Influence of atmospheric uncertainty, convective indicators, and cost-index on the leveled aircraft trajectory optimization problem
Introduction
The Air Traffic Management (ATM) system in the busiest airspaces in the world is currently being overhauled to deal with multiple capacity, socioeconomic, and environmental challenges. One major pillar of this process is the shift towards a concept of operations centered on aircraft trajectories instead of rigid airspace structures. However, its successful implementation rests on appropriate understanding and management of uncertainty.
Due to its complex socio-technical structure, the operation of the ATM system is heavily impacted by uncertainty, emerging from multiple sources and propagating through the interconnections between its subsystems. Any analysis of uncertainty in the ATM system should take into account different scales with their respective sources. Refer to (Cook and Rivas, 2016, Chapter 4) for a thorough description of uncertainties impacting the ATM system, with weather recognised as one of its major contributors. Due to its nonlinear and chaotic nature, some meteorological phenomena, especially those related to convection, cannot be forecasted with complete accuracy at any arbitrary lead time with the required accuracy. This then leads to a general uncertainty or disruption for individual air and ground operations, which then propagates through all ATM processes. It is, therefore, necessary to deal with meteorological uncertainty at multiple scales, and its impact on the trajectory prediction and planning processes and trajectory execution. The focus of the present paper is on flight uncertainty and pre-departure temporal scale (flight dispatching planning level, from two/three hours up to off-block time).
Ensemble Prediction Systems (EPS) provide probabilistic meteorological forecasts. They seek to provide an estimation of the uncertainty that is inherent to the Numerical Weather Prediction (NWP) process (Hacker et al., 2003), and thus to overcome the limitation of a single deterministic forecast. In an EPS, several runs of the NWP model are launched with slightly different characteristics to produce a set of (typically) 10 to 50 different forecasts or “members” of the ensemble. We refer to (Bauer et al., 2015) for a review of the status of NWP as well as the relevance of EPS in a wider meteorological context.
Recent attention has been put into analyzing meteorological hazards and their effects on flight planning. For instance, Kin et al. have considered combined effects of both winds and clear air turbulence (note that clear air turbulence is not considered in this paper) (Kim et al., 2015, Kim et al., 2016). Other works have recently focused on winds and its associated uncertainty. For instance, in (Gonzalez-Arribas et al., 2018), González-Arribas et al. studied the flight planning problem under wind uncertainty using robust optimal control. The same problem has been solved with two additional approaches: In (Franco et al., 2018) Franco et al. presented a hierarchical, bi-level flight planning algorithm in which they combined the Dijkstra algorithm (high-level) with a trajectory predictor (low-level) based on a probabilistic transformation; and Legrand et al. in (Legrand et al., 2018), who solve the problem with an approach based on dynamic programming.
If we focus on convective phenomena, there is substantial ongoing work on short-term hazard avoidance close to the encounter, when the information on the location and short-term evolution of the storm cell is available though still with some uncertainty. Just to cite a few, in (Hauf et al., 2013, Hentzen et al., 2018, González-Arribas et al., 2019) the focus is on en-route problems, whereas in (Yang, 2018, Serhan et al., 2019) the focus is on terminal airspaces. Indeed, tools that incorporate different path planning algorithms for tactical convective weather avoidance are in use today, e.g.. the so-term Dynamic Multi-Flight Common Route Advisories system (Sheth et al., 2016, Bilimoria et al., 2018) or the Convective Avoidance Weather Model (CWAM) (Taylor et al., 2018). On the contrary, the consideration of convective risk in flight planning algorithms (at a larger time scale of 1–3 h before departure) has not received enough attention so far.
This is based on the fact that for the time scales of 1–3 h we only know the area within which individual convective storms may develop. This area of potentially developing storms is referred to as convective area. The onset and the location of individual storms within a convective area, however, is currently and for the near future not possible to forecast at the given time scale. Nevertheless, some basic characteristics of those storms can be derived prior to any convective development, which are necessary but not sufficient conditions for the formation of storms (thermodynamic stability of the air mass). That information can be employed to create an index that estimates the probability of convection, i.e., an indicator of convection risk that can be used for trajectory planning. Convective areas may have a persistence up to 60 h for tropical latitudes, travelling with their surrounding air mass. They shall not necessarily be avoided but require a higher weather situation awareness by pilots and controllers. Trajectories, however, leading through convective areas might experience significant changes due to suddenly developing storms, which results in increased flight duration and delays. The dimension of the latter depends, among other factors, on the type of storms embedded in the convective area, density of cells, their orientation, the size of gaps separating the storms and the time of onset.
Preliminary work on robust optimal control with application to flight planning in which we consider both uncertainties associated with winds and convective areas was presented in (González-Arribas et al., 2019). Both the altitude and the true airspeed were considered constant. The main contribution of the present paper is to extend this work to the consideration of variable speed profiles, BADA4 aircraft performance modelling (which, contrary to BADA 3, incorporates compressibility effects to model drag forces) and the introduction of cost-index based operational cost. We applied this methodology to a case study assuming the altitude to be constant.
The paper is structured as follows: we introduce convection and its associated indicators in Section 2. The robust optimal control methodology and its application to flight planning is presented in Section 3. In Section 4, we present a case study, including the simulation results and a discussion. Finally, some conclusions are drawn in Section 5.
Section snippets
Ensemble Prediction Systems (EPS)
An eps produces a collection of forecasts for the same prediction time that constitutes a representative sample of the possible future states of the atmosphere. An ensemble is typically composed of 10 to 50 individual forecasts referred to as members. To produce the different members, nwp centers employ combinations of several techniques, including changing initial conditions in the most sensitive directions, changing the parameters of the simulation, combining different models or building
Robust trajectory planning methodology
We consider the problem of flight planning, i.e., 1–3 h before departure. To model uncertainties in the weather forecast, we rely on eps forecasts. The uncertainty will be represented with a quadrature rule where each member of the eps forecast corresponds to a quadrature point. Each scenario will be weighted equally; if the eps contains N members, then the weight of each member is . The approach is based on a robust optimal control approach to aircraft trajectory optimization problems (
Description
We consider a BADA4 A330 Aircraft model flying from the vertical of New York (−73.8 deg, 40.6 deg) to the vertical of Argel (3.2 deg, 36.7 deg) at constant barometric altitude 200 hPa. Initial mass and initial Mach have been set to 200 tons and M = 0.82, respectively. We use a forecast for a pressure of 200 hPa 9 h in advance for the 19th of December, 2016 from the ECMWF ensemble with 51 members. The relevant variables for our purposes are the values at isobaric levels of the temperature T, the
Conclusions and future work
A robust optimal control methodology has been used for computing efficient and predictable routes based on Ensemble Prediction Systems, including an approach to calculate the risk of convection. This risk, a necessary though not sufficient condition for the formation of storms, has been included in the objective functional of the robust optimal control problem. This cost combines other objectives, such as flight time predictability or a cost-index based operational performance. We have
Declaration of Competing Interest
None.
Acknowledgments
This work has been partially supported by project TBO-MET project (https://tbomet-h2020.com/), which has received funding from the SESAR JU under grant agreement No 699294 under European Union’s Horizon 2020 research and innovation programme.
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