The climate impact of non-CO

The aviation industry has experienced strong growth in recent years

Mitigating the climate impact associated with aviation-induced CO

Numerous studies have been proposed to reduce the climate impacts of non-CO

As for climate-optimal trajectory planning methods, various strategies ranging from mathematical programming (e.g.,

To quantify the non-CO

When accounting for the ensemble weather forecast in solving aircraft trajectory optimization, the computational time is an important issue that arises in addition to the capability to consider such uncertainties. This is due to the fact that, instead of taking one weather forecast and solving the trajectory optimization in a deterministic manner, the optimizer should be capable of considering

The focus of recent studies has been restricted to planning climate-optimal trajectories considering the concept of future free-route airspace (see last column Table

A classification of the recent studies in the literature proposed to reduce the climate impact of aircraft emissions with aircraft trajectory optimization.

Drawing upon the brief literature review and the presented open problems, we aim to address the problem of determining robust climate-optimal aircraft trajectories within the structured airspace in this study. Our main contributions are summarized as follows: (1) full 4D climate-optimal trajectory planning within the currently structured airspace, (2) accounting for uncertain meteorological conditions and uncertainty associated with initial flight conditions such as departure time and aircraft initial mass, and (3) determining the optimized trajectory computationally very fast. The uncertainty in weather forecast is characterized using the ensemble prediction system, and aviation's climate impacts are quantified by employing the latest version of aCCFs (V1.0A). The concept of robustness that we refer to is the determination of the aircraft trajectory considering all possible realizations of meteorological variables provided using an EPS weather forecast. In other words, instead of planning a trajectory based on one forecast in a deterministic manner, we aim to determine a trajectory that is optimal considering the overall performance obtained from using different members of an ensemble weather forecast. In this respect, from the operational point of view, the optimized trajectory is tracked as determined, and the effects of meteorological uncertainties are reflected in the flight performance variables such as flight time, fuel burn, and climate impact. Mathematically, the perturbations due to the meteorological uncertainty are considered in the dynamical model of aircraft, and the proposed trajectory optimization is generic in terms of the objective function, in which a wide range of objectives, such as flight time, fuel consumption, emissions, and climate impact, and different statistics including expected values and variance of the performance variables, can be considered. Such flexibility in defining the cost function allows for solving a multi-objective optimization problem. Moreover, by penalizing the mean and variance of the objectives, the effects of uncertainty on flight variables can be controlled. In this study, the flight planning objective is a weighted sum of the simple operating cost (as a function of flight time and fuel consumption) and climate impact. The focus is restricted to optimizing the expected performance since, as will be shown in the simulation results, minimizing the averaged performance leads to reducing the uncertainty ranges for the considered case studies.

We employ the probabilistic flight planning method firstly developed by

The paper is arranged as follows. The robust climate-optimal aircraft trajectory planning problem is stated and formulated in Sect.

Aviation-induced non-

The focus of this study is on determining robust flight plans taking into account meteorological uncertainty and uncertainty associated with the initial flight conditions. In this respect, we start by presenting a general formulation of the dynamic optimization problem with uncertainty in Sect.

We present a general formulation of a dynamic optimization problem with the inclusion of uncertainty effects, focusing mainly on the dynamical model and objective function.

Let us consider a class of dynamical systems with uncertainty as follows:

A general form of the cost functional considered for dynamic optimization problems with uncertainty is

The objective of the stated dynamic optimization problem is to find a feasible control policy (

The definition of the aircraft trajectory optimization problem mainly requires the aircraft dynamical model, flight planning objectives, and physical and operational constraints.
To consider climate impact within aircraft trajectory planning, information on the climate impacts of CO

To determine reliable aircraft trajectories, accurate aircraft dynamical models are necessary.
In this work, the point-mass model with the following equations of motion is used to represent the aircraft's dynamical behavior, as is usually considered within air traffic management studies.

As trajectory optimization in this study is performed within the structured airspace, the evolutions of aircraft states are constrained. In the following, we briefly present our proposed modeling of airspace structure and flight plan.

We consider a directed acyclic graph

Structure of airspace. The route graph is generated by processing the full airspace graph to include paths from the end of the SID to the beginning of the STAR to the destination airport.

The goals of the aircraft trajectory optimization problem are interpreted mathematically and defined as an objective function to be minimized (or maximized). In addition to the climate impact, the operating cost is a crucial aspect that needs to be considered as it is one of the main interests of airliners.
Generally, there is a trade-off between the operating cost and climate impacts. This is due to the fact that rerouting areas sensitive to climate increases operational costs as the aircraft tends to fly longer routes

There are various approaches to account for operating costs within aircraft path planning. Flight time and/or fuel are common objectives. However, more realistic cost metrics exist, which include additional costs such as flight crew, cabin crew, and landing fees. Interested readers are referred to

In this study, we use simple operating cost (SOC) as a metric expressing cost in USD with linear relation to flight time and fuel consumption:

In spite of considering only flight time and fuel consumption to represent the operating cost, it was reported in Table 4 of

Numerous approaches have been proposed in the literature to consider climate impact within aircraft trajectory planning strategies (see Table 1 and Fig. 2 of

aCCFs account for the temporal and spatial dependency of climate impacts associated with non-CO

aCCFs estimate the climate impact associated with aircraft emissions computationally in real time, making it well-suited for climate-optimal trajectory planning; and

aCCFs directly quantify climate impacts in average temperature change.

The V1.0A aCCFs are multiplied by some factors in order to be more suitable for the flight planning application.

For the aCCFs of (daytime and nighttime) contrails, the ice supersaturation is applied using temperature and relative humidity over ice in order to predict regions where persistent contrails are expected to form, called persistent contrail formation areas (PCFAs)

The geographical aCCF pattern of water vapor, NO

Algorithmic climate change functions of

To benefit from the spatial and temporal dependency of non-

The dynamical model of aircraft requires weather-related variables such as wind and temperature (see, e.g., Sect.

In this paper, the focus is on forecast-related uncertainties, which will be characterized by employing ensemble prediction system (EPS) weather forecasts, a numerical weather prediction method introduced to deal with uncertainty in weather forecast

To investigate the degree of uncertainty (or variability) in the meteorological variables provided by an EPS and its effects on the computed aCCFs, we take the standard deviation (SD) from 10 ensemble members of weather data obtained using the ERA5 reanalysis data products (

Variability (quantified using SD) of the meteorological conditions for an ensemble weather forecast with 10 ensemble members on 13 June 2018 at 00:00 UTC over European airspace for FL340.

Variability (quantified using SD) of aCCFs for an ensemble weather forecast with 10 ensemble members on 13 June 2018 at 00:00 UTC over European airspace for FL340.

Figure

In Sect.

Propagation of uncertainties (associated with initial flight conditions and meteorological variables) within climate-optimal aircraft trajectory planning.

The aircraft trajectory optimization problem formulated in Sect.

As mentioned in Sect.

To determine the performance of a flight plan and evaluate the cost function Eq. (

Relationship between wind, course, heading, airspeed, and ground speed.

To efficiently reflect the effects of wind uncertainty in flight performance variables, instead of time, the distance flown along the route (

According to the defined objective function (Eqs.

Heun's method is adopted for integrating the aircraft dynamical model along discretized segments of the route through each phase, i.e., climb, descent, and cruise

The expected values obtained from Eq. (

Calculation and evaluation of the expected performance for a given flight plan and an ensemble weather forecast.

The objective is to find a flight plan that minimizes Eq. (

To solve the optimization with hybrid decision variables in an efficient manner, instead of directly searching for the optimal flight plan, the optimization is conducted in the space of probability distributions defined over flight plans. In other words, instead of directly minimizing

In the probabilistic-execution flight plan approach, to sample the flight plan from a given

In this work, the V1 version of the augmented random search (ARS) algorithm adopted from

The effectiveness of the proposed optimization algorithm to plan robust climate-optimal aircraft trajectories with respect to uncertain meteorological conditions is analyzed for a flight from Frankfurt to Kyiv for two different days and departure times.

For the route graph, the full airspace graph of the considered days is filtered and processed to include all paths from the end of the standard instrument departures of the origin airport to the beginning of the standard instrument arrivals of the destination airport with the maximum length of 104 % of the shortest path length. The considered aircraft is an Airbus model A320-214, with the engine CFM56-5B4/P. Table

As for meteorological input data, due to ease of availability, the ERA5 Reanalysis data products containing 10 ensemble members are adopted for this study. It is worth mentioning that forecast data with more ensemble members can be similarly employed. Simulations are launched on the NVIDIA GeForce RTX 3090 graphics card, providing 10496 CUDA cores at a clock speed of 1.4 to 1.7 GHz.

The data obtained from the ICAO data bank to calculate the actual emission index of Airbus model A320-214, with the engine CFM56-5B4/P.

The weighting parameters of the objective function given in Eq. (

We consider a scenario in which the aircraft flies through warming contrails for the cost-optimal routing option. Before presenting the results, the performance of the proposed optimizer in terms of convergence and computational time is analyzed. Since the optimization approach is stochastic, different results may be obtained with different executions.
To explore the sensitivity of the optimization method, 50 different runs are performed with similar settings for pure cost- (i.e.,

Sensitivity of the convergence performance of the optimization algorithm to 50 different executions for the cost- and climate-optimal routing options (

Now, we proceed to present the obtained results.
The aircraft profiles and climate responses for different routing options are given in Fig.

Results of scenario 1 (13 June 2018, 00:00 UTC) for different routing options (i.e.,

By analyzing the contribution of each species to the net ATR for different

Lateral paths for scenario 1 (13 June 2018, 00:00 UTC) depicted with

Overall performance of the optimized trajectories in terms of ATR and SOC for scenario 1 (13 June 2018, 00:00 UTC).

In the next scenario, we analyze the mitigation potential when no persistent contrails are formed with the cost-optimal routing option.

For this case, aircraft profiles and climate responses are depicted in Fig.

In conclusion, climate impact reduction is achieved at the expense of a higher cost increase than in the previous scenario. Moreover, since no contrails are formed, the uncertainty in climate impact is almost negligible.

Results of scenario 2 (10 December 2018, 12:00 UTC) for different routing options (i.e.,

Lateral paths with

Overall performance of the optimized trajectories in terms of ATR and SOC for scenario 2 (10 December 2018, 12:00 UTC).

This paper presented a methodology to plan a robust climate-optimal aircraft trajectory under uncertain meteorological conditions. The climate-sensitive regions were identified using the prototype algorithmic climate change functions (version 1.1). The ensemble prediction system was employed to characterize uncertainty in weather forecasts. A heuristic algorithm was employed and implemented on graphics processing units to solve the proposed robust trajectory optimization in a computationally fast manner. The effectiveness of the proposed approach was explored in two scenarios. Discussion of the obtained results (mainly related to the achieved mitigation potentials, current limitations, and future lines of research) and some general remarks are presented in the following.

The obtained mitigation potentials for the considered scenarios were different due to the variability of meteorological conditions. In both cases, the climate-optimal routing options could reduce the climate impacts. The cost-optimal trajectories flew at higher altitudes compared to climate-optimal ones, as flying at higher altitudes is beneficial for reducing fuel consumption. This is also in line with related studies in the literature (e.g.,

In spite of considering the ensemble members in trajectory planning, a unique (or deterministic) flight plan is determined. This reflects the operational feasibility and applicability of this method since, in the flight planning context, the requirement is to determine a unique lateral route in latitude and longitude that starts and ends at predefined points in space and follows the real structure of airspace as well as having a fixed altitude profile and a fixed airspeed schedule. In this case, the effects of uncertainty are reflected in the aircraft performance variables. For instance, let us consider the first scenario. For

One of the next steps should be analyzing the feasibility of such a routing strategy for real traffic scenarios. In fact, air traffic management (ATM) is a complex multi-agent system that cannot be represented by individual elements but by their collective behavior at the network scale. It was shown in the paper that for the climate-optimal routing options, the aircraft tends to fly at relatively lower altitudes compared to the cost-optimal one. Such behavior to avoid climate-sensitive areas may result in more congested areas, raising some challenges, particularly increased workload, complexity, and conflicts. Thus, the mitigation potentials reported at the micro-level may not be achievable considering real traffic scenarios. Therefore, after generating climatically optimal flight plans, one needs to assess the resulting effects at the network scale and perform a resolution (typically modeled as an optimization problem) to re-stabilize the ATM system by compensating for the negative impacts while keeping the modified trajectories as close as possible to inputted climate-optimized ones. The assessment of manageability of climate-optimal trajectory planning in an ATM system is called macro-scale analysis and lies outside the scope of this paper (see

In this study, we only considered the minimization of the expected performance, e.g., expected climate impact. However, the concept of robustness is mainly related to having less uncertain results (i.e., also minimizing the uncertainty range). In the case of robustness to meteorological uncertainty, we need to find a flight plan that avoids areas of airspace with high variability among the ensemble members. For instance, in

Regarding the computational time and convergence performance, it was shown that they are scenario-dependent. For more complex problems, such as the case including climate impacts quantified by using aCCFs, the optimizer required more iterations to enhance the convergence compared to the cost-optimal routing option. It is worth mentioning that the distance between the origin and destination, available route graphs, and also parameters within the optimization algorithm can change the convergence performance and computational time. The number of iterations is a user-defined parameter that needs to be specified based on the required performance and availability of computational resources. In the performed simulations, we considered 4000 iterations. By looking at the Pareto frontiers, it is clear that the optimizer was able to find nearly optimal solutions. Thanks to the parallelization on GPUs, the computational time for achieving a nearly optimal solution is promising. There are several controlling parameters within the optimization algorithm of ROOST, including the number of search directions, the augmented random search (ARS) step size, and the Nesterov velocity factor (see

To explore the trade-off between climate impact and the operating cost, Pareto frontiers were generated. By changing the weighting parameter

As was explored in the paper, the mitigation of climate impact within the flight planning context is achieved only by accepting some extra costs due to avoidance of highly climate-sensitive regions, which is in line with related studies in the literature (e.g.,

The aCCFs used in this study represent a prototype formulation. The aCCF algorithms were developed for meteorological summer and winter conditions with a focus on the North Atlantic flight corridor. Thus, the usage of the aCCFs for different seasons and regions needs special caution. However, further development of the aCCFs and an expansion of their geographic scope and seasonal representation represent ongoing research.

The robust aircraft trajectory optimization technique presented in the paper is released as an open-source Python library called ROOST V1.0 (Robust Optimization of Structured Trajectories).
It is developed at

The ERA5 datasets used in this study can be freely accessed from the respective repositories after registration. ERA5 data were retrieved from the Copernicus Climate Data Store (

Conceptualization, AS and MS; development of the library (ROOST V1.0), DGA; incorporation of climate impacts to the library, AS; algorithmic climate change functions, SD, SM, and FY; writing – original draft, AS and MS; writing – review and editing, AS, MS, DGA, SM, VG, SD, SB, HY, FL, BL, MMM, FY, and FC. All authors have read and agreed to the published version of the paper.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was carried out as a part of the EU project FlyATM4E.

FlyATM4E has received funding from the SESAR Joint Undertaking under the European Union's Horizon 2020 research and innovation program (grant no. 891317). The JU receives support from the European Union's Horizon 2020 research and innovation program and the SESAR JU members other than the union.

This paper was edited by Andrea Stenke and reviewed by two anonymous referees.