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Reducing Carbon Dioxide Emissions for Transatlantic Flights via Optimal Control Theory

By Cathie Wells

If aviation were a country, it would rank among the top 10 emitters in terms of contributions to climate change [1]. Although reducing the number of flights would make an important difference in emissions, an ideal solution would instead ensure that commercial flight is available in a more sustainable way.

Changing aircraft design or fuel could theoretically help to lessen emissions. But the current fleet is already 54 percent more efficient than aircraft from the 1990s, meaning that additional emission reduction would require a drastic redesign — like the use of hydrogen-powered aircraft [2]. Biofuel could also massively reduce emissions, but this approach would require research into cheaper production and less land-intensive crops. Furthermore, current forecasts indicate that biofuel could only meet one-third of aviation fuel needs by 2050 [3].

While the aforementioned ideas are exciting projects for the future, they are high-risk, expensive, and long-term solutions. In contrast, changing the way in which flights are routed is an immediate alteration that could simultaneously reduce emissions and fuel usage. So why has this change not yet been implemented? Until recently, situational awareness across the North Atlantic was limited; people have instead relied on an organised track structure (OTS) wherein air navigation service providers (ANSPs) produce two sets of tracks across the North Atlantic every day that go westbound and eastbound. These tracks allow air traffic management to predict aircraft positions when out of radar range. 

However, a new satellite network that provides 100-percent global coverage now offers an opportunity for trajectory-based operations, in which every flight route can be individually designed and aircraft are permitted to fly farther while closer together. My collaborators and I thus created time- and fuel-minimal routes across the North Atlantic between John F. Kennedy (JFK) International Airport in New York and Heathrow Airport (LHR) in London: one of the busiest routes in the world.

We began by using the wind fields across the North Atlantic between JFK and LHR to reduce emissions when compared with flights that are limited to the OTS. The wind fields are dominated by the slowly evolving jet stream, which enabled us to plan flights at least one day in advance. 

We utilized a simplified model that considers the 91 days of winter from 2019 to 2020. Because the winds come from reanalysis data, we treated them as a set wind field each day. The altitude, airspeed, and mass of the aircraft in question remain constant on each trajectory. We found the air distance—the distance that the aircraft travels relative to the air around it—in order to measure route efficiency. Both emissions and fuel usage are proportional to this value at a fixed altitude and airspeed.

Next, we minimized the time of flight between JFK and LHR. The resulting dynamical system is based on several factors, including the rate of change of longitude with time \((\dot{\lambda})\): the sum of the zonal wind \(u\) and zonal component of airspeed \(V\cos\theta\) mapped conformally onto a sphere. It also incorporates the rate of change of latitude with time \((\dot{\phi})\) via the meridional component of wind \(v\) and airspeed \(V\sin\theta\). With \(R\) as the radius of the Earth, the dynamical system is thus

\[\dot{\lambda} = \frac{V \cos{\theta} + u\left(\lambda,\phi\right)}{R\cos{\phi}}\]

\[\dot{\phi} = \frac{V \sin{\theta} + v\left(\lambda,\phi\right)}{R}.\]

Based on these equations, we used Pontryagin’s minimum principle [6] to derive an expression for the rate of change of the heading angle \((\dot{\theta})\): 

\[\dot{\theta} = -\frac{1}{R\cos \phi} \left[ -\sin\theta\cos\theta \frac{\partial u}{\partial \lambda} + u {\cos}^2\theta\sin\phi + {{\cos}^2}\theta\cos\phi \frac{\partial u}{\partial \phi} - \frac{\partial v}{\partial \lambda} + {{\cos}^2}\lambda \frac{\partial v}{\partial \lambda}... \right.\]

\[ \left. + v\sin\theta\cos\theta\sin\phi + \sin\theta\cos\theta\cos\phi \frac{\partial v}{\partial \lambda} + V\cos\theta\sin\phi \right].\]

Animation 1. Most efficient routes between London’s Heathrow Airport (LHR) and John F. Kennedy International Airport (JFK) from both the organised track system (OTS) and the time minimal optimization. The Great Circle Route (GCR)—the shortest ground distance path between the airports—is also marked. The arrows depict the daily wind fields. Animation courtesy of Cathie Wells.
We solved these formulas via the Euler forward step method. As long as one chooses the correct initial heading angle via a bisection method, the trajectory will evolve forward in time to reach the destination airport.

We found that savings in air distance—and consequently in emissions—could occur on all days between December 1, 2019, and February 29, 2020. Animation 1 depicts the OTS routes (in pink and green) and the routes that are optimized for wind (in blue and red) with the shortest air distances in each direction for every day. 

While these routes are often similar, this animation is solely a comparison with the most efficient OTS track; eight or nine alternative tracks may exist in a given direction on each day. When compared to the worst tracks, our approach can save up to 16.4 percent of emissions. We used data from NATS (formerly National Air Traffic Services) about the number of aircraft that flew on each track for every day to find a weighted average. The average revealed that at the representative airspeed of 240 ms-1, our method could save up to 2.5 percent of emissions; this percentage amounts to 6.7 million kilograms of carbon dioxide (CO2) across the entire 2019-2020 winter season [7].

The utilization of time-minimal routes is not always practical for airlines, as planes that arrive too early to land will emit more CO2 while in holding stacks. Therefore, we next worked to set fixed durations for flights, with two different formulations that allowed us to compare the effects of fuel minimization. The first (OCP1) involves keeping airspeed constant and only altering the heading angle, and the second (OCP2) involves varying both the heading angle and airspeed. Our assumptions remained the same as before, with several important additions: time of flight is fixed, mass \((m)\) is changing, and the fuel burn \((g)\)—which depends on position, airspeed, and mass—will be minimized.

\[\dot \lambda = \frac{V\cos\theta + u\left({\lambda,\phi} \right)}{R\cos\phi}\]

\[\dot \phi = \frac{V\sin\theta + v\left({\lambda,\phi} \right)}{R}\]

\[\dot m = - g\left({\lambda,\phi,V,m} \right).\]

We use a formula from a recent study to calculate fuel burn for each time interval [4, 5].

Our method requires a heading angle for each timestep and an airspeed for OCP2. We fed in atmospheric data and solved the dynamical system numerically using the fourth-order Runge-Kutta numerical method. We calculated the fuel burn from the positions and airspeeds and compared the outcome with previous results to determine whether we had identified a minimum-fuel route. If not, we attempted new controls that closely resembled our previous attempt. If a result did pass the tests and was at least a local minimum, then we accepted it. To ensure that we identify a global (rather than local) minimum, we encased the system within a global search function that applies a whole selection of initial conditions.

Figure 1. Most efficient routes between London’s Heathrow Airport (LHR) and John F. Kennedy International Airport (JFK) from both OCP1 and OCP2 and the time minimal optimization. The Great Circle Route (GCR)—the shortest ground distance path between the airports—is also shown. The arrows depict the daily wind fields. 1a. Westbound routes on December 12, 2019. 1b. Eastbound routes on December 12, 2019. Figure courtesy of Cathie Wells.

Figure 1 illustrates the results for December 12, 2019. Altering an aircraft’s airspeed and heading angle allows it to fly a more direct route while accounting for the wind field. Including both controls (rather than just one) can save fuel, though the savings vary considerably each day due to the changing wind field. Controlling airspeed as well as the heading angle could yield a total extra fuel savings of 723,000 kilograms throughout the winter [8]. 

Modifying flight paths permits us to make substantial reductions in fuel consumption in the short term. NATS (one of the ANSPs) even mentioned our research in its announcement of several OTS-free trials [9]. The next stage of our work will compare time- and fuel-minimal flight cruise phases that we find using dynamic programming to data from actual flights between LHR and JFK. 


Cathie Wells presented this research during a contributed presentation at the 2022 SIAM Conference on Mathematics of Planet Earth (MPE22), which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., in July 2022. She received funding to attend MPE22 through a SIAM Student Travel Award. To learn more about Student Travel Awards and submit an application, visit the online page

References
[1] Directorate-General for Climate Action. (2018). Reducing emissions from aviation. European Commission. Retrieved from https://ec.europa.eu/clima/policies/transport/aviation_en.
[2] Graver, B., & Rutherford, D. (2018). Transatlantic airline fuel efficiency ranking, 2017. Washington, D.C.: International Council on Clean Transportation. Retrieved from https://theicct.org/sites/default/files/publications/Transatlantic_Fuel_Efficiency_Ranking_20180912.pdf.
[3] Haslam, C. (2019, May 12). Eco-friendly flights? That’s fuel for thought. The Sunday Times. Retrieved from https://www.thetimes.co.uk/article/eco-friendly-flights-thats-fuel-for-thought-t6brvcc0x.
[4] Poll, D.I.A., & Schumann, U. (2020). An estimation method for the fuel burn and other performance characteristics of civil transport aircraft during cruise: Part 2, determining the aircraft’s characteristic parameters. Aeronaut. J., 125(1284), 296-340. 
[5] Poll, D.I.A., & Schumann, U. (2020). An estimation method for the fuel burn and other performance characteristics of civil transport aircraft in the cruise. Part 1 fundamental quantities and governing relations for a general atmosphere. Aeronaut. J., 125(1284), 257-295. 
[6] Pontryagin, L., Boltyanskii, V.G., Gamkrelidze, R.V., & Mishchenko, E.F. (1962). The mathematical theory of optimal processes. New York, NY: John Wiley & Sons.
[7] Wells, C.A., Williams, P.D., Nichols, N.K., Kalise, D., & Poll, I. (2021). Reducing transatlantic flight emissions by fuel-optimised routing. Environ. Res. Lett., 16(2), 025002.
[8] Wells, C.A., Kalise, D., Nichols, N.K., Poll, I., & Williams, P.D. (2022). The role of airspeed variability in fixed-time, fuel-optimal aircraft trajectory planning. Optim. Eng
[9] Young, J. (2021, February 3). Is it time to disband the Organised Track Structure? NATS. Retrieved from https://nats.aero/blog/2021/02/is-it-time-to-disband-the-organised-track-structure.

Cathie Wells is a third-year Ph.D. student at the University of Reading, where she works with the Mathematics of Planet Earth Centre for Doctoral Training. Her research interests include optimal control theory and mathematical modeling, and she is particularly passionate about solving real-world problems. Wells works with Paul Williams and Nancy Nichols from the University of Reading, Ian Poll from Poll Aerosciences Ltd, and Dante Kalise from Imperial College London. 
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