Probabilistic aircraft trajectory prediction in cruise flight considering ensemble wind forecasts

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Probabilistic aircraft trajectory prediction in cruise flight considering ensemble wind forecasts

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Antonio Franco, Damián Rivas ∗ , Alfonso Valenzuela Department of Aerospace Engineering, Universidad de Sevilla, 41092 Sevilla, Spain

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Keywords: Stochastic trajectory prediction Fuel consumption uncertainty Flight predictability Wind uncertainty Ensemble weather forecasting

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The problem of aircraft trajectory prediction subject to wind uncertainty is addressed. In particular, a probabilistic analysis of aircraft flight time and fuel consumption in cruise flight is presented. The wind uncertainty is obtained from ensemble weather forecasts. The cruise is composed of a given number of segments subject to uncertain winds (both along-track winds and crosswinds). The resulting average ground speed of each segment is modeled as a random variable, assuming a Log-Normal distribution. The probabilistic trajectory predictor developed is based on the Probabilistic Transformation Method; the input is the probability density functions of the average ground speeds of the cruise segments, and the output is the probability density functions of the flight time and the fuel consumption. Results are presented for several aircraft of different categories (medium and heavy), for a given trans-oceanic route and a real ensemble weather forecast. The effects of wind uncertainty on flight predictability and on fuel loading are analyzed. A fuel penalty parameter is defined, and the cost of flight unpredictability is quantified. The sample variability of all the results has been quantified by means of standard errors. © 2018 Elsevier Masson SAS. All rights reserved.

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Article history: Received 5 September 2017 Received in revised form 5 July 2018 Accepted 19 September 2018 Available online xxxx

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1. Introduction

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The future Air Traffic Management (ATM) system must address the performance challenges posed by today’s aviation: the capacity and the efficiency of the system must be increased while preserving or augmenting the safety levels. To accomplish these goals it is required a paradigm shift in operations through innovative technology and research. In this future system the trajectory becomes the fundamental element of a new set of operating procedures, collectively referred to as Trajectory-Based Operations (TBO), which aim at evolving from the current airspace-based ATM system to a trajectory-based system designed to accommodate airspace users’ requests to the maximum extent possible [1]. One key factor that affects those challenges is uncertainty, which is an inherent property of real-world socio-technical complex systems, and ATM is clearly not an exception. Uncertainty is critical from different perspectives in air transport: safety, environmental impact and cost. Researchers must accept the fact that uncertainty is unavoidable and must be dealt with, rather than ignored. If the capacity of the ATM system is to be increased while maintaining high safety standards and improving the overall performance, uncertainty levels must be reduced and new strategies

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Corresponding author. E-mail address: [email protected] (D. Rivas).

https://doi.org/10.1016/j.ast.2018.09.020 1270-9638/© 2018 Elsevier Masson SAS. All rights reserved.

to deal with the remaining uncertainty must be found. In particular, procedures to integrate uncertainty information into the ATM planning process must be developed. In Rivas and Vazquez [2] one can find a review of all the uncertainty sources that affect the ATM system. Among those, weather uncertainty has a major impact on aircraft trajectory prediction. The analysis of weather uncertainty has been addressed by many authors, using different methods, considering adverse weather and/or wind uncertainty. For instance, Nilim et al. [3] consider a trajectory-based ATM scenario to minimize delays under weather uncertainty, where the weather processes are modeled as stationary Markov chains; they develop a dynamic routing strategy for the en-route portion of the flight, when subject to adverse weather. Pepper et al. [4] present a method, based on Bayesian decision networks, for taking into account uncertain weather information in air traffic flow management (TFM); they identify as key to improving TFM the understanding of how to account for uncertainty in the demand and the capacity of the system, which depend critically on the weather. Grabbe et al. [5] address the problem of developing TFM controls that account for system uncertainties such as imperfect weather information, designing a sequential optimization method with strategic and tactical control loops; the impact of adverse weather is translated into changes in the sector-capacity constraints. Clarke et al. [6] stress the need for TFM algorithms that utilize available stochastic weather information; they develop a methodology to determine the stochastic capacity for a volume

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Nomenclature

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A, B constants of the problem in the aircraft mass equation a speed of sound CD, CL drag and lift coefficients C D 0 , C D 2 coefficients of the drag polar specific fuel consumption cT C f 1 , C f 2 , C f ,cr coefficients of the specific fuel consumption D aerodynamic drag E[.] expectation fy probability density function of random variable y g gravity acceleration; also, speed-time segment transformation gF time-fuel transformation h altitude L lift M Mach number m aircraft mass mF (total) fuel consumption n number of ensemble members P uncertainty penalty p pressure altitude; also, number of cruise segments

r rf S S [.] T t, t f V V g, V g wT , w X W t

ε[.] μ, σ

Subindices f i j k

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study is focused on the cruise phase of a given route, composed of several cruise segments, and considers the wind uncertainty provided by the prediction technique known as Ensemble Weather Forecasting, which has proved to be an effective way to quantify weather prediction uncertainties. This study is relevant because wind is an important source of uncertainty in trajectory prediction, and because cruise uncertainties have a large impact on the overall flight since the cruise phase is the largest portion of the flight (at least for long-haul routes). The main objective is to understand how wind uncertainty affects flight predictability and fuel loading. A conceptual vision of the integration of ensemble-based, probabilistic weather information with ATM decision support tools, focused on convective storms, is presented in Steiner et al. [14]. The importance of weather uncertainty information in probabilistic TFM is shown in Steiner et al. [15], where the translation of ensemble forecasts into probabilistic air traffic capacity impact is described. These papers clearly show the importance of making use of ensemble forecasts to generate probabilistic weather information for aviation needs. Sloughter et al. [16] (and references therein) address the problem of the statistical postprocessing of the ensemble forecasts, considering issues like calibration, multimodality and underdispersion (which are not considered in this paper). In this work, a probabilistic trajectory predictor is presented which propagates the wind uncertainty along the aircraft trajectory. The method used for the uncertainty propagation is the Probabilistic Transformation Method (see Kadry [17] and Kadry and Smaily [18]) which is a non-parametric method according to the classification of Halder and Bhattacharya [19]. This method is used in Vazquez and Rivas [20] to study the propagation of uncertainty in the initial aircraft mass, and in Vazquez et al. [21] to study the fuel consumption in aircraft cruise subject to along-track wind uncertainty. The probabilistic trajectory predictor developed for a multi-segment cruise is built up by considering a sequential application of the method: first, the probability density function (PDF) of the ground speed in each cruise segment is evolved to give the PDF of the flight time in each segment; then, all of these PDFs are convolved to give the PDF of the total flight time; finally, the total flight time is evolved to give the PDF of the aircraft fuel consumption. Thus, the uncertainties both in flight time and in fuel

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of en-route airspace, based on probabilistic forecasts of adverse weather, and formulate a dynamic stochastic optimization algorithm for TFM, providing guidance for routing aircraft. The importance of modeling wind uncertainties has been recognized by the ATM community for a long time. In the early work of Menga and Erzberger [7] they develop a statistical model of the altitude-dependent mean wind profile for descent guidance by fitting a Markov process to historical wind statistics. More recently, Zheng and Zhao [8] develop a statistical model of wind forecast uncertainties for stochastic trajectory synthesis of straight, level flight trajectories, expressing the wind field as the sum of a nominal wind (predicted) plus the wind uncertainties, which are modeled from historical wind forecast errors. Kim et al. [9] consider different sources of uncertainty (wind and severe weather among them) and derive service time distributions for different flight phases, assessing traffic flow efficiency by means of queuing network models; they assume that the mean value of the wind speed is piecewise constant over the flight path, and, in each spatially discretized flight segment, they describe the wind field as the sum of the mean wind speed plus the wind speed uncertainty, which is assumed to be a white Gaussian noise; crosswinds are not considered in this analysis. Cheung et al. [10] analyze the sensitivity of flight duration to wind uncertainties defined by the ensemble weather forecast provided by the UK meteorological office; they consider a fixed route and a cruise flight at constant flight level and constant Mach number. Considering the use of ensemble weather forecasting, Gonzalez-Arribas et al. [11] perform an analysis of wind-optimal cruise trajectories using pseudospectral methods; Franco et al. [12] address the problem of optimal aircraft path planning considering wind uncertainty, using a stochastic approach based on Dijkstra’s algorithm; and Valenzuela et al. [13] present a probabilistic analysis of sector demand considering both wind and temperature uncertainties. From these works it is clear that wind forecast uncertainties must be taken into account in trajectory prediction and trajectory planning, because they directly affect the accuracy and reliability of planned and/or predicted trajectories; they manifest in dispersions of flight time and fuel consumption, hence negatively affecting flight predictability. In this paper a probabilistic approach to aircraft trajectory prediction taking into account wind uncertainty is presented. The

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final initial cruise segment running index ensemble member running index

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flown distance range wing surface area standard deviation thrust time, (total) flight time aircraft true airspeed aircraft ground speed, average ground speed along-track and cross-track wind speeds aircraft weight segment flight time standard error due to the sample variability parameters of the Log-Normal distribution

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Fig. 1. Ensemble approach. Legend: m – member, w – weather, x – trajectory.

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Fig. 2. Transformation approach. Legend: m – member, w – weather.

proaches (Arribas et al. [27]; Lu et al. [28]). Thus, the EPS generated by this technique is a representative sample of the possible (deterministic) realizations of the potential weather outcome [14]. An ensemble forecast is a collection of typically 10 to 50 weather forecasts (referred to as members). Cheung et al. [29] review various EPS: PEARP (form Meteo France), consisting of 35 members; MOGREPS (form the UK Met Office), with 12 members; the one provided by the European Centre for Medium-Range Weather Forcasts (ECMWF), with 51 members; and a multi-model ensemble (SUPER) constructed by combining the previous three forming a 98-member ensemble, designed so that it is more likely to capture outliers and give a higher degree of confidence in predicting future atmospheric evolution. Some examples of EPS from the US are MEPS (form the Air Force Weather Agency) with 10 members, and SREF (form the National Centers for Environmental Prediction) comprised of 21 members. Ensemble forecasting has proved to be an effective way to quantify weather prediction uncertainty. The uncertainty information is in the spread given by the members of the ensemble, and the hope is that this spread bracket the true weather outcome [14]. It is important to notice that for strategic planning the analysis of all the individual ensemble members must be included (rather than an ensemble mean) [15]. Two approaches are commonly used for trajectory prediction subject to uncertainty provided by EPS, which are described next: 1) Ensemble approach (see Fig. 1). In this case, for each member of the ensemble, a deterministic trajectory predictor (TP) is used, leading to an ensemble of trajectories from which probability distributions can be derived. This approach is used in [10,29]. 2) Transformation approach (see Fig. 2). Now, probability distributions of meteorological parameters of interest (such as wind) are evolved along the aircraft trajectory using a probabilistic trajectory predictor (pTP), leading to probability distributions of trajectory parameters of interest (such as flight time and fuel consumption). The pTP defined in this paper for multi-segment trajectories follows this probabilistic approach (as described in Section 4). The required input from the EPS to the trajectory predictors will depend on the ATM problem under consideration. In this work the trajectory uncertainty caused by wind uncertainty is studied,

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consumption are determined. A preliminary version of this approach for a one-segment cruise is presented in Rivas et al. [22]. The study of flight time uncertainty is relevant because the dispersion in flight time is a measure of the flight predictability, which is identified by ICAO as a key performance metric [23], and, therefore, its improvement is a clear performance objective for airlines and providers of air navigation services. On the other hand, the study of fuel consumption uncertainty is relevant as well because it can help in the determination of the contingency fuel, hence, allowing a more effective decision making. In this respect, Hao et al. [24] analyze the cost of carrying the additional discretionary fuel excessively loaded, above a reasonable and conservative buffer, to face flight unpredictability; and Ryerson et al. [25] stress the possibility of achieving substantial savings through the use of a reformed policy for discretionary fuel loading. A major contribution of this paper is a probabilistic study of the cost of flight unpredictability. Results are presented for a given trans-oceanic route, defined by several cruise segments, subject to uncertain winds (both alongtrack winds and crosswinds) defined by a real ensemble forecast. To take into account that the aircraft has to fly through areas of different look-ahead time, several forecasts are considered, released at the same time with time steps of 6 hours, using for each cruise segment the most appropriate forecast available at the prediction time. The probabilistic approach presented in this paper is applied to different aircraft, with different weights, including medium and heavy, considering the same route and the same ensemble forecast; the effects of wind uncertainty on flight predictability and on fuel loading are analyzed. A fuel penalty parameter is defined and analyzed, and the cost of flight unpredictability is quantified for given aircraft, route and wind forecast. A sample variability analysis of the input and output random variables analyzed in the problem is performed; in particular, standard errors are provided. The main improvements of this paper with respect to the previous work [21,22] are the following: multi-segment cruise (rather than one-segment), real forecasted winds (instead of synthetic wind distributions), several wind forecasts (rather than a fixed wind distribution for the whole cruise), Log-Normal distributions for the input variables (instead of uniform distributions), sample variability analysis of the results, and multi-aircraft application. In comparison, these improvements make this work more application oriented and more realistic. The outline of the paper is as follows: first, in Section 2 the problem of aircraft trajectory prediction considering ensemble weather uncertainty is reviewed; in Section 3 the calculation of flight time and fuel consumption is addressed; the probabilistic trajectory predictor is described in Section 4, and its input is defined in Section 5; the particular application considered is defined in Section 6; the results are discussed in Section 7; and finally some conclusions are drawn in Section 8.

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2. Trajectory prediction considering ensemble weather uncertainty

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To model weather for strategic planning horizons, a probabilistic approach is the appropriate one, so that the inherent weather uncertainty can be taken into account. The use of probabilistic forecasts is currently recomended by the American Meteorological Society [26]. Today’s trend is to use the Ensemble Prediction Systems (EPS), which attempt to characterize and quantify the inherent prediction uncertainty based on ensemble modeling. Ensemble forecasting is a prediction technique that consists in running an ensemble of weather forecasts by slightly altering the initial conditions and/or the parameters that model the atmospheric physical processes, and/or by considering time-lagged or multi-model ap-

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dr

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Fig. 3. Sketch of a multi-segment cruise (the wind and the initial aircraft mass are uncertain).

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L= γ where γa = 1.4 is the ratio of specific heats for air, p is the pressure altitude, S the wing surface area, C D is the drag coefficient, and C L is the lift coefficient. The aircraft models considered in this work are defined by Eurocontrol’s BADA database [31]: parabolic drag polar with constant coefficients C D = C D 0 + C D 2 C L2 , and specific fuel consumption linear on true airspeed   V [kt ] c T = C f ,cr C f 1 1 + (note that because V is constant, c T is C f2 constant as well). From the kinematic equation in (3) one obtains the segment flight time (t ) j

(r f ) j

(t ) j =

therefore, { w 1 , ... w n } in Figs. 1 and 2 represent the wind fields defined by each ensemble member, n being the number of ensemble members.

Equation (4) defines the transformation (t ) j = g j ( V g j ) relating segment flight time and ground speed. Hence, the total flight time t f is simply given by

3. Flight time and fuel consumption in cruise flight

p p   (r f ) j tf = (t ) j =

V gj

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As already indicated, in this paper the flight time and the fuel consumption in cruise flight are studied. The cruise is supposed to be formed by p cruise segments, each one of them defined by a constant course, and flown at constant Mach number and constant pressure altitude, as usually required by Air Traffic Control (ATC). The Earth is assumed to be spherical, and the atmosphere is supposed to be defined by the International Standard Atmosphere (ISA) model plus the winds given by the EPS. The true airspeed is also constant, defined by V = Ma, where a is the ISA speed of sound at the given flight pressure altitude. Sketches of a multisegment cruise and a generic cruise segment are given in Figs. 3 and 4, respectively (note that turns are assumed to be instantaneous, which is an acceptable simplifying hypothesis in cruise flight because the differences in length or time caused by not considering them are negligible). In cruise segment j, the flight is subject to an along-track wind w T j (r ), and a crosswind w X j (r ), which vary along the cruise (r is the distance flown by the aircraft). The effects of the crosswinds are analyzed by taking them into account in the kinematic equations, ignoring the lateral dynamics, and translating the crosswind into an equivalent headwind. This leads to a reduced ground speed, which for cruise segment j is given by

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V g j (r ) =



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V 2 − w 2X (r ) + w T j (r ) j

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(r f ) j

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Because w T j and w X j are uncertain, V g j is uncertain as well. In the analysis presented in this work, in each cruise segment the aircraft is supposed to have an average ground speed defined as

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where (r f ) j is the known range of cruise segment j. The equations of motion for cruise flight, for segment j, are (see [30])

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(6)

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where the positive constants A and B are given by

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Now, from the dynamic equations in (3) the following equation is obtained, which describes the evolution of the aircraft mass,

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(4)

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γa pM 2 S

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In this paper, the range of each cruise segment (r f ) j and the final aircraft mass (m f ) p = m f are given. Fixing m f (instead of the initial aircraft mass) is consistent with having a fixed landing weight. It also allows for a fair comparison for different values of the wind, which lead to different fuel loads and therefore to different values of the initial aircraft mass. Therefore, the whole cruise flight has to be computed backwards, starting from the last segment ( j = p) and ending at the first one ( j = 1). For each segment, Eq. (6) is to be solved backwards, from t = (t ) j to t = 0, with the boundary condition

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m((t ) j ) = (m f ) j

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Fig. 4. Sketch of a generic cruise segment (the ground speed and the initial and final aircraft masses are uncertain).

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(3)

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= −c T T

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where t is the time, T , D , L are the thrust, the aerodynamic drag and the lift, m is the aircraft mass, g = 9.80665 m/s2 is the acceleration of gravity, and c T is the specific fuel consumption. The drag and the lift can be written as D = 12 γa pM 2 SC D and

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= V gj,

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where (m f ) j = (mi ) j +1 , j < p, as dictated by mass continuity. This problem (Eqs. (6)–(8)) has the following solution for the segment initial aircraft mass (mi ) j



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Adding the solutions for all cruise segments, one can easily obtain the initial aircraft mass (mi )1 = mi . Then, the cruise fuel consumption follows from m F = mi − m f , namely



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Equation (10) defines the transformation m F = g F (t f ), relating cruise flight time and fuel consumption.

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4. Probabilistic trajectory predictor

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f z ( z) =

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expression that is valid only if the function g ( y ) is invertible on the domain of y. Moreover, the pTP needed in this work for multi-segment trajectories relies also on the well-known result in statistics that the PDF of the sum of independent variables is the convolution of the PDFs of the addends (see Grinstead [33]), that is

θ = x + y ⇒ f θ (θ) = f x ∗ f y

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where

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∞ fx ∗ f y =

∞ f x (x) f y (θ − x)dx =

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The pTP developed has two steps:

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Step 1: Computation of the PDF of the cruise flight time The diagram in Fig. 5 shows the procedure devised in this paper to obtain the PDF of the cruise flight time, namely f t f (t f ). The procedure is as follows: First, for each cruise segment, the ground speed is transformed into the flight time according to the transformation (t ) j = g j ( V g j ) defined by Eq. (4). Let f V g ( V g j ) be the PDF of the ground j

speed of cruise segment j (to be defined in Section 5); then, the PDF of the corresponding flight time is obtained by applying Eq. (11), namely

(r f ) j

(r f ) j f (t ) j ((t ) j ) = f ( ) (t )2j V g j (t ) j

(14)

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(15)

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which is computed from the following set of convolutions

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f θ1 (θ1 ) = f (t )1 ((t )1 ) ∗ f (t )2 ((t )2 )

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f (t )1 ((t )1 ) f (t )2 (θ1 − (t )1 )d(t )1

=

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f t f (t f ) = f (t )1 ∗ f (t )2 ∗ ... ∗ f (t ) p

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−∞

(11)

| g  ( g −1 ( z))|

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In the problem analyzed in this paper, in each cruise segment (except in the last one, which is integrated first) there are two uncertain variables: the ground speed and the final aircraft mass (see Fig. 4), therefore the probabilistic approach presented in [22] for a one-segment cruise, with only one uncertain variable, cannot be applied now. A different probabilistic approach has been developed, which is described in this section. As already indicated, the pTP is based on the Probabilistic Transformation Method (PTM). The basis of this method is the following theorem (see Canavos [32]): Given a random variable y with PDF f y ( y ), if one defines another random variable z using a transformation g such that z = g ( y ), then the PDF of z is given by

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Next, the PDF of the cruise flight time is obtained from the PDFs of the flight times of all the cruise segments. For this purpose, the ground speeds of the cruise segments (and, therefore, also the flight times) are considered to be independent of one another. Assuming the independence of the flight times, Eq. (12) can be applied, so that the PDF of the flight time follows from

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Fig. 5. Computation of the PDF of the cruise flight time. Trajectory with p segments.

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∞

f θ2 (θ2 ) = f θ1 (θ1 ) ∗ f (t )3 ((t )3 ) =

f θ1 (θ1 ) f (t )3 (θ2 − θ1 )dθ1

−∞

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.. .

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f t f (t f ) = f θ p−2 (θ p −2 ) ∗ f (t ) p ((t ) p )

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∞ =

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f θ p−2 (θ p −2 ) f (t ) p (t f − θ p −2 )dθ p −2

(16)

−∞

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Step 2: Computation of the PDF of the fuel consumption When the PDF of t f has already been computed, a transformation is applied to obtain the PDF of the fuel consumption (as outlined in Fig. 6); in particular, from Eqs. (10) and (11) it follows

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A + B (m f + m F )2



 arctan

B A

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1 ft f (g− F (m F ))

1 where the inverse transformation t f = g − F (m F ) is easily obtained from Eq. (10), and is given by

1 tf = √ AB

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where θ1 , ..., θ p −2 are intermediate variables.

f m F (m F ) =

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 (m f + m F ) − arctan



B A



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mf

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Therefore, the average ground speed of that segment has the following PDF

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⎧  ⎫ ⎨ log( V g / V ) − μ j 2 ⎬ 1 j ,i , f V g (V g j ) = exp − √ j ⎩ ⎭ 2σ j2 V g j σ j 2π

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V gj > 0

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Fig. 6. Computation of the PDF of the fuel consumption.

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E [t f ] =

⎡

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S [t f ] = ⎣

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2 (t f )dt f − E [t f ] ⎦

(20)

and analogously for the mean and the standard deviation of the fuel consumption.

{ V g j,1 , ..., V g j,n } be the average ground speeds of segment j given

by the n ensemble members. Now one must assume that they follow a particular distribution. This is not a minor point, and in fact is one of the open challenges in this problem; this task is heavily constrained by the fact that the ensemble has few samples, and because all the samples, even the outliers, are important and need to be taken into account. In this paper, the average ground speed of each segment nondimensionalized with the true airspeed is assumed to be distributed as a Log-Normal continuous variable, i.e., V g j / V ∼ log N (μ j , σ j ). The Log-Normal distribution is a common alternative in engineering to the Normal model (see Bury [34]) because, on one hand, its sample space is positive and, on the other hand, it shows a great shape flexibility. In this work, the Log-Normal model has been shown to fit the sample very well. In particular, for each flight segment, an Anderson–Darling goodness-of-fit test (see Anderson and Darling [35]) has been performed to the logarithm of the nondimensionalized ground speed with a significance level α = 0.05; in all cases, the results support the null hypothesis that the ground speed can be properly represented by a Log-Normal distribution. For cruise segment j, the defining parameters μ j and σ j are estimated from the sample by using the method of maximum likelihood (considering the unbiased version of σ j ), which gives n

μj =

n

log( V g j,i / V )

(21)

i =1

  n  1   2 σj =  log( V g j,i / V ) − μ j n−1 i =1

μ j + σ j2 /2

ε[σ j ] = √

(22)

76 77 78







exp

(24)



σ j2 − 1

n

In this section the input to the pTP is defined (see Fig. 5), that is, the probabilistic ground speeds. In the following, the approach to obtain the PDF of the ground speed of each cruise segment is described. The first step is to determine, for each member k of the ensemble and each segment j, the average ground speed V g j,k , computed from Eq. (2) using the winds w T j,k (r ) and w X j,k (r ). Let

1



μj + σ

2 j /2

σj ε[μ j ] = √

5. Probabilistic ground speed

60 61





σj

2(n − 1)

80 82 83 84 85 86 87 88 89 90 91

(26)

92 93 94

(27)

95 96

Any function of these parameters, generically denoted as ξ(μ1 , ..., μ p , σ1 , ..., σ p ), is also affected by the variability of the sample. The standard error of ξ(μ1 , ..., μ p , σ1 , ..., σ p ) can be approximated by the so-called error propagation formula (see [34]), taking into account that the estimators are uncorrelated to each other, which leads to



  2  2   p ∂ξ ∂ξ ε[ξ ] =  ε[μ j ] + ε[σ j ] ∂μ j ∂σ j

79 81

(25)

Due to the inherent randomness in the sample, any parameter estimate is subject to variability. This variability can be quantified by the square root of the sampling variance of the parameter estimator, which is commonly known as the standard error (see [34]). In particular, standard errors of the defining parameters μ j and σ j are given by

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E [ V g j ] = V exp

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26

t 2f f t f

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∞

⎤1/2

−∞

23 24

(19)

−∞

19 21

t f f t f (t f )dt f

74 75

S [ V g j ] = V exp

∞

73

RO

12

Notice that this analysis is valid because, as required by the PTM, all the transformation functions are invertible on their respective domains. Once the PDF of t f is known, one can compute the mean and the standard deviation, as follows

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Furthermore, the mean and the standard deviation of V g j are given as follows

DP

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68 70

(23)

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(28)

j =1

6. Application

In this section, the application considered in the paper is defined. Results are presented afterwards in Section 7. 6.1. Route

A cruise flight Washington–Rome flown at pressure altitude 200 hPa is considered (this corresponds to ISA altitude h = 11784 m, which is in the stratosphere, where the ISA speed of sound is a = 295.07 m/s). The route is composed of 9 segments, defined by the waypoints given in Table 1. The distances to be flown for each segment are given in Table 2 (notice that these distances are obtained from the ground distances between waypoints, multiplying them by the factor ( R E + h)/ R E , where R E = 6371.009 km is the mean Earth radius). The total range is r f = 7259.926 km. Results are presented in Section 7 also for the aircraft traveling the route Rome–Washington, with the same departure time and defined by the same waypoints (note that now for the westbound trajectory the cruise starts with segment 9 and ends with segment 1, as defined in Tables 1 and 2). Thus, one can analyze the difference in trajectory uncertainty between the cases of being in the presence of predominant tailwinds and headwinds.

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Table 1 Waypoints coordinates (eastbound numbering).

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Waypoint

1

2

Latitude Longitude

38°57 N 77°27 W

43°N 70°W

3

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46°N 60°W

5

48°N 50°W

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49°N 40°W

7

49°N 30°W

8

49°N 20°W

48°N 10°W

9

10

69

46°N 0°

41°48 N 12°15 E

70 71 72

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73

7

Table 2 Range for each cruise segment (eastbound numbering).

8 9

74 75

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Segment

1

2

3

4

5

6

7

8

9

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(r f ) j , km

771.863

861.745

791.624

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730.855

730.855

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Table 3 Aircraft data from BADA (ordered in increasing values of m f ). Code

31

A01 A02 A03 A04 A05 A06 A07 A08 A09 A10 A11 A12 A13 A14

32 33 34 35 36 37 38 39 40 41 42 43 44

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

133800 149700 154600 168000 173000 178000 183400 195000 209100 224500 232000 237900 247700 386000

S, m2

M 0.80 0.80 0.84 0.82 0.82 0.80 0.84 0.84 0.84 0.84 0.83 0.83 0.84 0.85

283.35 283.5 360.46 361.6 361.6 361.6 360.46 427.82 427.82 428.04 437.0 437.0 511.23 845.0

C D0 0.021112 0.017439 0.021782 0.018953 0.019805 0.019826 0.021871 0.018452 0.023205 0.016943 0.022634 0.022140 0.019945 0.018130

C D2

C f1 ,

0.042118 0.048227 0.036869 0.032965 0.031875 0.033947 0.034141 0.044604 0.041138 0.048939 0.036883 0.038022 0.049033 0.043198

The EPS chosen is PEARP, from Meteo France. Winds have been retrieved from the ECMWF database, corresponding to 5 May 2016, and for pressure altitude 200 hPa. To take into account that the aircraft has to fly through areas of different look-ahead time, several forecasts are considered, released at the same time with time steps of 6 hours. In this application, the following case is considered: the forecasts are released at 6:00, the flight departure time is 11:00, and the prediction is made 3 hours before departure (at 8:00). See sketch in Fig. 7. One can see that the forecasts closer in time to the flight are those for 6 and 12 hours (F 2 and F 3 in Fig. 7; F 2 forecasted for 12:00 and F 3 forecasted for 18:00). Hence, the first cruise segments flown are affected by F 2 and the last segments by F 3 . To decide which forecast affects each segment, the flight time is preliminary computed for the winds given by the control member of the ensemble (in a deterministic way using Eq. (4)). Especifically, one obtains the flight times at the beginning of all segments: the initial time of the first segment is the known departure time,

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6.2. Forecast

m f , kg

kg

min kN

TE

30

0.74220 0.62911 0.54198 0.59426 0.61503 0.61769 0.54660 0.54864 0.53638 0.49160 0.63894 0.64295 0.60040 0.54336

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29

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28

46

81 82 83 84 85 86 87 88 89 90 91

Fig. 7. Forecasting time scheme.

26

45

80

RO

18

79

C f 2 , kt

2060.5 851.87 1151.9 825.84 919.03 836.49 1198.1 817.5 861.36 487.93 1516.0 1468.9 880.35 866.22

92 93 94 95

kg

96

C f ,cr

cT ,

0.90048 0.91891 0.89710 0.95422 0.93655 0.95523 0.91377 0.93687 0.85686 0.88658 0.95095 0.95843 0.92241 0.93051

1.3620·10−5

97

1.1493·10−5 1.4834·10−5 1.4514·10−5 1.5229·10−5 1.1672·10−5 1.3616·10−5 1.1945·10−5 1.4437·10−5 1.3307·10−5 1.3599·10−5 1.4282·10−5 1.3170·10−5

99

sN

1.4825·10−5

98 100 101 102 103 104 105 106 107 108 109 110

and the initial time of any other segment is the sum of the computed flight times of the previous segments (note that the control member of F 2 is used for the segments with initial time closer to 12:00, and the control member of F 3 for the others). If for a given segment this initial time is closer to 12:00 than to 18:00, then F 2 is chosen, and if it is closer to 18:00, then F 3 is chosen. This is a simple approach that uses for each segment the most appropriate forecast available at the prediction time.

111 112 113 114 115 116 117 118 119

6.3. Aircraft

120 121 122

The probabilistic approach presented in this paper is going to be applied to 14 different aircraft (coded A01 to A14), with different weights, including medium and heavy, considering the same route and the same EPS. The parameters that define the aircraft and the flight Mach number are obtained from BADA; they are given in Table 3. Note that the final aircraft mass m f is defined as the sum of the minimum mass plus the maximum payload mass (both given by BADA). From these BADA data the specific fuel consumption c T is computed (for p = 200 hPa), which is also listed in Table 3.

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Segment Initial time Forecast

Eastbound route 1 2 11:00 11:47 F2 F2

Segment Initial time Forecast

Westbound route 9 8 11:00 12:22 F2 F2

47 48 49 50 51 53 54 55 56 57 58 59 60 61

7. Results

7 13:19 F2

4 13:24 F2

6 14:15 F2

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3 12:37 F2

First, the average ground speed, which is the input to the probabilistic approach, is presented in Section 7.1 for one particular case, namely M = 0.80. Then, results for the flight time and the fuel consumption are described in Section 7.2, considering all the aircraft listed in Table 3. Finally, in Section 7.3, a fuel penalty parameter is defined and analyzed, and the cost of flight unpredictability is quantified for all the aircraft considered.

62 63

7.1. Ground speed

64 65 66

From raw wind data, 315 values of average ground speed are obtained from Eq. (1) for the 9 segments and the 35 ensemble

5 14:09 F2

84

89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

Table 4 Deterministic initial times for each cruise segment (M = 0.80).

42

83

88

Fig. 8. Ground speeds PDFs (eastbound route). Bars: relative frequency histograms. Curves: Log-Normal distributions for the pTP approach. M = 0.80. (Gaps between bars are introduced only for aesthetical purposes.)

CO RR

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82

87

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80

85

DP

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108 109

6 14:52 F2

7 15:39 F3

8 16:29 F3

9 17:24 F3

110 111 112 113

5 15:13 F3

4 16:16 F3

3 17:20 F3

2 18:31 F3

1 19:51 F3

members. These values are not listed here, but they are represented for the different cruise segments in Fig. 8 in the form of relative frequency histograms, along with the corresponding PDFs (assuming Log-Normal distributions), for the eastbound route. Note that the forecast F 2 is used for the first 6 segments flown and the forecast F 3 for the other 3 (according to the initial times computed at the beginnings of the segments using the control member of the ensemble, which are given in Table 4). In Table 5, the mean and the standard deviation are listed for all segments, including their corresponding standard errors. Since the same scale for the abscissa is selected in all subfigures, it is easy to see that the spread of the average ground speed is different for the different segments, with the standard deviation ranging from 0.665 m/s for segment 6 to 3.370 m/s for segment 3 (as shown in Table 5).

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1 2 3

Segment

7 8 9 10 11 12

1 2 3 4 5 6 7 8 9

Eastbound route

Segment

E [ V g ], m/s

S [ V g ], m/s

274.92 ± 0.24 285.09 ± 0.41 281.65 ± 0.57 281.42 ± 0.24 279.73 ± 0.15 260.08 ± 0.11 249.03 ± 0.17 241.84 ± 0.21 251.82 ± 0.23

1.419 ± 0.172 2.443 ± 0.296 3.370 ± 0.409 1.416 ± 0.172 0.889 ± 0.108 0.665 ± 0.081 1.017 ± 0.123 1.228 ± 0.149 1.347 ± 0.163

9 8 7 6 5 4 3 2 1

13 14 15 16 17 19 20 21 22 23 25 26 27 28 29 30 32 33 34 35 37 38 39 40

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Ground speed distributions have also been obtained for the westbound route; they are represented for the different cruise segments in Fig. 9. In this case the forecast F 2 is used for the first 4 segments flown and the forecast F 3 for the other 5 (according to the initial times given in Table 4). Again, the mean and the standard deviation are listed in Table 5 for all segments, including their corresponding standard errors. Now the standard deviation ranges from 0.414 m/s for segment 9 to 4.170 m/s for segment 2. Comparing both routes, it is clear that the average ground speeds are larger for the eastbound cruise which is subject to predominant tailwinds.

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

Fig. 9. Ground speeds PDFs (westbound route). Bars: relative frequency histograms. Curves: Log-Normal distributions for the pTP approach. M = 0.80. (Gaps between bars are introduced only for aesthetical purposes.)

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0.414 ± 0.050 0.879 ± 0.107 0.570 ± 0.069 0.682 ± 0.083 2.751 ± 0.334 2.969 ± 0.360 2.788 ± 0.338 4.170 ± 0.506 2.645 ± 0.321

CO RR

41

46

221.59 ± 0.07 229.87 ± 0.15 221.40 ± 0.10 211.72 ± 0.12 193.21 ± 0.47 188.95 ± 0.50 183.86 ± 0.47 181.47 ± 0.70 196.77 ± 0.45

EC

36

45

S [ V g ], m/s

TE

31

44

E [ V g ], m/s

DP

24

43

69

RO

18

42

68

Westbound route

OF

6

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Table 5 Mean and standard deviation of average ground speeds (M = 0.80).

4 5

9

119 120 121

7.2. Analysis of flight time and fuel consumption uncertainties

122 123

In this section the PDFs of the flight time and the fuel consumption are analyzed for all the aircraft listed in Table 3. Once the input to the pTP is defined, the PDF of the flight time f t f (t f ) is obtained as described in Section 4 (and outlined in Fig. 5). The mean and the standard deviation of t f can be expressed in analytic form (even though the PDF cannot). Indeed, according to the Log-Normal’s reproductive property, the flight time of segment j, namely (t ) j , is also Log-Normal, with parameters

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A01 A02 A03 A04 A05 A06 A07 A08 A09 A10 A11 A12 A13 A14

23 24 25 26 27 28 29 30 31 32 33 34

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

S [t f ], s

455.17 ± 0.16 455.17 ± 0.16 435.72 ± 0.15 445.23 ± 0.15 445.23 ± 0.15 455.17 ± 0.16 435.72 ± 0.15 435.72 ± 0.15 435.72 ± 0.15 435.72 ± 0.15 440.43 ± 0.15 440.43 ± 0.15 435.72 ± 0.15 431.12 ± 0.15

57.2 ± 3.1 57.2 ± 3.1 52.5 ± 2.9 54.8 ± 3.0 54.8 ± 3.0 57.2 ± 3.1 52.5 ± 2.9 52.5 ± 2.9 52.5 ± 2.9 52.5 ± 2.9 53.7 ± 3.0 53.7 ± 3.0 52.5 ± 2.9 51.5 ± 2.8

E [t f ]

· 10

2.09± 0.12 2.09± 0.12 2.01± 0.11 2.05± 0.11 2.05± 0.11 2.09± 0.12 2.01± 0.11 2.01± 0.11 2.01± 0.11 2.01± 0.11 2.03± 0.11 2.03± 0.11 2.01± 0.11 1.99± 0.11



μt , j = log (r f ) j / V − μ j and σt , j = σ j ; therefore, as (t ) j are independent random variables, the mean and the standard deviation of the flight time are given by E [t f ] =

p  (r f ) j j =1

V



exp −μ j + σ j2 /2

(29)

  p # $



  (r f ) j 2 exp −2μ j + σ j2 exp σ j2 − 1 (30) S [t f ] =  j =1

V

E [t f ], min

3

598.26 ± 0.46 598.26 ± 0.46 565.41 ± 0.38 581.12 ± 0.43 581.12 ± 0.43 598.26 ± 0.46 565.41 ± 0.38 565.41 ± 0.38 565.41 ± 0.38 565.41 ± 0.38 572.91 ± 0.41 572.91 ± 0.41 565.41 ± 0.38 557.65 ± 0.37

82 83 84

The PDFs of the flight time for both east and westbound cruises and for each aircraft are shown in Fig. 10. Values of the means and the standard deviations in both cases are presented in Table 6, along with values of the quotient S [t f ]/ E [t f ]; this metric is a relative measure of uncertainty, and is termed the coefficient of variation (see [34]). The standard errors of these results are also provided in Table 6, which have been obtained by applying the error propagation formula in Eq. (28). Note that because V g j is the same for all aircraft flying at the same Mach number, they share the same PDF and, consequently, the same values of the mean and the standard deviation. As expected, for higher values of the Mach number the mean values of the total flight time are smaller. Moreover, the higher the flight time, the higher the standard deviation of the flight time, and the higher the coefficient of variation. For all the aircraft considered, the standard error of E [t f ] is shown to be very small as compared to E [t f ], whereas the relative standard errors of S [t f ] and S [t f ]/ E [t f ] are shown to be much larger. The PDFs of the fuel consumption f m F (m F ) are computed according to Eq. (17); these PDFs are shown in Fig. 11 for both eastbound and westbound cruises and for each aircraft. Values of the

S [t f ], s

161.8 ± 10.4 161.8 ± 10.4 135.4 ± 9.4 151.8 ± 9.7 151.8 ± 9.7 161.8 ± 10.4 135.4 ± 9.4 135.4 ± 9.4 135.4 ± 9.4 135.4 ± 9.4 147.2 ± 9.4 147.2 ± 9.4 135.4 ± 9.4 131.3 ± 9.1

85

S [t f ] E [t f ] 4.51 4.51 3.99 4.36 4.36 4.51 3.99 3.99 3.99 3.99 4.28 4.28 3.99 3.92

· 10

± ± ± ± ± ± ± ± ± ± ± ± ± ±

3

0.29 0.29 0.28 0.28 0.28 0.29 0.28 0.28 0.28 0.28 0.27 0.27 0.28 0.27

means, the standard deviations, and the coefficients of variation in both cases are presented in Table 7, along with their corresponding standard errors, which have also been obtained by applying the error propagation formula in Eq. (28). A general trend can be seen in Fig. 11 when comparing different aircraft: The higher the mean fuel consumption, the higher the standard deviation, which is clearly shown because the PDFs become shorter and wider. However, it can only be stated in a loose sense, as some pair of aircraft, with very close values of the mean, do not follow this rule (see also Table 7). What is clear is that for very different aircraft, the heavier they are, the higher the average fuel consumption and the higher the standard deviation; compare, for instance, the values for A03, A08 and A12 in Table 7. In contrast, the relative measure S [m F ]/ E [m F ] remains quite similar for all aircraft. The standard errors of the mean, the standard deviation, and the coefficient of variation of the fuel consumption show the same trends as for the flight time. These results also show that, for a given aircraft, for the westbound cruise one has larger values of the means (as expected, because in this case one has predominant headwinds), and also larger values of the standard deviations. For instance, for A01, S [t f ] increases from 57.2 s for the east route to 161.8 s for the west route (see Table 6), and S [m F ] increases from 76.5 kg to 230.7 kg (see Table 7). These results imply that the trajectory uncertainty is larger in the presence of headwinds (for the same wind uncertainty), as it is also shown in [21]. This trend also holds for the relative uncertainties: S [t f ]/ E [t f ] increases from 2.09·10−3 to 4.51·10−3 , and S [m F ]/ E [m F ] increases from 2.29·10−3 to 5.11·10−3 for the same aircraft; one can see that the relative uncertainty roughly doubles.

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E [t f ], min

S [t f ]

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Westbound route

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Eastbound route

DP

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Table 6 Mean, standard deviation, and coefficient of variation of flight time.

17

36

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Fig. 10. Flight time PDFs for M = 0.80, 082, 0.83, 0.84 and 0.85.

15

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Table 7 Mean, standard deviation, and coefficient of variation of fuel consumption.

19

Eastbound route E [m F ], kg

21 22 23 24 25 26 27 28 29 30 31 32 33

A01 A02 A03 A04 A05 A06 A07 A08 A09 A10 A11 A12 A13 A14

33339 38988 32010 38488 39065 43391 35119 44803 46379 53457 53310 55488 64886 83006

± ± ± ± ± ± ± ± ± ± ± ± ± ±

13 16 11 14 14 17 13 16 17 20 20 21 24 30

S [m F ], kg 76.5 ± 4.2 92.7 ± 5.1 67.8 ± 3.7 84.7 ± 4.7 85.8 ± 4.7 99.0 ± 5.5 74.8 ± 4.1 97.6 ± 5.4 99.9 ± 5.5 119.6 ± 6.6 117.0 ± 6.5 122.6 ± 6.8 144.1 ± 8.0 177.5 ± 9.8

S [m F ] E [m F ] 2.29 2.38 2.12 2.20 2.20 2.28 2.13 2.18 2.15 2.24 2.20 2.21 2.22 2.14

· 103

± 0.13 ± 0.13 ± 0.12 ± 0.12 ± 0.12 ± 0.13 ± 0.12 ± 0.12 ± 0.12 ± 0.12 ± 0.12 ± 0.12 ± 0.12 ± 0.12

34 35

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

82 83 84 85

E [m F ], kg

S [m F ], kg

45192 ± 39 53545 ± 49 42220 ± 31 51397 ± 42 52134 ± 42 58698 ± 50 46400 ± 34 59636 ± 45 61514 ± 46 71810 ± 56 71105 ± 57 74168 ± 60 86943 ± 67 109806 ± 80

230.7 ± 14.8 287.2 ± 18.4 180.7 ± 12.5 246.3 ± 15.8 249.2 ± 15.9 297.4 ± 19.1 200.0 ± 13.9 265.0 ± 18.4 269.4 ± 18.7 331.0 ± 23.0 338.3 ± 21.6 356.0 ± 22.8 396.9 ± 27.6 474.6 ± 32.9

S [m F ] E [m F ] 5.11 5.36 4.28 4.79 4.78 5.07 4.31 4.44 4.38 4.61 4.76 4.80 4.56 4.32

86

· 103

± 0.33 ± 0.34 ± 0.30 ± 0.31 ± 0.31 ± 0.32 ± 0.30 ± 0.31 ± 0.30 ± 0.32 ± 0.30 ± 0.31 ± 0.32 ± 0.30

7.3. Cost of flight unpredictability

In this section a fuel penalty parameter is defined and analyzed. Results are presented for the different aircraft defined in Table 3, and for both routes (eastbound and westbound). The penalty P is defined as the extra fuel mass that has to be loaded, additional to the mean fuel mass, to account for 99% of the possible weather realizations, as sketched in Fig. 12; according to this definition, the likelihood that the fuel mass required be higher than the mean plus the penalty is 1%. A relative penalty, defined as the ratio of the penalty to the mean fuel mass, i.e. P / E [m F ], is represented in Fig. 13. The superimposed straight lines go from one standard error below to one standard error above the relative penalty. One can see that, for a given aircraft, the relative penalty is higher for the westbound trajectory (with predominant headwinds) than for the eastbound trajectory (with predominant tailwinds): for some aircraft it is about 0.52% for the east route and about 1.1% for the west route (it roughly doubles). Although a negative correlation between the relative penalty and the aircraft final mass exists, it is quite small. Hence, the dependence of P / E [m F ] on m f is weak, whereas the dependence on the prevailing winds is strong. The variability analysis indicates that the highlighted trends are valid even though the

87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

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Because a measure of predictability is the dispersion of the flight time, one can quantify the flight unpredictability by S [t f ]; hence, the results show that the flight unpredictability is larger in the presence of headwinds (for the same wind uncertainty). As a consequence, one has that more extra fuel needs to be loaded in the presence of headwinds, for a given standard for safe operation (this is analyzed further in Section 7.3).

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Fig. 11. Fuel consumption PDFs for all aircraft listed in Table 3.

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103 104 105 106 107 108 109 110 111 112 113 114 115 116

Fig. 12. Definition of the fuel penalty parameter, P .

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standard errors are significant. Some numerical results are given in Table 8. Our analysis allows one to quantify the percentage of extra fuel to be loaded due to wind uncertainty for given aircraft, route and wind forecast. Moreover, this analysis also provides the corresponding standard error, i.e., a quantitative measure on how accurate this extra fuel is expected to be. Now the ratio of the penalty to the standard deviation of the total flight time, namely P / S [t f ], is analyzed. This specific penalty is represented in Fig. 14, where a visualization of the corresponding standard error is not included, because the specific penalty is shown to be very little affected by the variability of the groundspeed sample. It is interesting to note that, for each aircraft, the specific penalty is somewhat similar for both the westbound and the eastbound trajectories (slightly higher in the presence of head-

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A01 A02 A03 A04 A05 A06 A07 A08 A09 A10 A11 A12 A13 A14

E [m F ]

178.6 ± 9.9 216.5 ± 12.0 158.3 ± 8.8 197.9 ± 11.0 200.4 ± 11.1 231.2 ± 12.8 174.6 ± 9.7 227.8 ± 12.7 233.3 ± 13.0 279.3 ± 15.5 273.3 ± 15.2 286.3 ± 15.9 336.5 ± 18.7 414.6 ± 23.1

0.536 0.555 0.494 0.514 0.513 0.533 0.497 0.509 0.503 0.523 0.513 0.516 0.519 0.499

,%

± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.030 0.031 0.028 0.029 0.029 0.030 0.028 0.028 0.028 0.029 0.029 0.029 0.029 0.028

P

kg

, S [t f ] min

P , kg

187.4 ± 0.1 227.1 ± 0.2 180.7 ± 0.1 216.6 ± 0.1 219.5 ± 0.1 235.6 ± 0.2 199.4 ± 0.1 260.1 ± 0.2 266.4 ± 0.2 318.9 ± 0.2 305.6 ± 0.2 320.1 ± 0.2 384.2 ± 0.3 483.2 ± 0.3

542.5 ± 35.6 675.5 ± 44.3 424.7 ± 30.1 578.7 ± 27.8 585.4 ± 38.3 699.1 ± 45.8 470.1 ± 33.3 623.1 ± 44.2 633.3 ± 44.9 778.4 ± 55.2 794.9 ± 51.9 836.4 ± 54.6 933.3 ± 66.2 1115.7 ± 79.1

19 21 22 23

26 27 28 29 30 31

Fig. 13. Relative penalty vs aircraft final mass. Squares: eastbound route; circles: westbound route. Error bars: standard error.

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46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Fig. 14. Specific penalty vs aircraft final mass. Squares: eastbound route; circles: westbound route. (Almost negligible standard errors are not depicted.)

winds). On the other hand, there exists now a strong positive correlation between the specific penalty and the aircraft final mass. Hence, the dependence of P / S [t f ] on m f is strong, whereas the dependence on the prevailing winds is weak. This ratio represents the extra fuel to be loaded per unit of flight time dispersion (as caused by wind uncertainty); it is clear that larger aircraft require larger loads of extra fuel. Some numerical results are given in Table 8. This ratio quantifies the cost of unpredictability; conversely, it quantifies the savings in fuel to be loaded per minute of increase of flight predictability, ranging from about 200 kg for medium aircraft to about 400 kg for heavy aircraft and around 500 kg for A388. Our analysis allows one to quantify this figure for given aircraft, route and wind forecast, and it shows that the standard error of this figure is almost negligible (see Table 8).

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± 0.079 ± 0.083 ± 0.071 ± 0.074 ± 0.073 ± 0.078 ± 0.072 ± 0.074 ± 0.073 ± 0.077 ± 0.073 ± 0.074 ± 0.076 ± 0.072

kg

, S [t f ] min 201.2 ± 0.4 250.5 ± 0.5 188.2 ± 0.3 228.7 ± 0.4 231.3 ± 0.4 259.3 ± 0.5 208.4 ± 0.4 276.2 ± 0.5 280.7 ± 0.5 345.1 ± 0.6 324.0 ± 0.5 340.9 ± 0.6 413.7 ± 0.8 509.8 ± 0.9

The general framework for this paper is the development of a methodology to manage weather uncertainty suitable to be integrated into the trajectory planning process. This work has provided an assessment of the impact of wind uncertainty on aircraft trajectory prediction. The uncertainties in flight time and fuel consumption have been studied in the case of cruise flight at constant Mach and constant pressure altitude, following a given route defined by several segments, subject to forecasted uncertain winds (both along-track winds and crosswinds) provided by ensemble prediction systems. The probabilistic analysis presented allows one to quantify the effects of wind uncertainty on flight predictability and on fuel loading for given aircraft, route and wind forecast. The analysis is based on the Probabilistic Transformation Method, which provides not only the mean and the standard deviation, but the complete PDF of the stochastic solution, as opposed to other methods (such as Generalized Polynomial Chaos) which do not provide the PDF. The results have shown that, for given wind uncertainty, the uncertainties in flight time and fuel consumption are larger for prevailing headwinds than for tailwinds; this result also holds if one considers a relative measure of the uncertainty, such as the metric S [.]/ E [.]. Considering different aircraft of different categories, one has the general trend that the higher the fuel consumption, the higher the uncertainty in fuel consumption, but with S [m F ]/ E [m F ] roughly constant. A fuel penalty parameter has been defined and analyzed, which allows one to quantify the cost of flight unpredictability in terms of extra fuel load per minute of flight time dispersion (for given aircraft, route and wind forecast). The sample variability of all the results has been quantified by means of standard errors. Note that the larger the values of the horizontal distance traveled in each segment, the more realistic the assumption that the ground speeds in the cruise segments (and, therefore, also the flight times) are independent of one another, but the less appropriate the consideration of constant average ground speeds. Therefore, a trade-off between these two effects has to be considered when selecting the segment lengths. Note also that the probabilistic trajectory predictor presented in this paper is capable of taking as input any type of ground speed distribution. In this work, LogNormal distributions have been considered, although other types of distribution could be considered as well. The consideration of temperature uncertainty, also provided by ensemble forecasts, is left for future work (as cruise segments are usually flown at constant Mach number and constant pressure altitude, the main effect of the temperature distribution is a change in true airspeed, due to the change in the speed of sound, which

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1.200 1.262 1.006 1.126 1.123 1.191 1.013 1.045 1.030 1.084 1.118 1.128 1.074 1.016

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Westbound route P

P , kg

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68

Eastbound route

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Table 8 Relative and specific penalties for the different aircraft.

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leads to changes in ground speed and specific fuel consumption). Moreover, the route optimization problem (trajectory planning), not considered in this paper, is also left for future work as a next step in this research.

[15]

5 6

[16]

Conflict of interest statement

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[17]

The authors declare that they have no conflicts of interest associated with this work.

[18]

10 11

Acknowledgements [19]

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The authors gratefully acknowledge the financial support of the Spanish Ministerio de Economía y Competitividad through grant TRA2014-58413-C2-1-R, co-financed with FEDER funds.

[20] [21]

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