Analysis of aircraft trajectory uncertainty using Ensemble Weather Forecasts

Transcription

1 DOI: /EUCASS TH EUROPEAN CONERENCE OR AERONAUTICS AND SPACE SCIENCES (EUCASS Analysis o aircrat traectory uncertainty using Ensemble Weather orecasts Damián Rivas, Antonio ranco, and Alonso Valenzuela Department o Aerospace Engineering Escuela Técnica Superior de Ingeniería, Universidad de Sevilla 4109 Sevilla, Spain Abstract The problem o aircrat traectory prediction subect to weather uncertainty is addressed. In particular, a probabilistic analysis o aircrat uel consumption taking into account wind uncertainty is presented. The wind uncertainty is obtained rom ensemble weather orecasts. The analysis is ocused on the cruise light, which is composed o several segments. Each segment is subect to both an average constant along-track wind and an average constant crosswind. The resulting ground speed is modeled as a random variable. A probabilistic traectory predictor is presented, based on the Probabilistic Transormation Method. Results are presented or a given trans-oceanic route and a real ensemble weather orecast. 1. Introduction The uture Air Traic Management (ATM system must address the perormance challenges posed by today's airspace: the capacity and the eiciency o the system must be increased while preserving or augmenting the saety levels. To accomplish these goals, in this uture system the traectory becomes the undamental element o a new set o operating procedures, collectively reerred to as Traectory-Based Operations (TBO [1]. One key actor that aects those challenges is uncertainty, which is an inherent property o real-world socio-technical complex systems, and ATM is clearly not an exception. Uncertainty is critical rom dierent perspectives in air transport: saety, environment and cost. Researchers must accept the act that uncertainty is unavoidable and must be dealt with, rather than ignored. I the capacity o the ATM system is to be increased while maintaining high saety standards and improving the overall perormance, uncertainty levels must be reduced and new strategies to deal with the remaining uncertainty must be ound. In particular, procedures to integrate uncertainty inormation into the ATM planning process must be developed. In Rivas and Vazquez [] one can ind a review o all the uncertainty sources that aect the ATM system. Among those, weather has perhaps the greatest impact. The analysis o weather uncertainty has been addressed by many authors, using dierent methods. Among many others, Zheng and Zhao [3] develop a statistical model o wind uncertainties and apply it to stochastic traectory prediction in the case o straight, level light traectories. The general ramework or this paper is the development o a methodology to manage weather uncertainty suitable to be integrated into the traectory planning process. In this paper a probabilistic analysis o aircrat uel consumption taking into account wind uncertainty is presented. The study is ocused on the cruise phase and considers the wind uncertainty provided by Ensemble Prediction Systems (EPS, which have proved to be an eective way to quantiy weather uncertainties. An analysis o wind-optimal cruise traectories using ensemble probabilistic orecasts together with pseudospectral methods is perormed in Gonzalez- Arribas et al. [4]. A conceptual vision o the integration o ensemble-based, probabilistic weather inormation with ATM decision support tools, ocused on convective storms, is presented in Steiner et al. [5]. The importance o weather uncertainty inormation in probabilistic air traic low management is shown in Steiner et al. [6], where the translation o ensemble weather orecasts into probabilistic air traic capacity impact is described. These papers clearly show the importance o making use o ensemble weather orecasts to generate probabilistic weather inormation or aviation needs. Copyright 017 by D. Rivas, A. ranco, and A. Valenzuela. Published by the EUCASS association with permission.

2 DOI: /EUCASS D. Rivas, A. ranco, and A. Valenzuela The two main approaches to the analysis o aircrat traectory uncertainty using ensemble weather orecasts are considered: ensemble traectory prediction (etp and probabilistic traectory prediction (ptp. The ensemble TP is the approach used by Cheung et al. [7]. The probabilistic TP considered has been presented or a one-segment cruise in Rivas et al. [8]. In this second approach the wind uncertainty is propagated along the aircrat traectory. The method used or the uncertainty propagation is the Probabilistic Transormation Method (see Kadry [9] and Kadry and Smaily [10]. Both approaches are applied to traectories composed o a given number o cruise segments, taking into account the wind distributions obtained rom a real EPS. The parameters that deine the aircrat and the light Mach number are obtained rom Eurocontrol BADA data base. This study is relevant because wind is one o the main sources o uncertainty in traectory prediction, and because cruise uncertainties have a large impact on the overall light since the cruise phase is the largest portion o the light (at least or long-haul routes. In particular it is expected that this study be relevant or the determination o the contingency uel, and, hence, or allowing a more eective decision making.. Ensemble weather orecasting To model weather or strategic planning horizons, a probabilistic approach is the appropriate one, so that the inherent weather uncertainty can be taken into account. Today's trend is to use Ensemble Prediction Systems (EPS, which attempt to characterize and quantiy the inherent prediction uncertainty based on ensemble modeling. Ensemble orecasting is a prediction technique that consists in running an Ensemble o Weather orecasts (EW 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 approaches. Thus, this technique generates a representative sample o the possible (deterministic realizations o the potential weather outcome [5]. An ensemble orecast is a collection o typically 10 to 50 weather orecasts (reerred to as members. Cheung et al. [11] review various EPSs: PEARP (rom Météo rance, consisting o 35 members; MOGREPS (rom the UK Met Oice, with 1 members; the European ECMW, with 51 members; and a multi-model ensemble (SUPER constructed by combining the previous three orming a 98-member ensemble, designed so that it is more likely to capture outliers and give a higher degree o conidence in predicting uture atmospheric evolution. Ensemble orecasting has proved to be an eective way to quantiy weather prediction uncertainty. The uncertainty inormation is on the spread o the solutions in the ensemble, and the hope is that this spread bracket the true weather outcome [5]. It is important to notice that or strategic planning the analysis o all the individual ensemble members must be included (rather than an ensemble mean [6]. 3. Traectory prediction considering ensemble weather uncertainty In this section, the two approaches or traectory prediction subect to uncertainty provided by ensemble weather orecasts are described. 1 Ensemble traectory prediction (see ig. 1. In this case, or each member o the ensemble, a deterministic traectory predictor (TP is used, leading to an ensemble o traectories rom which probability distributions can be derived. This approach is used in [7, 11]. Probabilistic traectory prediction (see ig.. Now, probability distributions o meteorological parameters o interest (such as wind are evolved along the aircrat traectory using a probabilistic traectory predictor (ptp, leading to probability distributions o traectory parameters o interest (such as uel consumption. The ptp deined in this paper or multi-segment traectories ollows this probabilistic approach (as described in Section 6. The required input rom the EW to the traectory predictors will depend on the ATM problem under consideration. In this work the uel consumption in cruise light is studied, subect to wind uncertainty; thereore, w 1, w,... represent the wind ields deined by each ensemble member. w n

3 DOI: /EUCASS ANALYSIS O AIRCRAT TRAJECTORY UNCERTAINTY USING ENSEMBLE WEATHER ORECASTS igure 1: Ensemble traectory prediction. Legend: m - member, w - weather, x - traectory igure : Probabilistic traectory prediction. Legend: m - member, w - weather 4. uel consumption in cruise light As already indicated, in this paper the uel consumption in cruise light is studied. The cruise is supposed to be ormed by cruise segments, each one o them deined by a constant heading, and lown at constant speed and constant altitude, as usually required by Air Traic Control (ATC. Sketches o a multi-segment cruise and a generic cruise segment can be ound in ig. 3 and ig. 4, respectively. p In a cruise segment, the light is supposed to be subect to both constant average along-track winds, average crosswinds, the cruise. w c w, and constant. These can be dierent or the dierent segments, thus modeling the wind variation along igure 3: Sketch o a multi-segment cruise (uncertain variables in red 3

4 DOI: /EUCASS D. Rivas, A. ranco, and A. Valenzuela igure 4: Sketch o a generic cruise segment (uncertain variables in red The eects o 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 or a cruise segment, which is given by V g V w c w where V g is the ground speed and V is the airspeed. Because w and w c are uncertain, V g is uncertain as well. Assuming symmetric light and the lat Earth model, the equations o motion or a cruise segment are dx V dt g, T D, L mg dm ct dt where x is the horizontal distance, t is the time, T is the thrust, D and L are the aerodynamic drag and the lit, m is the aircrat mass, g 9.8 m / s is the acceleration o gravity, and c is the speciic uel consumption, which can be taken as a unction o altitude and speed, and it is thereore constant under the given cruise condition. The drag can be written as coeicient C L V S L D V SC D, where C D is modeled by a parabolic polar (, and the coeicients C D0 and is the air density, D D0 D L C C C C, where deinitions and Eq. (, the ollowing equation is obtained or the aircrat mass C D A Bm S is the wing surace area, and the drag C L is the lit coeicient given by are constant under the given cruise condition. Using these dm Vg dx where the positive constants A and B are deined as c ccd g A V SCD, B 0 V S Equation (3 is a nonlinear equation describing the evolution o the aircrat mass as a unction o distance. Even though this model is quite simple, it is adequate to describe the cruise light o commercial transport aircrat, since they usually ly segments o constant Mach number ( M and constant altitude ( h ollowing ATC practice, and it is assumed that 4

5 DOI: /EUCASS ANALYSIS O AIRCRAT TRAJECTORY UNCERTAINTY USING ENSEMBLE WEATHER ORECASTS the constant values o the parameters o the aircrat model ( set or the light. C D0, C D, and c correspond to the values o M and h In this work, the range o each cruise segment ( x and the inal aircrat mass ( m p m are given. ixing m (instead o the initial aircrat mass is consistent with having a ixed landing weight. It also allows or a air comparison or dierent values o the wind, which lead to dierent uel loads and thereore to dierent values o the initial aircrat mass. Thereore, the whole cruise light has to be computed backwards, starting rom the last segment ( at the irst one ( boundary condition 1. or each segment, Eq. (3 is to be solved backwards, rom ( x to p ( x 0 0 and ending, with the m( ( x ( m (5 where ( m ( mi 1, p, as dictated by mass continuity. This problem has the ollowing explicit solution B B arctan m arctan m AB t where g i A A t x V is the light time corresponding to the cruise segment transormation t g ( V. g. This solution deines the Combining the solutions or all the cruise segments, one can easily obtain the initial aircrat mass ( mi 1 m. Then, the cruise uel consumption ollows rom m mi m. Hence, one has i A B m tan arctan m AB t m B A where t t p. This solution deines the transormation m ˆ g( t Ensemble traectory predictor In this section the ensemble TP is described. Suppose that the ensemble has n members, then, the irst step is to determine, or each member k o the ensemble and each segment average crosswind, say w c. Next, the ground speed k, corresponding light time is determined by the transormation t g k Vg, V g k, ensemble, the cruise light time ollows rom k, t p 1 t, the average along-track wind, say w k,, and the c k, V w w has to be computed. Then, the, k (. inally, or the member k o the, k k, and the cruise uel consumption is given by the transormation m gˆ( t. Thereore, the inal result is a set o n values o the cruise uel consumption ( m m n k k, data that needs some postprocessing to help the decision making process. 1, 5

6 DOI: /EUCASS D. Rivas, A. ranco, and A. Valenzuela 6. Probabilistic traectory predictor The ptp is described in this section. The input is the pd o the aircrat ground speed at each segment and the output is the pd o the uel consumption, see sketch in ig. 5. As already indicated, the ptp is based on the Probability Transormation Method (PTM. The basis o this method is the ollowing theorem (see Canavos [1]: Given a random variable g y such that with probability density unction z g( y y ( y, then the probability density unction o, i one deines another random variable z is given by z using a transormation ( z z g z 1 y ( ( 1 g ( g ( z, expression that is valid only i the unction g( y is invertible on the domain o y. The ptp needed in this work or multi-segment traectories relies on the well-known result in statistics that the pd o the sum o independent variables is the convolution o the pds o the addends where X Y ( x y, ( x ( x dx x y x y x ( y ( y dy In this paper, the ollowing procedure is proposed, see sketch in ig. 5: y. 1 In each cruise segment, the ground speed is transormed into the light time according to the transormation t g V. Let ( V be the pd o the ground speed o the cruise segment (to be deined in Section ( g Vg g 7; then, the pd o the corresponding light time is obtained by applying Eq. (8 ( t (( t Vg ( t ( t Aterwards, the pd o the cruise light time, namely ( ( x ( x. (11 t, is obtained rom the pds o the light times corresponding to the cruise segments. or that purpose, the ground speed in the cruise segments (and, thereore, also the light times are considered to be independent o one another. Assuming the independence o the light times, Eq. (9 can be applied and the pd o the cruise light is given by t. t ( t ( t 1 ( t ( t p 3 inally, the pd o the uel consumption ollows rom Eq. (8 m ( m 1 ( gˆ ( m t A B( m m, where 1 gˆ ( m is easily obtained rom Eq. (7. This analysis is valid because all the transormation unctions are invertible on their respective domains. Once the pd is known, one can compute the mean and the standard deviation, as ollows 6

7 DOI: /EUCASS ANALYSIS O AIRCRAT TRAJECTORY UNCERTAINTY USING ENSEMBLE WEATHER ORECASTS E[ m ] m ( m dm m m [ m ] m ( m dm E[ m ( w] 1/ igure 5: Probabilistic traectory predictor (ptp. Traectory with p segments 7. Probabilistic wind model In this section the input to the ptp is deined (see ig. 5, that is, the probabilistic ground speed that aects the aircrat traectory. In the ollowing, the approach to obtain the pd o the ground speed in each cruise segment is described. or an ensemble that has members, let Vg,, V,1 g be the ground speeds or segment, as obtained in, n Section 5. Now one must assume that they ollow a particular distribution. This is not a minor point, and in act is one o the open challenges in this problem. In this paper, to obtain the pd o the ground speed in each segment, it is assumed that the ground speed is distributed as a uniorm continuous variable in the interval [ V, V ], where n g, m g, M and V are estimated rom the sample by using the method o moments. Thereore, the ground speed along the g M, cruise segment has the ollowing pd V g m, 1( V V, V [ V, V ] 0, ] g, M g, m g g, m g, M V ( V g g Vg [ V, g V, m g, M The mean and the standard deviation o Vg are given as ollows E[ V ] ( V V g g, M g, m [ V ] ( V V g g, M g, m ( 3 7

8 DOI: /EUCASS D. Rivas, A. ranco, and A. Valenzuela 8. Analysis o uel consumption uncertainty In the ollowing, results are presented or a given aircrat and a given cruise light with the ollowing parameters: D kg/m ( h m, V 36 m/s ( p 9 M 0.8 segments deined by, C , C , c s/m, S 83.5 m, and m kg. The selected route (westbound is described in Tables 1 and, where the waypoints that deine it and the horizontal distances or each segment are given (note that or the eastbound traectory the cruise starts with segment 9 and ends with segment 1. Table 1: Route waypoints (westbound numbering D Latitude 43º 37.5 N 46º N 48º N 49º N 49º N 49º N 48º N 46º N 4º N 40º 38.4 N Longitude 6º 9.84 E 0º 10º W 0º W 30º W 40º W 50º W 60º W 70º W 73º W Table : Horizontal distance travelled in each segment (westbound numbering Horizontal distance (km ( x Cruise segment Results are presented or the aircrat travelling the route both westbound and eastbound. Thus, one can analyse the dierence in traectory uncertainty between the cases o being in the presence o tailwinds and headwinds. The EPS chosen has been PEARPS, rom Météo rance, and winds have been retrieved rom the ECMW database, corresponding to 5 May 016, a release time o 6:00, a time step o 0 hours, and or pressure level 00 HPa ( h m. Raw wind data have been processed to give a constant wind per segment and per ensemble member. The resulting averaged winds are a set o 35 9 =630 values o along-track and cross-track winds, not included or brevity. rom this weather input, 35 9=315 values o ground speed are obtained or the westbound route, which are not listed here, but they are represented or the dierent cruise segments in ig. 6, in the orm o relative requency histograms, along with the corresponding pds (assuming uniorm distributions. One can see that empirical data do not clearly ollow any common statistical distribution; however, the uniorm distribution turns out to be a good proposal because it is simple, yet roughly matches the data. Since the same scale or the abscissa is selected in all subigures, it is easy to see that the spread o the ground speed is dierent or the dierent segments, with the sample standard deviation ranging rom m/s or segment 6 to 0.95 m/s or segment 1. Ground speed distributions have also been obtained or the eastbound route; however, they are not represented or the sake o brevity, because they are quite similar to the ones presented. It is important to note that there are not substantial dierences in the spread o the distributions between both cases. or the eastbound traectory, the average along-track and cross-track winds are the same as or the westbound traectory in magnitude, ust with the opposite signs. 8

9 DOI: /EUCASS ANALYSIS O AIRCRAT TRAJECTORY UNCERTAINTY USING ENSEMBLE WEATHER ORECASTS igure 6: Ground speed distributions corresponding to the cruise segments (westbound route. Blue bars: relative requency histograms. Black curves: uniorm distributions or the ptp approach. (Gaps between bars are introduced only or aesthetical purposes. 8.1 Results rom ensemble TP As already known, the result rom the ensemble TP is a set o n values o the cruise uel consumption (, m n. They are listed in Table 3 and represented in the orm o relative requency histograms in ig. 7. The same width has been considered or all the histogram bins, so that one can compare the spread in the results. Values o the mean and the standard deviation are presented in Table 4. m 1 9

10 DOI: /EUCASS D. Rivas, A. ranco, and A. Valenzuela Table 3: uel consumption (in kg or each EPS member (westbound/eastbound 1/34186 /556 /3416 /5579 3/3416 /555 4/34150 /5581 5/34137 /5574 6/34175 /5531 7/34155 /5551 8/34157 /5554 9/34181 / /34130 / /341 /556 1/34099 / /34113 / /34199 /554 15/34131 / /34181 / /34169 / /34144 / /34150 /5570 0/3416 /5535 1/34077 /5585 /3435 /550 3/34131 /5567 4/34181 /5538 5/34 /5546 6/3409 /5559 7/3413 /5571 8/34188 /5534 9/34195 / /34117 / /34156 /5544 3/34155 / /3417 / /34140 / /34155 / Results rom probabilistic TP Once the input to the ptp is deined (given by Eq. (15, the pd o the uel consumption m ( m is computed ollowing the procedure deined in Section 6 (and outlined in ig. 5. The pds o the aircrat uel consumption obtained with the ptp approach are shown in ig. 7 or both westbound and eastbound cruises. Values o the mean and the standard deviation are presented in Table 4. These results show that or the westbound cruise one has larger values o the mean (as expected, because in this case one has predominant headwinds, and also larger values o the standard deviation, implying that the traectory uncertainty is larger in the presence o headwinds. This trend also holds or the relative uncertainty [ m] / E[ m], increasing rom to (it roughly increases 50%. Thereore, more extra uel needs to be loaded in the presence o headwinds, or a given standard or sae operation. The results also show that [ m] / E[ m] is approximately constant or dierent aircrat (medium and heavy, which is in line with the common practice o loading the same percentage o the trip uel or contingencies. That is, our analysis shows that the same percentage o contingency uel should be loaded due to wind uncertainty or all aircrat (or the given route and EPS, percentage that can be quantiied or the particular wind orecast under consideration. igure 7: uel mass distributions or westbound cruise (let and eastbound (right. Blue bars: relative requency histograms (ensemble TP. Black curves: pds (probabilistic TP 10

11 DOI: /EUCASS ANALYSIS O AIRCRAT TRAJECTORY UNCERTAINTY USING ENSEMBLE WEATHER ORECASTS Table 4: Mean and standard deviation o uel consumption E[ m ] Westbound (kg [ m ] (kg E[ m ] Eastbound (kg [ m ] (kg etp ptp inal remarks This work has provided an assessment o the impact o wind uncertainty on aircrat traectory, and in particular on the cruise uel load. It is expected that by considering the weather uncertainty in the traectory prediction process, one could adust the contingency uel depending on the uncertainty obtained or the uel consumption. Note that the larger the values o the horizontal distance travelled in each segment, the more realistic the assumption that the ground speeds in the cruise segments (and, thereore, also the light times are independent o one another, but the less appropriate the consideration o constant average winds. Thereore, a trade-o between these two eects has to be considered when selecting the segment lengths (or, equivalently, the location o the waypoints. The consideration o temperature uncertainty, also provided by EW, is let or uture work. As cruise segments are usually lown at constant Mach number and constant pressure altitude, the main eect o the temperature distribution is a change in true airspeed (due to the change in the speed o sound, which leads to changes in ground speed and speciic uel consumption. The probabilistic traectory predictor presented in this paper is capable o taking as input any type o ground speed distribution. In this work, simple uniorm distributions have been considered, although other types o distributions could be considered as well. It is clear that the determination o the ground speed pd rom the uncertainty inormation contained in the EPSs is an open challenge in this problem. This issue poses a multidisciplinary task to be addressed ointly by meteorologists, statisticians and ATM experts. Acknowledgements The authors grateully acknowledge the inancial support o the Spanish Ministerio de Economía y Competitividad through grant TRA C-1-R, co-inanced with EDER unds. Reerences [1] SESAR Consortium The ATM target concept - SESAR deinition phase, deliverable 3. September 007. [] D. Rivas, and R. Vazquez Uncertainty. In Complexity Science in Air Traic Management, A. Cook and D. Rivas Ed., Ashgate Publishing Limited, Chap. 4. [3] Q.M. Zheng, and Y.J. Zhao Modeling wind uncertainties or stochastic traectory synthesis. 11th AIAA Aviation Technology, Integration and Operations (ATIO Conerence, paper AIAA , pp. 1. [4] D. Gonzalez Arribas, M. Soler Arnedo, and M. Sanuro Rivo Wind-optimal cruise traectories using pseudospectral methods and ensemble probabilistic orecasts. Proc. 5th International Conerence on Application and Theory o Automation in Command and Control Systems (ATACCS015, pp [5] M. Steiner, C.K. Mueller, G. Davidson, and J.A. Krozel Integration o probabilistic weather inormation with air traic management decision support tools: a conceptual vision or the uture. Proc. 13th Conerence on Aviation, Range and Aerospace Meteorology, pp [6] M. Steiner, R. Bateman, D. Megenhardt, Y. Liu, M. Xu, M. Pocernich, and J.A. Krozel Translation o ensemble weather orecasts into probabilistic air traic capacity impact. Air Traic Control Quarterly, 18: [7] J. Cheung, J.-L. Brenguier, J. Heistek, A. Marsman, and H. Wells Sensitivity o light durations to uncertainties in numerical weather predictions. Proc. SESAR Innovation Days (SID014, pp

12 DOI: /EUCASS D. Rivas, A. ranco, and A. Valenzuela [8] D. Rivas, R. Vazquez, and A. ranco Probabilistic Analysis o Aircrat uel Consumption using Ensemble Weather orecasts. Proc. 7th International Conerence on Research in Air Transportation (ICRAT, pp [9] S. Kadry On the generalization o probabilistic transormation method. Applied Mathematics and Computation, 190: [10] S. Kadry, and K. Smaily Using the transormation method to evaluate the probability density unction o z x y. International Journal o Applied Economics and inance, 4: [11] J. Cheung, A. Hally, J. Heistek, A. Marsman, and J.-L. Brenguier Recommendations on traectory selection in light planning based on weather uncertainty. Proc. SESAR Innovation Days (SID015, pp [1] G. Canavos Applied Probability and Statistical Methods. Little, Brown, and Company, Boston, p