff. However, to adapt to any required change in the trajectory, it would be advantageous if airborne equipment could perform trajectory optimization calculation both before takeoff and while airborne. The airborne equipment in charge of performing this calculation and that assists pilots in managing the flight trajectory is the Flight Management System (FMS). Because of their low computation power, some FMSs might not be able to use EOM, but

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rather a set of experimental lookup tables called a Performance Database (PDB). Since lookup tables contain discrete input values, solving the VNAV reference trajectory optimization (the trajectory that burns the least fuel) can be seen as a combinatorial optimization problem. To improve trajectory calculation time, cruise cost estimators have been developed25 , as well as a method to calculate a complete flight using a PDB.26 However, in order to fully solve the VNAV problem, researchers have focused on techniques to reduce the search space or by using genetic algorithms.27 Gagn´e et al. calculated the cost of available climbs, evaluated the combinations of the first Mach available with all the altitudes, considering the climb cost, after selected the most economical altitudes for the rest of the available Mach.28 Murrieta et al. reduced the space search by finding an optimal candidate Mach/altitude cruise pair and calculating the rest of the trajectories around that candidate.29 In addition to methods to reduce the search space, the golden search method has also been implemented to calculate the optimal trajectory.30, 31 Researchers have also worked on Lateral Navigation (LNAV) optimization. Tail winds and low temperatures are preferable to headwinds and high temperatures. Metaheuristic algorithms such as taboo to avoid obstacles such as bad weather32, 33 and genetic algorithms to profit from winds have also been investigated.34 Classical deterministic algorithms such as Dijstkras algorithm has been implemented as a means to profit not only from winds but also from low temperatures.35 Coupling VNAV and LNAV has also been an avenue to investigate. After calculating the optimal VNAV for a given LNAV, Murrieta36 evaluated the same VNAV for four LNAV trajectories parallel to the original LNAV. The best of the five routes was taken as the optimal one. Genetic algorithms have also been used to couple VNAV and LNAV37 . The option of performing step climbs was introduced later,38 where its effect in a real flight was studied. Depending on the available aircraft model, trajectory optimization has been treated as a continuous, discrete or mixed optimization problem. This paper uses a PDB as the aircraft performance model. The combinations of speeds in climb, speeds in descent, altitudes, and Mach in cruise available in the PDB are modeled as a graph in the form of a decision ”tree”. To find the most economical combination, a branch and bound-based algorithm was developed. By analyzing the potential of each node (speed or altitude) a decision is made if the node is to be expanded (if it has enough potential) or discarded (if it does not have enough potential). The structure of this document is as follows: Section 2 describes the Performance Database and explains how the decision tree is constructed. Section 3 describes a typical Branch and Bound (B & B) algorithm and the methods used to travel within the decision tree. Section 4 presents the numerical simulations and observations found.

II.

The Performance Database

The PDB is a performance numerical model of an aircraft and is divided into seven lookup tables, one for every typical commercial flight phase: IAS climb, acceleration, Mach climb, Mach cruise, Mach descent, deceleration, and IAS descent. Table 1 describes the inputs and outputs of the different sub-databases. Outputs are usually fuel burn, distance travelled (useful for flight time computations), and the Altitude required to perform a given acceleration. An interpolations-based algorithm to compute the cost and flight time using a PDB is described39 and used in this paper to calculate the real trajectory cost with satisfactory accuracy. All of the inputs must be available in order to obtain data from the PDB. Weight and temperature cannot be selected by the algorithm since the former depends on the number of passengers, cargo, fuel, etc. of a given flight and so it changes every time and is thus out of the control of flight optimization. Temperature is a completely stochastic value that is constantly changing and so cannot be controlled. However, forecasts may be used to calculate the temperature deviation, and if flying in the International Standard Atmosphere (ISA), a constant value of 0 can be used. With the elimination of weight and temperature, the remaining input variables are Altitude, IAS and Mach as design variables. Speed and altitude values can only take the discrete input values available in the PDB; the problem is finding the best combinations of altitudes and speeds that compose the VNAV reference trajectory. This becomes a combinatorial optimization problem. One way of solving these problems is by using a graph search. For this problem, the available speeds and altitudes can be modeled as a decision tree-like graph.

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

Acceleration

Climb Mach

Cruise

Descent Mach

Deceleration

Descent IAS

1.

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)

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)

Graph Construction

Before constructing the tree, some conditions are imposed: • Flights always begin at 2,000 ft at an initial IAS of 250 kts. • Acceleration can only happen once at 10,000 ft. • Cruise Mach is equal to climb Mach and descent Mach. • While descending, IAS must be at or below 250 kts at 10,000 ft. • The flight trajectory always ends at 2,000 ft. With these conditions, the optimal trajectory solution is given in the V N AVOP T IM AL vector shown in Eq (1), where optimal means the combination that consumes the least fuel. V N AVOP T = [ClimbIAS

M ach

Altitude

DescentIAS ]

(1)

The tree-graph can then be constructed, starting by defining the root node as the IAS of 250 kts. The first level of the tree is composed of N odesClimb (IAS climb), the children of N odesClimb define the second level which is composed of N odesM ach , and the third level, N odesM ach ’s children, are defined by N odesALT . The last node, called a leaf, is at 250 IAS, however, since at the end of each trajectory the cost changes, this is not a common node as is the root node. Therefore, the graph has many different leaves. In this tree every node represents a partial solution of the problem. Every node has a cost. A leaf, the last nodes in the

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tree, which are special kind of nodes, represents a complete solution; this means a given VNAV reference trajectory. The real cost can be calculated after only by reaching a leaf. The graph takes the form shown in Fig (1).

Figure 1. VNAV Combinations tree

III.

Branch and Bound based algorithm

Branch and Bound (B & B) is an optimization method normally used in integer problems. However, it can be applied on a decision tree-graph. Every node in the tree is associated with an optimist cost estimation that considers the whole flight. This estimation is considered to be contained with a combination. The name given to this estimation is called a bound. This bound is used to evaluate if a node is promising or not. The algorithm developed in this paper aims to reduce costs. Thus, the lower the bound the more promising it is. This means that the node itself, along with some of its descendants, and its father if available, potentially compose the bound cost. If a node is not promising it is pruned from the tree along with all its children-descendants without even visiting them. It is this particularity that makes B & B useful, as discarding nodes reduces the number of available combinations. It is desirable to prune the highest number of nodes as possible to obtain the optimal solution with the minimal of leaves visited/evaluated. However, one must be careful not to prune potential nodes. A node is pruned when its bound is higher than the current best solution. This means that the most optimistic cost of the combination of a node’s children will not give a more economical solution than the current one. The first best solution is calculated the first time a leaf is visited. To calculate the bound of a node and decide if it should be pruned, a bounding function is needed. A.

Bounding function

The bounding function is the key element of this algorithm, since it is critical to provide a good optimistic estimation, but not one that is so optimistic that it will not allow the algorithm to prune branches. Optimistic here means that the estimated cost is lower than the real cost. This function determines the bound for each node, which serves to determine if a node is promising. The bounding function makes an estimation of every flight phase cost and adds these phases costs together to provide a Total bound cost, as shown in Eq (2). BoundOU T = CLIM BIAS + CLIM BM ach + CRU ISE + DESCEN TM ach + DESCEN TIAS

(2)

The bounding function is used on the first two levels (IAS and Mach), where not all flight phases values are known. Only at the lowest level (leaves) are all the parameters of the solution in Eq (1) known, and thus 5 of 11 American Institute of Aeronautics and Astronautics

the real cost of the given profile is calculated using a PDB based method.39 Notice that for a given altitude, all of the descent combinations are calculated and the most economical one is selected. 1.

First Level (IAS)

At the first level, the IAS is fixed with the value of the current node. However, Mach, cruise altitude, and descent IAS are still unknown for the total flight cost estimation. Fixing random values to these nodes could result in an unrealistic estimation for the many different flights that an aircraft is able to perform. To be able to identify up to what altitude the speed reference is at IAS (the crossover altitude), the Mach speed should be known. However, since Mach is known only at the next tree level, the IAS altitude is defined by selecting the smallest crossover altitude of the available IAS/Mach. This will lead to an inexpensive Climb IAS, and thus an optimistic cost. The optimism coefficient (Copt ) is introduced in the bounding function in order to influence the optimism level while calculating the bound. This coefficient can have any value from zero (0) to one (1). A Copt = 0 corresponds to a pessimistic bounding function, and Copt = 1 corresponds to an optimistic bounding function. Taking this coefficient into account, the cost estimation is performed using eq (3). CostEstimation = mean − Copt ∗ (mean − P DBmin )

(3)

where CostEstimation is the cost estimation for the flight phase, P DBmin is the minimal value obtained from the PDB for the current flight phase, Copt is the optimism coefficient, and mean is the mean of the maximal and minimal value found in the PDB for the given flight condition. For the climb phase, the travel distance is provided by the PDB, while for the cruise phase, the PDB only provides fuel flow (kg/hr); the distance is required in order to know the flight time and to calculate the fuel burn. To determine the estimated cruise distance, the estimated distance during climb (taken from the PDB) and a distance of 90 nm (typical descent distance for the test aircraft) are subtracted from the total distance from point A to point B. The maximal available Mach is used to calculate the flight time. CLIM BM ach , CRU ISE, DESCEN TM ach and DESCEN TIAS are estimated using Eq (2), and CLIM BIAS is calculated determining the fuel burn and horizontal distance traveled value directly from the PDB with the node value and with the consideration discussed above. 2.

Second Level (Mach)

Once in the second level, CLIM BIAS and CLIM BM ach are known. Since the Mach of climb is the same as the Cruise Mach and Descent Mach, and all available Descent IASs are calculated with the complete model, only altitude is left to be estimated in the bounding function. Knowing both CLIM BIAS and CLIM BM ach brings the advantage of knowing the crossover altitude. As a consequence, a more accurate altitude is known and thus a more realistic CLIM BIAS cost can be obtained from the PDB. The CLIM BM ach for a known Mach cost can be obtained from the PDB, providing an even more accurate estimation. During bounding, the top of climb (TOC) altitude selected to estimate a cost is independent from the cruise altitude. The one selected is the next altitude after the crossover. This will result in a small Mach climb, providing optimistic results. The cruise altitude is determined to be the highest available altitude. This may lead to unfeasible trajectories, which does not represent an issue to the bounding function, since its goal is to predict if any of a node’s children may have a better cost, and as a general rule, the higher the flight, the less fuel it requires. In addition to determining if a node should be branched or pruned, a methodology needs to be defined for traveling within the tree.

B.

Traveling within the tree

The methodology to travel within a tree is a mix between the known methods of Best-First Search and Depth-First Search. The first makes a list of all the visited nodes, giving priority to the most promising first, and then it expands the one that has the best cost thus far, regardless of its position in the tree. This process is repeated until all the nodes have been visited. The latter method expands the nodes at a given level and tries to arrive at the leaves as quickly as possible. Once all the nodes in the current level have been explored, it comes back to the preceding level and explores the next available node. This process is repeated

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until all the nodes have been visited.40 For this paper, it is desirable to arrive at the leaves as soon as possible in order to obtain the first complete solution (incumbent solution) of the form given in Eq (1). Obtaining this solution quickly is important because it is only in a leaf that the real cost of the flight is calculated, with an accurate but time-consuming method. Nevertheless, this real cost presents the opportunity for pruning, which suggests the use of DepthFirst Search. However, it is similarly desired to expand the most promising node first in order to find the optimal solution faster. These methods are combined to obtain the desired characteristics with a minimal effort and as quickly as possible. In the first level after the root node, all nodes are evaluated using the bounding function. A first queue list (ListIAS ) is sorted with the most promising nodes first. The best first node is selected and expanded. For the second level a second queue list is created (ListM ach ), and again the best node is selected and expanded for the last level, the leaves. Once all the leaves have been calculated, the algorithm comes back to the precedent node and goes to the ListM ach . It repeats this process until the list is empty and then begins to work with ListIAS . Eventually the ListIAS is completed, finishing the available nodes. The nature of the bounding function is to be less optimistic as it gets closer to the leaves, as more values of the solution become known. Since the function is less optimistic, it tends to prune nodes, thereby reducing calculation time.

C.

The Algorithm

The pseudo code of the algorithm developed in this project is as follows: 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35:

Inputs = coordinates, takeoff weight, Copt Step 1: first level of the tree For each ClimbIAS do Bound If promising node then Insert ClimbIAS into queue ListIAS Else Pruning End if End for Step 2: Second level and leafs While ListIAS is not empty do Take the first ClimbIAS in ListIAS For each ClimbM ach do Bound If promising node then Insert ClimbM ach into queue ListM ach Else Pruning End If While ListM ach is not empty do Take the first ClimbM ach in ListM ach For each Altitude and DescentIAS do Compute Real cost If Real cost < current best cost then Current best cost = Real cost Current best trajectory = tested leaf End if End for Remove tested ClimbM ach of ListM ach End while Remove tested ClimbIAS of ListIAS End for End While Display best cost and V N AVOP T IM AL

IV.

Results

The results found by the algorithm were compared to an exhaustive search algorithm using a PDB-based method39 to test the optimal VNAV profiles provided by the algorithm. Since all the options were evaluated, it can be concluded that it finds the optimal combination. The aircraft considered for the experiments was

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a 3-engine aircraft with a maximal take-off weight of 225,000 kg, a maximal cruise altitude of 42,000 ft, and a minimal and maximal Mach of 0.78 and 0.84, respectively.

A.

Weight Sensitivity

The first tests were performed by changing the take-off weight with a Copt = 0.5. The test flights were from Montreal to Vancouver with a distance of 1992 nm. For this test, 4 different takeoff weights were evaluated, 170,000 kg, 180,000 kg, 190,000 kg, and 200,000 kg. In all cases, the B & B-based algorithm found the same solution as the exhaustive search. Table 2 shows the results of the pruned branches.

Table 2. Pruned branches and Weight Take-off Weight (kg) Pruned branches Level 1 Pruned branches Level 2 Total leaves not evaluated

170,000 0 82 4920

180,000 0 16 960

190,000 0 20 1200

200,000 0 25 1500

As can be observed in Table 2, nodes in the first level are rarely pruned, which can be explained by the value of the optimism coefficient, as will be shown in other sets of results below. However, for Level 2, the number of branches pruned is much higher. This is due to the fact that at the second level, the bounding function is more realistic and thus less optimistic, allowing more branches to be pruned. If the algorithm was not this optimistic at the first stage, the optimal solution might have been pruned early in the process. The objective of not arriving at the leaves too quickly is to reduce the number of times the complete trajectory optimization algorithm must be used. Although pruning at the first level would have been desired, pruning at the second level brings an important reduction in the number of calculations to perform. The total number of leaves available is around 10,200. For these tests, it can be seen that the number of leaves (total trajectories) not evaluated can be as high as close to 50%. A relationship between weight and branches is found for values over 170,000 kg; the higher the weight, the more branches are pruned. The 170,000 kg weight is a particular case. Many branches were cut due to unfeasible solutions (i.e. there was not enough fuel to get to the destination). In order to have more branches pruned, a less optimistic bounding function is required. The first flight was explored further by varying the optimism coefficient, as seen in Table 3.

Table 3. Pruned Branches and the Optimism Coefficient Optimism Coefficient (Copt ) Pruned Branches Level 1 Pruned branches Level 2 Total leaves not evaluated

0 12 166 9960

0.2 12 165 9900

0.3 12 164 9840

0.4 0 95 5700

0.5 0 82 4920

0.6 0 69 4140

0.7 0 61 3660

0.8 0 46 2760

0.9 0 37 2220

1 0 28 1680

In Table 3, the optimal solution was found for Copt ≥ 0.5, while for 0 ≥ Copt ≤ 0.4 the solutions were sub-optimal. With a Copt = 1, it can be seen that the number of branches cut was the lowest, thus the number of leaves evaluated was the highest. For a Copt = 0, the highest number of branches were cut, thus the number of leaves not evaluated was at its highest level. Fewer leaves to be evaluated is desired, as that means less calculations to be done by the complete time-consuming trajectory method. A constant reduction in the number of branches pruned can be observed as the coefficient is more optimistic. For 0 ≥ Copt ≤ 0.3 the number of pruned branches in the first level is 12. This number, along with the branches pruned in the second level, brings a reduction of around 91% of the available branches to evaluate. As stated earlier, the counterpart to this level of reduction is that the optimal profile was not found. However, the question arises of how far the cost found differs from that of the optimal. An analysis of this particular flight was done using data from Table 3, comparing the optimal profile calculated by an

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exhaustive search and the results provided by 0 ≥ Copt ≤ 0.4.

Table 4. Calculation accuracy — comparisons Copt 0.4 0.3 0.2 0

Optimal cost difference 0.005 % 0.007 % 0.24 % 0.40 %

The difference from the optimal is so small that even a Copt = 0 provides an interesting trajectory with a small deviation from the optimal profile. This difference is due to the fact that the branch with the optimal profile was pruned due to a lack of optimism. As the optimism gets better, the difference from the optimal tends to be 0, but as shown above, incrementing the number of iterations is required to find the optimal. In the test computer , for a Copt = 0.5 it took 210 s to find the optimal trajectory and for a Copt = 0 it took 7.3s to find a sub-optimal. The optimism coefficient allows a trade-off between accuracy and computation time. It can be fixed depending on the system where it is to be implemented. Due to the low computation power available in a conventional FMS, it might be desirable to place it somewhere between 0.3 and 0.6, since that value will allow a short computation time to find accurate solutions. The last item evaluated was the effect of cruise distance on the computations. For this evaluation, Montreal (YUL) Minneapolis (MSP) with a distance of around 800 nm, Montreal Winnipeg (YWG) , with a distance of around 1000 nm, and Los Angeles (LAX) Minneapolis with a distance of around 1300 nm were evaluated with a Copt = 0.5. In all cases the optimal trajectory was found by the B & B based algorithm (compared to the exhaustive search method). Table 5 shows the results and the branches pruned. 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 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. Table 5. Flight Distance and Pruned Branches Flight Pruned Branches Level 1 Pruned Branches Level 2 Leaves not evaluated

YUL - YWG 0 46 2760

YUL-MSP 0 52 3120

LAX - MSP 0 46 2760

The behavior of the branches pruned and the level where this operation was executed concur with the behavior of the tests presented in Table 1, which suggests that the estimation performed in the bounding function to compute the cruise is accurate enough, regardless of the flight distance. This is an important observation, since cruise is the longest stage of flight, and it is the one that contributes the most to the total fuel burn. According to the set of results presented in Tables 2-5, it can be seen that Copt is the parameter that has the highest influence in selecting the optimal solution and that most directly influences the number of calculations required to find that solution.

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V.

Conclusion

The VNAV trajectory optimization problem was modeled as a tree-graph. This method is made possible because, from a PDB perspective, there are only discrete values, making it a combinatory optimization problem. Combinatory optimization problems are difficult to solve as there are normally a very high number of combinations to evaluate. To solve this problem, a B &B -based algorithm was developed, which combined two search methods: best first search and depth-first search. This combination brings the advantages of both methods to the VNAV optimization problem: to find an incumbent solution early in the algorithm to obtain a real cost and begin pruning to make the algorithm quicker, and to focus the algorithms attention on the most promising nodes first. A crucial part of this algorithm is the bounding function, an estimation of the cost which is performed to decide if a node should be explored or pruned. The bounding function adapts to the known parts of the solutions and introduces the optimism coefficient to estimate the parts of the solution where a value has not yet been defined. The test performed used the algorithm suggested that with a correct selection of Copt , the optimal solution, or a close sub-optimal can be found. It was shown that this coefficient has a direct influence on the speed of the algorithm, since less-optimistic bounding tend to prune more branches. This work does not take into account the time of flight or the influence of the weather. Time is an important parameter in flight cost, and weather (winds) and has a direct impact on flight time. Ongoing research is being conducted to incorporate the flight time cost into the bounding function. Weather is to be implemented in both the bounding function and the complete trajectory computation.

Acknowledgments This research was conducted in the Research Laboratory in Active Controls, Avionics and Aeroservoelasticity (LARCASE) for the global project Optimized Descent and Cruise with funds from the business-led Network of Centers of Excellence Green Aviation Research & Development Network (GARDN). The authors would like to thank Mr. Rex Haygate, Dominique Labour and Yvan Blondeau from CMC-Electronics Esterline, and Mr. Oscar Carranza from LARCASE. A.M.M would like to thank CONACYT.

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