Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014 November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-37568

METHOD TO CALCULATE AIRCRAFT VNAV TRAJECTORY COST USING A PERFORMANCE DATABASE

Alejandro Murrieta-Mendoza Ruxandra Botez University of Quebec (ÉTS) - LARCASE University of Quebec (ÉTS) - LARCASE Montreal, Quebec, Canada Montreal, Quebec, Canada http://larcase.etsmtl.ca [email protected]

ABSTRACT there will be almost 40,000 commercial airplanes in operation, Vertical Navigation (VNAV) trajectory optimization has doubling the number of airplanes available in 2010. been identified as a means to reduce fuel consumption. Due to the computing power limitations of devices such as Flight Aircraft use fossil fuel; among the multiple associated Management Systems (FMSs), it is very desirable to implement disadvantages is its release contaminant emissions such as a fast method for calculating trajectory cost using optimization carbon dioxide (CO2), nitrogen oxides, (NOx), hydrocarbons, algorithms. Conventional trajectory optimization methods solve and water vapor [2]. From these emissions, CO2 is of special a set of differential equations called the aircraft equations of interest due to its contribution to global warming. The motions to find the optimal flight profile. Many FMSs do not aeronautical industry is responsible for 2% of the CO2 released use these equations, but rather a set of lookup tables with to the atmosphere. The aeronautical industry has set a goal of experimental, or pre-calculated data, called a Performance reducing their CO2 emissions in 2050 to 50% of the emissions Database (PDB). This paper proposes a method to calculate a recorded in 2005 [3]. full trajectory flight cost using a PDB. The trajectory to be calculated is composed of climb, acceleration, cruise, descent Different technologies are being developed to reduce and deceleration flight phases. The influence of the crossover emissions, such as engine improvements [4], the use of altitude during climb and step climbs in cruise were considered biofuels [5], weight reduction by means of advanced avionics for these calculations. Since the PDB is a set of discrete data, and by replacing heavy materials with composites, and aircraft Lagrange linear interpolations were performed within the PDB design improvements such as winglets [6]. to calculate the required values. Given a takeoff weight, the initial and final coordinates and the desired flight plan, the Optimal flight planning and optimal flight trajectory trajectory model provides the Top of Climb coordinates, the calculation have been identified as ways to reduce fuel Top of Descent coordinates, the fuel burned and the flight time consumption, thus reducing polluting emissions and saving on needed to follow the given flight plan. The accuracy of the costs. Air authorities in North America and Europe have trajectory costs calculated with the proposed method was considered this challenge and are redefining their airspace with validated for two aircraft; one with an aerodynamic model in the Next Generation Air Transportation System (NextGen) [7] FlightSIM, software developed by Presagis, and the other using and the Single European Sky (SESAR). the trajectory generated by the reference FMS. There are two trajectories that can be optimized, Vertical INTRODUCTION Navigation (VNAV) and Lateral Navigation (LNAV). VNAV is According to the Air Transport Action Group (ATAG), air composed of the speeds and altitudes to follow during flight, transportation has become an important pillar of economical and LNAV is the geographical coordinates the aircraft has to interchange, transporting close to 35% of the world’s trade follow. value. Air transportation is also one of the preferred ways to travel, with occupancy at 78% in 2011 [1]. This activity has Due to the proximity of airports to cities, the descent phase motivated air transportation providers to increment their fleets. has been of great concern regarding trajectory optimization. T he ATAG, in the same study, estimated that for the year 2030, Murrieta et al [8] and Dancila et al. [9], using information from the European Monitoring and Evacuation Programme and

1 Copyright © 2014 by ASME the European Environment Agency (EMEP/EEA) emission technique called “Moving Search Space Dynamic inventory guidebook developed methods to calculate the fuel Programming” to calculate the optimal trajectory of a flight needed and the pollution generated by the execution of the considering weather and required arrival time (RTA) [22]. missed approach (go-around) procedure. Celis et al. [10] proposed a model to predict the polluting emissions of gas Solving a set of differential equations (equations of turbine combustors for optimization processes. motion) to find the optimal trajectory in a device with limited processing power such as the FMS can be time consuming. Stell [11] developed a way to estimate the location of the Normally, FMSs do not use these equations, but a set of lookup Top of Descent in order to execute a Continuous Descent tables with experimental data called a Performance Database Approach (CDA) at a three-degree slope. Kwok-On et al. [12] (PDB). Hagelauer and Mora-Camino considered equations of measured the fuel-burned reduction effectiveness of the CDA at motion to implement dynamic programming with neuronal the Louisville Airport (Kentucky, USA). Clarke et al [13] networks to optimize the trajectory [23], revealing that the proposed a method to implement the CDA at Los Angeles FMS uses databases (PDBs) instead of equations of motion. To International Airport. emulate these databases using this approach, the equations of motion were discretized. In the 1970’s, a device called the Flight Management System (FMS) was developed, and became widely introduced Various authors have considered using a PDB to calculate in aircraft in the 1980’s [14]. This device, among other tasks, an aircraft’s optimal trajectory. Felix et al used a PDB, is responsible for flight plan management by computing the implementing step climbs, the golden section search and optimal trajectory, thereby helping to reduce the crew’s genetic algorithms to optimize the VNAV profile of an aircraft, workload. [15]. Ground teams also calculate the flight encompassing all of the flight phases for three different trajectory before an aircraft is airborne, however, it is ideal commercial aircraft [24, 26]. Dancila et al. developed a method provide the FMS with the autonomy to calculate the optimal to estimate the fuel burn for cruise segments at constant trajectory because pre-calculated routes can change before or altitudes using a PDB [27, 28]. while airborne. The FMS’s processing power available to perform all of its tasks is limited, and only a fraction of this Gagné et al. used PDBs to develop an optimization processing power is dedicated to finding the optimal trajectory. algorithm by means of a semi-exhaustive search for a Thus, methods to rapidly perform trajectory calculations are commercial aircraft, considering weather information along the needed desirable. trajectory. All of the flight stages were analyzed, and step climb opportunities were identified and performed during Lidén was one the first researchers to study flight trajectory cruise to optimize the trajectory cost. Murrieta et al. in [30] calculations with FMSs. Lidén evaluated using speed control to developed an algorithm to reduce the search space and minimize flight costs, considering the effects of arriving too calculate the optimal VNAV profile. late to a given trajectory, and also discussed a way to estimate the Cost Index (CI) [16]. Felix et al. used genetic algorithms and information from a PDB to develop an algorithm to find the optimal LNAV Fays et al., using information from Boeing, implemented trajectory at constant altitude, including taking advantage of metaheuristic algorithms to find the optimal trajectory while tailwinds [31]. avoiding obstacles [17, 18]. Sidibé implemented a conventional dynamic programming Khardi proposed a dynamic method to optimize the CDA algorithm to find the optimal VNAV by using the numerical using an Ordinary Differential Equations system [19]. This model provided by the PDB. [32] work took into account jet noise, fuel consumption, and operational aircraft constraints. Researchers have also tried to couple LNAV and VNAV, as Murrieta did in [2], where five lateral parallel trajectories were Franco and Rivas used the equations of motions of an evaluated, taking advantage of weather parameters after aircraft at constant altitude and constant heading to calculate its determining t he V N AV. F e l i x et al. calculated the optimal optimal profile in the presence of winds [20]. For the flight VNAV profile, and then coupled the optimal LNAV trajectory, cost, arriving early or late at a given destination was modeled which was determined using genetic algorithms [33]. as a penalty. This paper suggested the use of singular optimal control to find the optimal flight profile. Despite the number of optimization algorithms that use a PDB, only a few provide an overview of the method for Sridhar et al. utilized equations of motion to develop an calculating the trajectory cost. Instead, they mostly focus in the algorithm to calculate an optimal trajectory considering wind optimization method. Felix et al. describes a way to reduce and avoiding regions favorable to contrail formation [21]. interpolations, but only explains a small part of the climb Miyazawa et al. used the equations of motion to propose a calculations.

2 Copyright © 2014 by ASME This paper presents a method to calculate the cost of the METHODOLOGY full trajectory using a PDB. The PDBs were provided by our industrial partner and were used for commercial FMSs. The I.- The Conventional Flight trajectory method is composed by a climb, acceleration, cruise, The flight evaluated here is a standard commercial flight descent and deceleration. Concepts such as crossover altitude composed of constant climb at 250 kts Indicated Airspeed effects during climb and step climbs in cruise were considered. (IAS). If needed, after 10,000ft, an acceleration phase to the Since the PDB is a set of discrete data, linear interpolations are desired constant climb speed is performed. At a given altitude performed within the PDB tables to calculate the trajectory the Airspeed Indicator (ASI) reference is changed from IAS to cost. mach, and the plane continues climbing in mach until the Top of Climb (TOC). Step climbs (change of altitudes) may be This paper is organized as follows: First, the conventional executed during cruise. From the Top of Descent, the descent is flight to be calculated is defined. The PDB, as well as all of its calculated in the inverse order as the climb. Figure 1 shows the sub-databases are then described. The Flight Cost and its order of these flight phases. variables are discussed, followed by description of the trajectory calculation method. Finally, the simulation results are Step Climb Top of descent discussed and presented. Top of climb

MACH climb MACH decent NOMENCLATURE Crossover altitude

A1 Next multiple of 1,000ft altitude after acc. KIAS climb KIAS descent A0 Altitude after acceleration Altitude (ft) ah Sound speed at a given altitude Acceleration Deceleration acc Acceleration 10,000 ft 10,000 ft ASI Airspeed Indicator KIAS climb @ 250 KIAS KIAS descent @ 250 KIAS ATAG Air Transport Air Group Distance (nm) CDA Continuous Descent Approach FIGURE 1. CONVENTIONAL COMMERCIAL FLIGHT CI Cost Index CO2 Carbon dioxide II.- The Performance Database (PDB) FMS Flight Management System The information needed to perform all of the required GS Ground Speed calculations is contained in the PDB. The PDB is a numerical IAS Indicated Airspeed model of the aircraft, divided into seven sub-databases, one for ISA International Standard Atmosphere each flight phase All of the input parameters have to be LAX Los Angeles International Airport code provided in order to obtain the information from the PDBs,. LN AV Lateral Navigation Table 1 describes the inputs and outputs of the different sub- MSP Minneapolis--Saint Paul International Airport databases. NOx Nitrogen oxides P0 Stagnation pressure The sub-databases are defined in a discrete form, and so P1 Static pressure at a given altitude only certain parameters can be found in the PDB inputs. For Ps Sea level pressure example, the International Standard Atmosphere (ISA) PDB Performance Database deviation temperature may be given in steps of 10 degrees, PTT Part Task Trainer gross weight in steps of 15,000 kg, and so on. For those cases RTA Required Time of Arrival where the values to be introduced in the inputs are not exactly SESAR Single European Sky in the same form as that available in the PDB, interpolations TAS True Airspeed between the outputs of the available data must be performed. TAS IAS True Airspeed in IAS TAS mach True Airspeed in mach TABLE 1. PDB SUB DATABASES TOC Top of Climb Sub- Inputs Outputs TOD Top of Descent database Climb IAS IAS (knots) Fuel burn (kg) V N AV Vertical Navigation Gross weight (kg) Horizontal traveled distance YUL Montreal-Trudeau International Airport code ISA deviation temperature (ºC) (nm) Altitude (ft) YVR Vancouver International Airport code Climb Gross weight Fuel burn (kg) YYZ Toronto International Airport code acceleration Initial IAS (knots) Horizontal traveled distance λ Specific heat of air Acceleration altitude (ft) (nm) Delta speed to accelerate Altitude needed (ft) (knots) Climb Mach Mach Fuel burn (kg) Gross weight (kg) Horizontal traveled distance ISA deviation temperature (ºC) (nm) Altitude (ft)

3 Copyright © 2014 by ASME TABLE 1. PDB SUB DATABASES (Continue) Knowing the TAS, the Flight_Time can be calculated from Cruise Mach Fuel flow (kg/hr) the Horizontal_traveled_distance distance obtained from the Mach Gross weight (kg) ISA deviation temperature (ºC) PDB (Table 1), as shown in eq (5). Altitude (ft) Descent Mach Fuel burn (kg) Mach Gross weight (kg) Horizontal traveled distance (nm) _ = _ _ (5) ISA deviation temperature (ºC)

Altitude (ft) 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹ℎ𝑡𝑡 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑇𝑇𝑇𝑇𝑇𝑇 Deceleration Gross weight Fuel burn (kg) During cruise, the Horizontal_traveled_distance�𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 is pre-defined deceleration Initial IAS (knots) Horizontal traveled distance (nm) Deceleration altitude (ft) Altitude needed (ft) by the user, as explained below in the cruise section. Delta speed to accelerate (knots) Descent IAS IAS (knots) Fuel burn (kg) IV.- Trajectory Calculations Method Gross weight (kg) Horizontal traveled distance (nm) ISA deviation temperature (ºC) Variables such as weight and the ISA deviation Altitude (ft) temperature are rarely exact discrete values in the PDB. Therefore, as mentioned above, interpolations in the outputs of III.- Flight Cost the PDB are required. The trajectory procedure uses the The typical equation used in the literature to define flight Lagrange interpolations from the PDB outputs at the discrete cost is: input containing the desired value. This interpolation method was selected because it is the same interpolation that has been = _ + _ (1) successfully implemented in the algorithms defined in the literature. Equation (6) describes the Langrage interpolation. where𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 Fuel_Burned𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐵𝐵 (kg)𝐵𝐵𝐵𝐵𝐵𝐵 𝐵𝐵is𝐵𝐵 the total𝐶𝐶𝐶𝐶 ∗ fuel𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 burnedℎ𝑡𝑡 𝑇𝑇𝑇𝑇𝑇𝑇 during𝑇𝑇 flight, Flight_Time (hr) is the total flight duration, and CI (kg/hr) is ( ) = + (6) the Cost Index. CI is the variable that converts the cost of time 1 0 into fuel burned terms. A higher CI will give priority to a lower 1 𝑥𝑥 − 𝑥𝑥 0 𝑥𝑥 − 𝑥𝑥 1 𝑝𝑝 𝑥𝑥 0 1 𝑓𝑓 1 0 𝑓𝑓 flight time without considering fuel burned and a low CI will The required interpolation𝑥𝑥 − 𝑥𝑥 path𝑥𝑥 −for𝑥𝑥 the majority of the give priority to fuel burned without considering flight time. In PDB tables, with exception of acceleration and deceleration, is this paper, the CI will always be assumed to be zero. graphically shown in Figure 2. The word “limit” in this figure refers to the discrete interpolation values taken from the PDB A key objective is to develop a method to calculate the that contain the desired value to interpolate. The desired trajectory cost in terms of the variables present in eq (1): fuel outputs are fuel consumption and horizontal traveled distance. burned and flight time. Fuel burned can be obtained directly For cruise, only fuel flow is calculated. from the PDB. However, to calculate Flight Time the Aircraft Ground Speed (GS) must be known. In order to calculate the This method requires the following inputs: aircraft total GS, the atmosphere is considered to be the ISA and the winds weight, a VNAV/LNAV flight plan containing the waypoints, to be null. Under these circumstances the GS is equal to the altitudes and speeds to follow, and the ISA Deviation Standard True Air Speed (TAS). When the speed is given in the IAS, Temperature at each waypoint. If flying in ISA conditions, the TAS is calculated with equation (2) ISA Deviation Standard is considered to be 0 throughout the flight. 2 ( )/ = 1 (2) Each flight stage has some particularities, which will be 21 𝛾𝛾−1 𝛾𝛾 𝑎𝑎ℎ 𝑃𝑃0 described next. At the end of each phase, the Fuel burned and 𝑇𝑇𝑇𝑇𝑇𝑇𝐼𝐼𝐼𝐼𝐼𝐼 � �� � − � the required Flight Time are accumulated to their respective 𝛾𝛾 − 𝑃𝑃1 where ah is the speed of sound (kts) at the given altitude, γ is variables in Eq (1). the specific heat of air, typically 1.4, P1 is the static pressure at the given altitude and P is the stagnation pressure in the Pitot ISA DEV limit 1 0 from PDB tube calculated as in eq. (3), where Ps is the sea level pressure. Interpolation for ISA ISA DEV to DEV using PDB interpolate /( ) Weight limit 1 ( 1) Weight limit 1 from PDB = + 1 𝛾𝛾 𝛾𝛾−1 + (3) ISA DEV limit 2 (2) 22 from PDB Interpolation for Weight to 𝐼𝐼𝐼𝐼𝐼𝐼 𝛾𝛾 − weight using ISA Desired Output 0 𝑠𝑠 1 interpolate 2 DEV interpolations 𝑃𝑃 𝑃𝑃 �� ℎ � � 𝑃𝑃 ISA DEV limit 1 When the speed is given𝑎𝑎 in mach, the TAS is obtained with eq from PDB Weight limit 2 from PDB (4). Interpolation for ISA ISA DEV to DEV using PDB interpolate Weight limit 2 = (4) ISA DEV limit 2 from PDB 𝑚𝑚𝑚𝑚𝑚𝑚ℎ ℎ 𝑇𝑇𝑇𝑇𝑇𝑇 𝑚𝑚𝑚𝑚𝑚𝑚ℎ ∙ 𝑎𝑎 FIGURE 2. INTERPOLATIONS FLOWCHART

4 Copyright © 2014 by ASME i.- Climb IAS Figure 4 shows an acceleration from 250 IAS to 300 IAS at For this phase, the sub-database “Climb IAS” from Table 1 10,000 ft, finishing at an altitude of 12,520 ft. With the gross is used. Because the Take Off procedure is not considered, the weight after acceleration, a climb at the constant new altitude flight always initiates at 2,000 ft. At every 1,000 ft, an from 12,000 to 13,000 is calculated. The percentage of the interpolation is executed to calculate the horizontal distance remaining climb (480 ft or 48% of the climb) is added to the traveled and the fuel burned to climb to a higher flight level. previously calculated acceleration fuel burned and/or the Using the TAS from eq (2) and the horizontal distance traveled, horizontal distance traveled. the segment flight time is calculated with eq (5).

The default speed used to perform the climb from 2,000 ft to 10,000 ft is normally at or below 250 IAS. This is in accordance with the Code of Federal Regulations 91.117 in the CLIMB @ 300 IAS 14,000 ft United States and the Canadian Aviation Regulation 602.32 limiting speed below 10,000 ft to 250 IAS. 13,000 ft

12,520 ft ii.- Acceleration Acceleration Climb speed after 10,000 ft is normally higher than 250 250 – 300 IAS

IAS, and thus an acceleration is required. The acceleration (ft) Altitude phase has two different stages: The first computes the 10,000 ft acceleration using the acceleration sub-database in the PDB, the CLIMB @ 250 second is a small climb at the new constant IAS using the IAS “climb IAS” sub-database to reach the next multiple of 1,000 ft in altitude.

Figure 3 describes the interpolations needed to calculate Distance (nm) the acceleration outputs. The required speed incremental (delta speed) that the aircraft has to accelerate is determined, and the FIGURE 4. ACCELERATION EXAMPLE interpolations for the required increment speed are performed. If the PDB initial speeds differ from 250 IAS, speed Equation 7 expresses the total fuel cost of the fuel added to interpolations may be required. This first set of interpolations the acceleration fuel burned. are performed for the lower step weight, followed by the same set of interpolations for the highest weight step. An = + ( ) (7) interpolation between the results for both weights is then 11,0000 𝐴𝐴 − 𝐴𝐴 performed using the weight of the aircraft at the beginning of 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑙𝑙𝑓𝑓𝑓𝑓𝑓𝑓𝑙𝑙푎푐푐 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎 � � 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝐼𝐼𝐼𝐼𝐼𝐼 the acceleration phase. For Flight Time calculation, speed is the A0 is the altitude after the acceleration, A1 is the next multiple average of the initial and the final speed. The weight of the fuel of 1,000 ft altitude after A0, fuelacc (kg) is the fuel obtained burned is reduced from total gross weight. after the acceleration interpolations, and fuelIAS is the fuel needed to climb from altitude (A1 – 1000 ft) to A1. A similar The second part of the acceleration phase computations equation is used to obtain the total horizontal distance traveled. begins after calculating the required altitude. This altitude is rarely a multiple of 1,000 ft. Thus a small climb is calculated iii.- Climb mach using the methodology described in Climb IAS. An example is When the crossover altitude is reached, the PDB table is shown in Figure 4. changed from “Climb IAS” to “Climb mach”. Keeping the IAS reference may lead the aircraft to fly at higher speeds than Delta speed step 1 desired, approximating to mach speed. Commercial aircraft are PDB initial Required delta W speed 1 interpolation not normally designed to fly at these speeds. The crossover E Initial speed step 1 I Delta speed step 2 Delta altitude can be defined as the altitude where the TAS of the Initial requiered G speed speed interpolation H needed IAS equals the TAS of the scheduled mach. An approximation T Delta speed step 1 Initial speed step 2 PDB weight 1 of the crossover altitudes for the Sukhoi Super Jet 100 is Required delta PDB initial 1 interpolation Fuel burned (kg) speed 2 Required weight Horizontal Distance (nm) shown in Table 2. If the crossover altitudes are not multiples of interpolation Delta speed step 2 Altitude needed (ft) 1,000 ft, one part of the climb has to be calculated using the PDB weight 2

Idem for weight 2 “Climb IAS” sub-database and the other part with the “Climb Mach” sub-database. Calculations similar to those in the FIGURE 3. ACCELERATION INTERPOLATIONS acceleration phase are performed to define the cost percentage of Climb in IAS and the percentage of climb in Mach.

5 Copyright © 2014 by ASME TABLE 2. IAS / MACH CROSSOVER Weight ALTITUDE APROXIMATION Initial Speed IAS / 260 270 280 290 300 310 Coordinates Mach 0.50 14000 12000 10000 10000 10000 10000 Climb @ 250 IAS to 0.56 20000 18000 16000 14000 12000 11000 next altitude 0.59 22000 21000 19000 17000 15000 13000 Altitude + 1,000 ft 0.62 25000 23000 21000 20000 18000 16000 0.65 27000 26000 24000 22000 20000 19000 No Altitude == 0.68 30000 28000 26000 24000 23000 21000 10,000 ft 0.71 32000 30000 28000 27000 25000 23000 Yes 0.74 34000 32000 30000 29000 27000 26000

Using the TAS from eq (4), the flight time for this phase is Yes Target speed == No Accelerate to calculated with eq (5). Initial speed needed speed Climb in IAS to the Climb finishes when the Top of Descent or the maximal next altitude altitude is reached. Figure 5 provides a flow chart of the steps followed to calculate the trajectory climb. Cimb in IAS @ needed speed iv.- Descent Estimation Altitude + 1,000 ft Before cruise, an approximate descent horizontal distance traveled is required in order to define an estimated Top of Altitude > No Descent (TOD). The estimation is performed by fetching the Crossover altitude horizontal distance traveled for the given altitude and the given Mach directly from the Mach climb PDB. A more precise Yes descent is calculated at the end of cruise. Cimb in KIAS in the first part before the v.- Cruise crossover Cruise is normally the longest flight phase; it begins at the Cimb in MACH after TOC and ends at the TOD. During this phase, the waypoints to crossover until next be followed have to be proposed or obtained from a proposed multiple of 1,000 ft LN AV route. For this paper, equidistant great circle waypoints altitude are considered as waypoints. The great circle, or geodesic, is Climb in MACH to the shortest distance between two points in a sphere. next altitude

At every waypoint, the fuel burned to attain the present Altitude + 1,000 ft waypoint is reduced from the total aircraft weight. This step increases the accuracy of the calculations. The more waypoints No TOD or Maximal Altitude and the shorter the distance between them, the better the accuracy of the calculations. However, a high resolution will also increase the computation time. Yes Climb End

Interpolations in weight and ISA Temperature Deviation FIGURE 5. CLIMB PROCEDURE FLOWCHART such as those described in Fig (2) are performed in this stage. vi.- Step Climb “Step climb” is the change of cruise altitude to a higher one, as an attempt to emulate an ideal constant climb cruise [35]. This change in altitude was studied by Lovegren [36] and has been proven to save fuel, especially on long haul flights. This change in altitude requires a climb in Mach, and so the step climb cost is calculated using the “Climb Mach” database at the desired geographical point. The climb step is normally executed in 2,000 ft steps. This respects the normal 1,000 ft separation between airways, where aircraft fly at opposing

6 Copyright © 2014 by ASME headings. The first waypoint at the new altitude is calculated destination point, the missing or surpassed distance is added or with the climb horizontal distance obtained from the PDB. removed from the cruise phase, the TOD is redefined and the descent is recalculated. This process is repeated until the Once the desired altitude has been reached, the aircraft ends within the imposed limits. computations are continued using the “Cruise Mach” sub- database. The step climb procedure can be executed as many Figure 7 is a description of the final descent procedure and times as required or until the maximal altitude is reached. the coupling with the cruise phase. The flight trajectory procedure ends here. At this point, fuel burned and flight time Figure 6 presents a flowchart of the method to follow to were calculated for each section and accumulated in the fuel perform the required cruise calculations with “step climbs” in and the time variable in eq (1). the cruise and step climb phases.. Last segment Weight of cruise Est. TOD Altitude MACH TOD reached

Switch to Cruise Calculate 1st waypoint at Descent in Mach Recaculate TOD Mach PDB new altitude

Descent in IAS Waypoint reached

Switch to Climb Mach PDB Flight to next Yes Deceleration to Altitude = 10,000 ft Step climb? Calculate Fuel burned and waypoint =< 250 IAS IAS =< 250 horizontal distance No No Correct last cruise Yes segment distance No Descent to 2,000 ft TOD reached?

Yes Cruise end Is the aircraft within No FIGURE 6. CRUISE CALCULATIONS the destination limits? PROCEDURE FLOW CHART vi.- Final Descent Yes Final Descent is assumed to be in the CDA, and begins Descent end when the TOD has been reached. This phase is highly FIGURE 7. DESCENT PROCEDURE FLOWCHART dependent on the cruise, since the weight to be used is the one at the end of cruise, and there could be TOD location errors. RESULTS The last waypoint of cruise is not equidistant. It is the distance The parameters to be evaluated for the flight trajectory that remains from the (TOD - 1) waypoint to the estimated cost are flight time and fuel consumption, as they are the TOD. variables that define eq (1). Different flights were executed with the method described here, and the same flights were Descent is similar to climb, but the order of the databases calculated with the reference method. The results were then used and the order of the calculations are in reverse. First the compared. For Aircraft A (Aircraft Max Weight = 245,000 kg), Mach descent is calculated until the crossover altitude is tests were validated with a full dynamic model simulated with reached. Next, a descent in IAS is calculated followed by a the software FlightSIM from Presagis. The results of deceleration. Finally, at 10,000 ft, the final speed must be 250 FlightSIM simulations are considered to be close to reality. For IAS, as required by legislation such as the Federal Aviation Aircraft B (Aircraft Max Weight = 47,000 kg), the results were Administration and Transport Canada. Finally, the descent is validated with the Part Task Trainer (PTT) of the reference executed at this constant speed of 250 IAS, and ends at an FMS. The PTT uses the PDB to calculate flight cost with a altitude of 2,000 ft. The same considerations utilized in climb commercial algorithm. For the tests performed for both are also assumed during the descent phase. The final location of aircraft, the CI was considered to be zero. the aircraft is then compared to the final point of the trajectory. If the final position of the aircraft is located after the destination Five flights were tested to validate the method; Tables 3 point, or if it is not located within a given distance before the and 4 summarize the results for Aircraft A. The first three

7 Copyright © 2014 by ASME flights were Montreal (YUL) to Toronto (YYZ) with a distance is kept constant regardless of the distance and the weight of the of 272 nm. The other two were from Los Angeles (LAX) to aircraft. These results indicate that the speed equations and the Minneapolis (MSP), with a distance of 1324 nm. Take off method’s considerations allow for accurate time calculations. aircraft gross weight and speeds were changed between flights. Figure 10 and Figure 11 are the results for Aircraft B. The Aircraft A first three flights are from LAX to MSP and the last two are from YUL to Vancouver (YVR), with a distance of 571.85 Kg 570.58 Kg 20000.00 3.10% 3.17% approximately 2000 nm each. The difference was obtained 18000.00 using the results of the PTT as a reference. 16000.00 14000.00 In Figure 10, it can be seen that the fuel burned has a 12000.00 similar difference percentage for all flights. This average 10000.00 FlightSIM difference is only 0.97%, which indicates the stability of the 8000.00 83.2 Kg 72.21 Kg 95.55 Kg Fuel Burned (Kg) Fuel Burned 6000.00 1.75% 1.49% 2.02 % Algorithm algorithm’s computations. Similar to the explanations above, 4000.00 reducing the distance between the waypoints would improve 2000.00 the fuel consumption. The Flight Time in Figure 11 shows a 0.00 very little difference, and so can be neglected. 1 2 3 4 5 Flight Number Aircraft B FIGURE 8. AIRCRAFT A CALCULATED FUEL BURNED 8000.00 64.61 Kg 72.03 Kg 0.93% 1.01% 7000.00

6000.00 49.96 Kg 40.54 Kg 42.68 Kg 1.04% Aircraft A 5000.00 0.92% 0.97% 3.50 4000.00 0.37 % 0.52 % PTT 3.00 3000.00 Algorithm 2.50 Fuel Burned (Kg) 2000.00 1000.00 2.00 0.00 1.50 FlightSIM 1 2 3 4 5 Algorithm Flight Time (Hr) 1.00 0.02% 0.94% 0.44 % Flight Number 0.50 TABLE 5. AIRCRAFT B CALCULATED 0.00 FUEL BURNED 1 2 3 4 5

Flight Number Aircraft B FIGURE 9. AIRCRAFT A CALCULATED FLIGHT TIME 5.00 0.37 % 0.52 % 4.50 Figures 8 and 9 for Aircraft A serve to validate the results 4.00 obtained with the method explained in this paper. Figure 8 3.50 0.02% 0.94% 0.44 % 3.00 shows that, for all cases, the results from this method calculated 2.50 PTT the need for more fuel than FlightSIM. The differences are 2.00 mostly attributed to its procedure of not instantaneously Flight Time (Hr) 1.50 Algorithm reducing the fuel burned from the gross weight. This difference 1.00 can be seen most clearly in flights 4 and 5, which present 0.50 higher percentage differences than the first three flights. Both 0.00 1 2 3 4 5 flights 4 and 5 have longer cruise phases. For these tests, the Flight Number total weight was upgraded every 25 nm. Reducing the distances between weight upgrading would reduce the fuel burned TABLE 6. AIRCRAFT B CALCULATED FLIGHT TIME differences in Figure 8. It is difficult to verify if the algorithm is more accurate Figure 9 shows that the differences in flight time are than the PTT. To better analyze this, from Figure 8 it can be somewhat constant for all flights. The distances between seen that, for Aircraft A, the algorithm calculated a higher fuel waypoints during cruise are of little importance, since the speed burned than FlightSIM. On the other hand, Figure 10 shows

8 Copyright © 2014 by ASME that for Aircraft B, the PTT calculated more fuel burned than [3] IATA, "Vision 2050," International Air Transport with the method described in this paper. This suggests that the Association, Singapore2011. algorithm provides more accurate results than the PTT. [4] N. Salvat, A. Batailly, and M. Legrand, "Modeling of However, due to the lack of a dynamic model with which to Abradable Coating Removal in Aircraft Engines compare the results, it is not possible to guarantee that the Through Delay Differential Equations," Journal of algorithm provides more accurate computations. Engineering for Gas Turbines and Power, vol. 135, p. 102102. CONCLUSION [5] IATA, " Beginner’s Guide to Aviation Biofuels," In this paper, a method to calculate flight time and fuel International Air Transport Association, Geneva, consumption using a Performance Database was developed Swissetzrland2009. using the Lagrange Interpolations and considering [6] W. Freitag, Terry, Shulze E., "Blended Winglets particularities of each stage of flight such as the crossover Improve Performance," Aeromagazine Boeing, pp. 9- altitude, acceleration and “step climbs”. 12, 2009. [7] E. Theunissen, R. Rademaker, and T. Lambregts, The algorithm results were compared to a dynamic model "Navigation system autonomy and integration in developed in FlightSIM, and showed that the accuracy of the NextGen: Challenges and solutions," in Digital calculations is reliable. Their reliability was also confirmed by Avionics Systems Conference (DASC), 2011 comparing results to the calculation generated by Aircraft B’s IEEE/AIAA 30th, 2011, pp. 1-19. PTT. FMSs contain PDBs, from which they obtain their [8] A. Murrieta-Mendoza, R. Botez, and S. Ford, required parameters, and not only fuel burned and flight time. "Estimation of Fuel Consumption and Polluting This method may be directly implemented in an FMS (or used Emissions Generated during the Missed Approach in simulations) as a basis for trajectory calculation for Procedure," presented at the The 33nd IASTED optimization algorithms whenever the aircraft model is given in International Conference on Modelling, Identification, the form of a database. Using a PDB to calculate required and Control (MIC 2014) Innsbruck, Austria, 2014. parameters has the advantage of not needing to be concerned [9] R. Dancila, R. M. Botez, and S. Ford, "Fuel burn and about aerodynamic parameters such as lift and drag. emissions evaluation for a missed approach procedure performed by a B737-400," presented at the 2013 During testing, it was observed that the frequency of Aviation Technology, Integration, and Operations incorporating the fuel burned into the total weight during cruise Conference, Los Angeles, USA, 2013. has a direct effect on the accuracy of the calculations. As future [10] C. Celis, B. Moss, and P. Pilidis, "Emissions work it would be desirable to determine the best compromise modelling for the optimization of greener aircraft between the frequency of total weight upgrades to improve operations," in ASME Turbo Expo 2009: Power for accuracy and the need to keep computation time within Land, Sea, and Air, 2009, pp. 167-178. acceptable limits. [11] L. Stell, "Predictability of Top of Descent Location for Operational Idle-Thrust Descents," presented at ACKNOWLEDGMENTS the 10th AIAA Aviation Technology, Integration, and This research was conducted in the Research Laboratory in Operations (ATIO) Conference, Fort Worth USA, Active Controls, Avionics and Aeroservoelasticity (LARCASE) 2010. for the global project “Optimized Descent and Cruise”. Funds [12] T. Kwok-On, W. Anthony, and B. John, "Continuous for this project were provided by the business-led Network of Descent Approach Procedure Development for Noise Centers of Excellence Green Aviation Research & Abatement Tests at Louisville International Airport, Development Network (GARDN). We would like to thank Rex K Y, " i n AIAA's 3rd Annual Aviation Technology, Haygate, Dominique Labour and Yvan Blondeau from CMC- Integration, and Operations (ATIO) Forum, ed: Electronics – Esterline, and Oscar Carranza from LARCASE. 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