Open AccessArticle

Cooperative Low-Carbon Trajectory Planning of Multi-Arrival Aircraft for Continuous Descent Operation

Cun Feng

¹,

Chao Wang

^1,*

Hanlu Chen

²,

Chenyang Xu

¹ and

Jinpeng Wang

College of Air Traffic Management, Civil Aviation University of China, No. 2898 Jinbei Road, Tianjin 300300, China

Huanghua International Airport, Changsha 410137, China

School of Electrical Engineering and Automation, Luoyang Institute of Science and Technology, Luoyang 471023, China

Author to whom correspondence should be addressed.

Aerospace 2024, 11(12), 1024; https://doi.org/10.3390/aerospace11121024

Submission received: 17 October 2024 / Revised: 21 November 2024 / Accepted: 13 December 2024 / Published: 15 December 2024

(This article belongs to the Section Air Traffic and Transportation)

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Browse Figures

Figure 1
A typical CDO process of an arrival aircraft. "> Figure 2
The explicit guidance for aircraft speed control. "> Figure 3
A simplified standard terminal arrival route for busy terminal areas. "> Figure 4
(a) Traditional open path arrival route structure to downwind leg; (b) T-shaped arrival route structure. "> Figure 5
Alternative route assembly schematic. "> Figure 6
Alternative set of 4D trajectories based on downwind leg segmentation. "> Figure 7
Correspondence between flight distance and time of critical waypoint. "> Figure 8
The chromosome model of decision variables in the MIP planning model. "> Figure 9
Diagram illustrating priority landing for aircraft on a direct final approach. "> Figure 10
Standard arrival flight procedures of ZSQD TMA. "> Figure 11
Alternative routes of T-shaped arrival route structure (schematic diagram not to scale). "> Figure 12
Actual and optimized vertical profile of B737-800. "> Figure 13
Variation in flight time and fuel consumption with different optimization objectives. (a) Flight time distribution; (b) fuel consumption distribution. "> Figure 14
Space–time diagram of multi-aircraft trajectory planning. Analysis of selected alternative routes and waiting times with the objective of minimizing total cost. "> Figure 15
Horizontal trajectory comparison. (a) Actual horizontal trajectories; (b) optimized horizontal trajectories. "> Figure 16
Vertical profile comparison. (a) Actual trajectory vertical profile; (b) optimized altitude profile. "> Figure 17
Fuel flow comparison of Aircraft 11. (a) Actual trajectory fuel profile; (b) optimized trajectory fuel profile. "> Figure 18
Comparison of fuel consumption of the 22 aircraft. "> Figure 19
The distribution of flight times under different numbers of arrival flights. ">

Versions Notes

Abstract

To address the technical challenges of implementing Continuous Descent Operations (CDO) in high-traffic-density terminal control areas, we propose a cooperative low-carbon trajectory planning method for multiple arriving aircraft. Firstly, this study analyzes the CDO phases of aircraft in the terminal area, establishes a multi-phase optimal control model for the vertical profile, and introduces a novel vertical profile optimization method for CDO based on a genetic algorithm. Secondly, to tackle the challenges of CDO in busy terminal areas, a T-shaped arrival route structure is designed to provide alternative paths and to generate a set of four-dimensional (4D) alternative trajectories. A Mixed Integer Programming (MIP) model is constructed for the 4D trajectory planning of multiple aircraft, aiming to maximize the efficiency of arrival traffic flow while considering conflict constraints. The complex constrained MIP problem is transformed into an unconstrained problem using a penalty function method. Finally, experiments were conducted to evaluate the implementation of CDO in busy terminal areas. The results show that, compared to actual operations, the proposed optimization model significantly reduces the total aircraft operating time, fuel consumption, CO₂ emissions, SO₂ emissions, and NO_x emissions. Specifically, with the optimization objective of minimizing total cost, the proposed method reduces the total operation time by 22.4%; fuel consumption, CO₂ emissions, SO₂ emissions by 22.9%, and NO_x emissions by 23.7%. The method proposed in this paper not only produces efficient aircraft sequencing results, but also provides a feasible low-carbon trajectory for achieving optimal sequencing.

Keywords:

air traffic management; continuous descent operations; four-dimensional trajectory optimization; optimal control theory; sequential scheduling

1. Introduction

In recent years, the rapid development of aviation demand has led to a notable increase in air traffic flow, resulting in the saturation of the capacity of many high-density airports. Concurrently, the ecological issues associated with aviation have become increasingly pronounced. The suboptimal operational efficiency of traffic flow in terminal areas has resulted in augmented fuel consumption and pollutant emissions. The reduction in energy consumption and emissions in the context of air traffic has emerged as a pivotal challenge in the field of aviation operations.

The Continuous Descent Operation (CDO) concept, as proposed by the International Civil Aviation Organization (ICAO), represents a significant advancement in the optimization of aircraft descent. It has emerged as the preferred green operation mode due to its notable effectiveness in reducing both energy consumption and emissions. However, the CDO trajectory of an aircraft is subject to uncertainty due to factors such as air traffic flow [1]. As efforts to explore emission reduction strategies through air traffic operation optimization continue, the fuel-saving and emission-reduction benefits of CDO during non-busy periods have been empirically validated [2]. However, in high-traffic-density terminal areas, aircraft scheduling still largely relies on the decisions of air traffic controllers. Due to the limitations of human cognitive abilities, air traffic controllers are unable to efficiently manage aircraft trajectories in such dense environments, making the implementation of CDO in high-density areas a challenge that still needs to be addressed.

Scholars have conducted extensive research on the sequencing of arrival traffic and the allocation of slot resources. The study of arrival sequencing has evolved significantly, transitioning from static [3] to dynamic [4], from deterministic [5] to stochastic [6], from single-objective [7] to multi-objective [8], and from directly optimizing landing times [9] to first determining landing order and then optimizing landing times [10]. This progression has spanned from model-driven to data-driven methods [11]. Various sequencing strategies have been established, including First-Come-First-Served (FCFS), constrained position shifting, time windows, and flight priority.

It has been increasingly recognized that the nature of the aircraft sequencing problem is that of a combinatorial optimization problem. In order to address this issue, scholars have developed a range of aircraft optimization sequencing models and related resource scheduling models, which are based on mathematical concepts such as integer planning, mixed integer planning, static planning, and dynamic planning [12]. These models have been solved using both exact algorithms and heuristic algorithms [13]. While exact algorithms can provide optimal solutions to models [6], they are not well suited to large-scale problems, where combinatorial explosion can occur. In such cases, heuristic algorithms offer a means of efficiently obtaining approximate solutions. For example, Dabin Xue proposed an innovative ADS-B-based heuristic search method to address the Aircraft Landing Problem (ALP), aiming to reduce flight time while adhering to the time separation standards set by the ICAO [14]. The most commonly employed heuristic algorithms for arrival flight optimization include particle swarm algorithms, annealing simulation algorithms, genetic algorithms, and empire competition algorithms.

In addition to theoretical studies, Air Navigation Service Providers (ANSPs) have developed a series of assisted decision-making systems for the optimal sequencing of arriving aircraft. Examples of such systems include the CTAS system in the United States [15] and the AMAN system in Europe [16]. The Arrival Decision Support Systems assist air traffic controllers in the efficient management of the arrival traffic flow by providing optimized landing sequences and estimated arrival times.

The aforementioned study provides air traffic controllers with the optimized order and expected arrival time of arriving aircraft. However, it fails to provide specific flight trajectory guidelines on how to achieve the expected arrival time. Furthermore, it still relies on the manual guidance of the air traffic controller, which is an inadequate method for completely solving the problem of automatic air traffic control.

Numerous scholars have conducted extensive and in-depth research on the aircraft trajectory optimization problem and have widely recognized that the core of the trajectory optimization problem is a highly complex nonlinear optimal control problem that incorporates process constraints, terminal constraints, and control constraints [17]. Currently, four-dimensional (4D) trajectory optimization methodologies are typically categorized into three primary methods, namely indirect methods, direct methods, and meta-heuristic algorithms.

The indirect method, based on Pontryagin’s principle, transforms the optimal control problem into a two-point boundary value problem, which is primarily used for the optimization of simple systems [18]. However, due to the complex dynamic models and numerous constraints involved in the 4D trajectory optimization problem, the indirect method encounters significant difficulties in solving such problems [19].

The direct method discretizes continuous optimal control problems into nonlinear programming problems, which can more effectively parameterize constraint conditions compared to the indirect method, making it suitable for solving more complex trajectory optimization problems [20]. Garcia-Heras et al. compared the polynomial approximation method of the indirect method with the pseudospectral method of the direct method in trajectory optimization problems, finding that the pseudospectral method outperforms the polynomial approximation method in terms of solution accuracy, while the polynomial approximation method has an advantage in solving speed [21]. Park et al. also compared the trajectory optimization results of the indirect and direct methods, demonstrating that the direct method possesses a certain superiority in accuracy [22].

Compared to other direct methods, the pseudospectral method does not require numerical integration, resulting in a higher solving efficiency and potential for online application. Park et al. formulated the trajectory optimization problem as a multi-stage optimal control problem and used the Legendre–Gauss pseudospectral method to compute the optimal descent profile [23]. Luchinskiy et al. aimed to minimize total energy consumption by optimizing the descent phase trajectory using the Gauss pseudospectral method [24]. Meanwhile, Ma et al. utilized the Legendre–Gauss pseudospectral method to generate optimized trajectories with the objectives of minimizing delay time, NPD noise values, and fuel consumption [25]. Most of these studies rely on the GPOPS toolbox for solving, but GPOPS has the drawback of being sensitive to the setting of parameter ranges [26].

With the advancement of swarm intelligence optimization techniques, some scholars have proposed the use of heuristic algorithms to solve the optimal control problem of trajectory vertical profile optimization. Yang et al. addressed the challenge of providing time-continuous control inputs in trajectory optimization by transforming the optimal control problem into a nonlinear programming problem. They selected thrust and load factors as the optimization variables to construct a nonlinear programming model and solved it using the IPSO-SQP algorithm [27]. Among the many swarm intelligence optimization algorithms, the genetic algorithm is widely used due to its global optimization capability and lack of special requirements for problem convexity, making it suitable for the parameter optimization of flight profiles [28]. These studies consider the optimization and planning of trajectory vertical profiles from the perspective of environmental impact, but do not integrate the 4D trajectory optimization problem with new aviation emission reduction technologies, thus having limitations in energy conservation and emission reduction.

Focusing on new emission reduction technologies for aircraft operations, some scholars have conducted trajectory optimization research for Continuous Descent Operations (CDO). Park and Clarke used the pseudospectral method to study the multi-stage CDO trajectory generation problem with the objectives of minimizing flight time and fuel consumption [23]. Ma et al. minimized fuel consumption and noise levels in the terminal area using a pseudospectral-based CDO technique [25]. These studies confirmed the effectiveness of CDO technology in reducing emissions and noise, but they did not fully consider the practical feasibility under complex terminal area operational conditions. Raul et al. used the direct method to calculate CDO trajectories that meet the required arrival times, finding that even with the path extension strategy, CDO trajectories still offer advantages in energy savings and emission reductions compared to planned flight routes [29].

The current research on the optimization of air traffic operations and the reduction in emissions primarily concentrates on the optimization of arrival aircraft sequencing and the development of 4D trajectory optimization, which are being developed independently and in parallel. In recent years, the two have gradually converged to form a new trend in the integrated optimization of arrival sequencing and trajectory planning, which collectively facilitate the advancement of air traffic management systems. Therefore, recent research has gradually combined the optimization of arriving aircraft scheduling with CDO technology. Some scholars have generated optimized CDO trajectories by determining the Required Time of Arrival (RTA) for arriving flights, facilitating the implementation of CDO in busy terminal areas. Pawelek et al. optimized the scheduling and sequencing of arriving aircraft using a Mixed Integer Programming (MIP) method, assigned an RTA to each aircraft, and computed the CDO profile based on the RTA [30]. Toratani et al. optimized the sequencing of arriving aircraft using a rolling horizon control method and constructed an optimal control model to generate CDO trajectories [31]. These studies first optimize the sequencing and then generate the flight trajectories based on the RTA. However, due to the lack of structured route constraints, the horizontal trajectories obtained to comply with the RTA lack orderliness and are difficult to apply to actual operational requirements.

Some scholars are dedicated to researching novel arrival procedures to support the implementation of CDO in high-density terminal areas. Hong et al. introduced a Point Merge System into arrival scheduling, considering the uncertainties during continuous descent, and proposed a robust solution strategy [32]. Liang and Delahaye combined CDO with a Point Merge structure, designed a multi-layer Point Merge arrival route structure, and used a simulated annealing algorithm to optimize converging traffic trajectories [33]. This study achieved the simultaneous optimization of aircraft sequencing and trajectories but lacked theoretical model support for optimizing the vertical profile of aircraft. Based on existing airspace structures and flight procedures, other scholars have adjusted the arrival slots of aircraft by altering flight paths, thereby enhancing the success rate of CDO in the airspace of busy terminals. Sáez et al. proposed the design of an open downwind leg structure to construct a set of alternative arrival paths, combining CDO with standard arrival procedures; they used a MIP model to achieve multi-aircraft scheduling trajectory optimization [34]. Furthermore, Sáez et al. introduced a pre-sequencing area in the terminal airspace, based on extended arrival management, to implement CDO at busy airports [35]; however, the presence of the pre-sequencing area may increase flight time. These path adjustment-based methods are more aligned with actual operational needs and are easier to apply and implement in various airspace environments.

By introducing a set of alternative arrival paths based on path adjustments, the above studies have clarified that future trajectory-based operation (TBO) scheduling and sequencing are essentially multi-aircraft trajectory planning problems. However, the existing research mainly focuses on the idealized application scenarios of traffic flows entering the extended downwind leg, without fully considering flight strategies for directly entering the final approach in busy terminal areas, leading to deficiencies in improving operational efficiency and practical application. In practice, there is a real demand for direct arrival and landing from the final approach. However, under the traditional air traffic control paradigm, stepwise convergence is often employed to reduce the complexity of air traffic management; this results in the design of guiding aircraft, which could have landed directly, into the downwind leg procedure, leading to unnecessary flight distances and increased greenhouse gas emissions. This study explores the trajectory planning problem for converging aircraft based on the concept of TBO. The 4D trajectories of arriving aircraft in busy terminal areas are optimized and generated, eliminating the complexity associated with traditional controller-based decision-making. This approach enables CDO to become feasible in busy terminal areas.

To address the aforementioned limitations and to implement CDO in busy terminal areas while estimating its CO₂ reduction effects and establishing the minimum carbon emission benchmarks for arriving aircraft, we explore a low-carbon trajectory optimization method for multiple aircraft, as well as an integrated optimization strategy for sequencing and trajectory planning. The differences between this study and existing studies are reflected in two specific aspects. Firstly, addressing the limitation of traditional extended downwind leg structures that do not consider direct final approach flight strategies, a novel T-shaped arrival route structure is proposed. This study investigates coordinated trajectory planning for multiple arriving aircraft to achieve CDO sequencing, effectively reducing the overall fuel consumption and emission levels of arriving aircraft, enhancing the environmental benefits of terminal area arrival traffic flow, promoting the full automation of a low-carbon air traffic control system, and ultimately achieving trajectory-based operations. Secondly, a genetic algorithm-driven CDO 4D trajectory optimization algorithm was developed, which simultaneously achieves aircraft sequencing and trajectory planning for arrivals. Moreover, this method not only provides the most operationally efficient aircraft sequencing results, but also offers specific low-carbon flight trajectories to achieve these sequencing outcomes.

To fully leverage the energy-saving and emission-reduction benefits of CDO and to establish a minimum carbon emission baseline for multi-aircraft arrival traffic flows, we introduce a low-carbon cooperative trajectory planning method and construct a T-shaped arrival structure. This method models the multi-aircraft convergence scheduling problem as a MIP problem, optimizing sequencing, vertical profiles, and horizontal trajectories. We adopt a genetic algorithm-driven 4D trajectory optimization algorithm for CDO and use it to solve the optimal control model for the CDO of multi-aircraft vertical profiles, which leads to the formation of a set of 4D trajectory alternatives. Finally, we find the eligible trajectories using a combinatorial optimization method. This method was then applied to a real terminal airspace scenario, and experimental results demonstrated its feasibility, as well as establishing the minimum carbon emission baseline for multi-aircraft arrivals.

2. Methodology

2.1. Optimal Control Model for CDO

To achieve the objective of low-carbon 4D trajectory planning for the CDO of aircraft, we propose an optimal control model specifically designed for CDO in the vertical profile, based on the typical arrival horizontal flight path method.

2.1.1. CDO Phase

This paper presents a four-phase division of the continuous descent operation within the terminal area, as illustrated in Figure 1. Phase 1 is defined as the level flight phase, during which the aircraft maintains a constant altitude and speed. Furthermore, in accordance with the flight operations of large commercial aircraft, the corrected airspeed is limited below 250 knots when the aircraft is below 10,000 ft. Consequently, phases 2 and 3 are divided according to this constraint. Subsequently, phase 3 and phase 4 are divided according to different flap configurations.

In Figure 1, the left axis represents the flight altitude

h

; the bottom axis represents the horizontal distance

s

of the aircraft from the point approach gate (AG);

p

corresponds to the phase in Figure 1;

s^{(p)}

is the horizontal distance of the aircraft from the point AG at the end of

p

;

h_{cr}

is the cruise altitude of the aircraft upon entering the terminal area; and

h_{AG}

is the altitude restriction for the aircraft when reaching the AG.

2.1.2. Mathematical Model

(1): Objective Function

CDO has been demonstrated to have positive effects with regard to energy conservation and the reduction in emissions. Accordingly, in order to optimize the advantages of CDO, we have established three objective functions with the objectives of minimizing fuel consumption (

{\min J}_{fuel}

), minimizing flight time (

{\min J}_{time}

), and minimizing the comprehensive total cost (

{\min J}_{DOC}

), as illustrated in Equation (1).

\{\begin{cases} \min J_{fuel} = f^{(1)} \times \frac{(s^{(1)} - s^{(0)})}{v_{T}^{(1)}} + \sum_{p = 2}^{4} \int_{t^{(p - 1)}}^{t^{(p)}} f^{(p)} d t \\ \min J_{time} = t^{(4)} \\ J_{DOC} = c_{f} \cdot J_{fuel} + c_{t} \cdot J_{time} \\ \min J_{DOC} = f^{(1)} \times \frac{(s^{(1)} - s^{(0)})}{v_{T}^{(1)}} + \sum_{p = 2}^{4} \int_{t^{(p - 1)}}^{t^{(p)}} f^{(p)} d t + C I \cdot t^{(4)} \end{cases}

(1)

where

f^{(p)}

is the fuel flow rate of the

p

-phase;

v_{T}^{(1)}

is the vacuum speed of the aircraft in the leveling-off phase;

s^{(p)}

is the horizontal distance from the end of the

p

-phase to the AG point;

t^{(p)}

is the ending moment of the

p

-phase;

c_{f}

is the cost per unit mass of fuel; and

c_{t}

is the cost per unit time.

{CI = c}_{t} / c_{f}

(2): Control Variables

In the optimal control model, the aircraft state

X

can be defined by the horizontal flight distance

s

, the aircraft altitude

h

, and the vacuum speed

v_{T}

, where

X = {[s h v_{T}]}^{'}

, as demonstrated in Equation (2). In the optimal control model, the sole control variable is the slip angle

γ

\{\begin{array}{l} \frac{d v_{T}}{d t} = \frac{T - D}{m} - g \sin γ \\ \frac{d h}{d t} = v_{T} \sin γ \\ \frac{d s}{d t} = v_{T} \cos γ \end{array}

(2)

where

v_{T}

is the vacuum speed of the aircraft;

T

is the engine thrust;

D

is the drag;

m

is the mass of the aircraft;

g

is the gravitational acceleration;

γ

is the flight downward angle;

s

is the projected distance from the ground of the flight trajectory; and

h

is the flight altitude.

(3): Constraint Conditions

In consideration of the boundary constraints, the descent rate and descent speed constraints, the phase connection constraints, the control variable constraints, and the constraints of the optimal control model are demonstrated in Equation (3).

\{\begin{matrix} s^{(4)} = 0 \\ \begin{array}{l} \begin{array}{l} h_{0}^{(1)} = h_{cr} \\ h_{f}^{(2)} = 10, 000 ft \\ h_{f}^{(4)} = h_{AG} \end{array} \\ \begin{array}{l} v_{s} \leq v_{C}^{(p)} \leq v_{MO}, p \in [1, 2] \\ \begin{array}{l} v_{C}^{AG} \leq v_{C}^{(p)} \leq 250 kn, p \in [3, 4] \\ v_{f}^{(4)} = v_{C}^{AG} \end{array} \end{array} \\ γ_{\min} \leq γ \leq γ_{\max} \\ X_{f}^{(p - 1)} = X_{0}^{(p)} \end{array} \end{matrix}

(3)

where

h_{0}^{(p)}

is the initial altitude of the

p

-phase;

h_{f}^{(p)}

is the altitude at the end of the

p

-phase;

h_{cr}

is the handover altitude when the aircraft enters the terminal area;

h_{AG}

is the limiting altitude at the AG point;

v_{C}^{(p)}

is the corrected airspeed of the aircraft during the

p

-phase;

v_{s}

is the stall speed of the aircraft;

v_{MO}

is the maximum corrected airspeed of the aircraft;

v_{f}^{(p)}

is the corrected airspeed at the end of the

p

-phase;

v_{C}^{AG}

is the corrected airspeed at the AG point;

γ_{\min}

and

γ_{\max}

are the upper and lower limits of the flight slip angle; and

X_{0}^{(p)}

and

X_{f}^{(p)}

are the initial state and final state of the aircraft in the

p

-phase, respectively.

2.1.3. CDO Profile Solving Model

This paper addresses the conventional GPOPS parameter-sensitive issue by proposing a genetic algorithm-driven 4D trajectory optimization algorithm for Continuous Descent Operations. In the absence of knowledge regarding the flight time, but with a known horizontal flight distance, this paper models the optimal control problem in terms of the horizontal flight distance, thereby transforming the optimal control problem into a nonlinear optimization problem.

In order to enhance the economic efficiency, safety, and passenger comfort of aviation operations, airlines have developed operating procedures according to different aircraft types, which include clear guidance for aircraft speed control, as illustrated in Figure 2. In this study, six decision variables were selected for the simulation of the speed control process. Variable

v_{1}

represents the extreme value of corrected airspeed in phase 2; variable

d_{1}

represents the flight distance in the level flight phase; variable

d_{2}

is the flight distance from the initial speed change to

v_{1}

; variable

d_{3}

represents the flight distance to maintain

v_{1}

; variable

d_{4}

is the flight distance from the extreme value of corrected airspeed change to 250 kn; and variable

d_{5}

is the flight distance for the aircraft to maintain the corrected airspeed of 250 kn.

In accordance with the prescribed decision variables, the optimization model, based on a genetic algorithm, has been constructed as illustrated in Equation (4).

\begin{matrix} \min f * (Γ) \\ s . t . \{\begin{array}{l} \begin{array}{l} d_{1} + d_{2} + d_{3} + d_{4} + d_{5} \leq s^{(0)} - 2 \\ 1 \leq d_{q} \leq 80, q = 1, 2, 3, 4, 5 \end{array} \\ \begin{array}{l} v_{\min} \leq v_{1} \leq v_{\max} \\ d_{q} \in Z, q = 1, 2, 3, 4, 5 \\ v_{1} \in Z \end{array} \end{array} \end{matrix}

(4)

where

f * (𝛤), (𝛤) = [d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, v_{1}]

is the objective function, and

s^{(0)} - 2

indicates that the minimum horizontal flight distance in phase 4 is set to 2 km; to ensure the practicality of aircraft maneuvering, the decision variables are usually set to integer values.

In order to ascertain whether the flight slip angle is within the prescribed range, it is necessary to introduce the slip angle constraint function ξ(

γ_{k}

), as illustrated in Equation (5).

ξ (γ_{k}) = \{\begin{array}{l} 0, - 6 < γ_{k} < - 2 \\ 1, else \end{array}

(5)

As illustrated in Figure 2, the aircraft adheres to the prescribed altitude constraints at points

s^{(2)}

and

s^{(4)}

, as defined in Equation (3). It should be noted that these constraints are identical to those previously outlined in Equation (3). In order to ascertain whether the altitude

h^{(q)}

corresponding to the critical distance point

s^{(q)}, q = 2, 4

is within a specified range of its constrained altitude

h_{f}^{(q)}

, it is necessary to introduce the altitude constraint functions

τ_{1} (h^{(2)})

and

τ_{2} (h^{(4)})

. The altitude constraint functions

τ_{1} (h^{(2)})

and

τ_{2} (h^{(4)})

for the critical distance points

s^{(2)}

and

s^{(4)}

are presented in Equation (6).

\{\begin{cases} τ_{1} (h^{(2)}) = \{\begin{array}{l} 0, 10, 000 ft - δ < h^{(2)} < 10, 000 ft + δ \\ 1, else \end{array} \\ τ_{2} (h^{(4)}) = \{\begin{array}{l} 0, h_{AG} - δ < h^{(4)} < h_{AG} + δ \\ 1, else \end{array} \end{cases}

(6)

where

δ

is the setup’s allowable height tolerance and is a positive integer.

In the fitness evaluation, we incorporated the constraints of the optimal control problem as a penalty function,

Ω

, into the fitness function. This ensures that the solution is reasonable and follows the constraints. The penalty function is shown in Equation (7).

Ω = M (\sum_{k = 1}^{K} ζ (γ_{k}) + τ_{1} (h^{(2)}) + τ_{2} (h^{(4)}))

(7)

where M is a sufficiently large positive number.

Three adaptation degree functions are established, each with a different objective, namely, to minimize fuel consumption, flight time, and comprehensive total cost. These functions are presented in Equation (8).

\{\begin{cases} \begin{matrix} \min & F_{fuel} (Γ) = f^{(1)} \times \frac{d_{1}}{v_{T}^{(1)}} + \sum_{k = 1}^{K} f_{k} \cdot Δ t + Ω \end{matrix} \\ \begin{matrix} \min & F_{time} (Γ) = t_{K} + Ω \end{matrix} \\ \begin{matrix} \min & F_{DOC} (Γ) = f^{(1)} \times \frac{d_{1}}{v_{T}^{(1)}} + \sum_{k = 1}^{K} f_{k} \cdot Δ t + C I \cdot t_{K} + Ω \end{matrix} \end{cases}

(8)

The application of the GA algorithm effectively searches for the variables

(d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, v_{1})

that satisfy the constraints, thereby determining the corrected airspeed at key points and its corresponding horizontal flight distance from the runway threshold. The obtained distance variables

d_{1} ~ d_{5}

were discretized, and linear interpolation was applied to compute the corrected airspeed corresponding to each distance step ds, resulting in a set of discrete data points approximating the actual corrected airspeed curve. This is a numerical solving algorithm that generates a set of discrete data points and iteratively computes the final result [36]. This profile was then used to perform the adaptation evaluation according to Equation (8), as well as to make dynamic adjustments to the start descent time, velocity profile, and descent angle for different optimization objectives, ultimately leading to the optimal continuous descent operation profile.

2.2. Low-Carbon Multi-Arrival Aircraft 4D Trajectory Cooperative Planning

2.2.1. T-Shaped Arrival Routes

The arrival routes to terminal airspace vary at different airports, but the primary conflicts in the terminal area can be summarized as the same traffic flow trailing interval conflict and the different traffic flow convergence conflict. Figure 3 illustrates the simplified arrival routes of a busy terminal area, exemplifying the potential conflicts inherent to typical routes such as STAR-01, STAR-02, STAR-03, and STAR-04. In these routes, aircraft entering at the same handover point form a single traffic flow along a designated route. In Figure 3, the gray dashed line represents the terminal area boundary, the black solid line represents the standard arrival route,

P_{AG}

represents the approach gate position,

P_{ent}

represents where the air traffic controller begins deploying the landing interval,

P_{con}

represents where different traffic flows converge, and

P_{in}

represents the handover point position.

In high-density operational scenarios, implementing open structured routes effectively reduces arrival path uncertainty, while simultaneously decreasing additional fuel consumption and aircraft emissions. This has the beneficial effect of improving the environmental management of the terminal area. Typically, the entry point of an open structured route is situated at

P_{ent}

, the point at which air traffic controllers commence the deployment of landing intervals, as illustrated in Figure 3.

Aircraft following the traditional extended downwind procedure achieve orderly convergence at AG, as shown in Figure 4a, making AG a suitable reference point for aircraft sequencing. In this procedure, arriving aircraft from different directions complete their flight paths along the standard instrument arrival. However, in the busy terminal area environment depicted in Figure 3, aircraft arriving from STAR-03 and STAR-04 are required to fly significantly longer distances to join the downwind leg (as illustrated in Figure 4a), resulting in a notable decrease in flight efficiency. In fact, aircraft arriving from STAR-03 and STAR-04 can directly enter the final approach from specific points, avoiding the need to execute the full downwind leg and base leg, thereby improving airspace utilization. Consequently, it is necessary to improve the traditional extended downwind procedure to meet the operational demands of busy terminal areas. Compared to the traditional procedure, the T-shaped procedure introduces entry points into the final approach and provides a complete flight path for the final approach, better aligning with actual airspace requirements, as illustrated in Figure 4b.

2.2.2. Alternative Route Generation

To achieve the pre-tactical planning of arrival routes, this study employs an equidistant segmentation method for the downwind leg to create several alternative routes and solves for the optimized descent profiles of these alternative routes, thereby constructing a set of alternative green 4D trajectories for arriving aircraft. The segmented and constructed set of alternative routes

J

is illustrated in Figure 5. The red and blue dashed lines in Figure 5 represent the flight paths of the aircraft when selecting an alternative route.

Assuming the length of the left downwind leg is

D

and the length of the right downwind leg is

D^{'}

, with the segment lengths between adjacent division points on the left and right downwind legs being

∆ D

and

∆ D^{'}

, respectively, the number of division points on the left downwind leg

n

(corresponding to point

P_{1}, \dots, P_{n}

in Figure 5) and the number of division points on the right downwind leg

n^{'}

(corresponding to point

P_{n + 1}, \dots, P_{n + n^{'}}

in Figure 5) can, thus, be determined.

Taking the terminal area operation scenario shown in Figure 3 as an example, the alternative arrival route set

J

j \in J = {1, 2, \dots, n, n + 1, \dots, n + n^{'}, \dots, n + n^{'} + 2}

is constructed based on the division points and the length of each segment. Among them, there are a total of

n

alternative routes on the left downwind leg,

n^{'}

alternative routes on the right downwind leg, and 2 direct final approach routes. When

j \in {1, 2, \dots, n}

, it indicates alternative arrival routes related to the left downwind leg, with route

j

consisting of a sequence of points

P_{in, 01} P_{1} \dots P_{j} P_{AG}

, as shown by the red route in Figure 5 where j = 2. When

j \in {n + 1, n + 2, \dots, n + n^{'}}

, it indicates alternative arrival routes related to the right downwind leg, with route

j

consisting of a sequence of points

P_{in, 02} P_{n + 1} \dots P_{j} P_{AG}

, as shown by the blue route in Figure 5 where

j = n + 3

. When

j \in {n + n^{'} + 1, n + n^{'} + 2}

, it indicates alternative arrival routes that fly directly along the final approach, with route

j

consisting of a sequence of points

P_{in, 03} P_{ent} P_{AG}

P_{in, 04} P_{ent} P_{AG}

By employing the equidistant segmentation of the downwind leg, it is possible to provide multiple selectable horizontal routes for each aircraft entering the downwind leg. For each alternative route

j

, the optimal vertical profile of the aircraft flying along that route can be determined through the application of an optimal control model. The set of horizontal alternative routes and their corresponding optimal vertical profiles collectively constitute a set of alternative 4D trajectories, as illustrated in Figure 6.

Figure 7 illustrates the correspondence between key waypoints

P_{in}

P_{con}, P_{ent}, and P_{AG}

, and each phase of the CDO trajectory. The alternative arrival routes are given, meaning that the distances from aircraft

i

to each critical waypoint on alternative route

j

are known. According to the calculation formula in [36] and the model proposed in Section 2.1, the times

t_{ij}^{(con)}

t_{ij}^{(ent)}

, and

t_{ij}^{(4)}

for aircraft

i

to arrive at each critical waypoint from the handover point

P_{in}

on alternative route

j

can be calculated from the known distances

s_{ij}^{(con)}

s_{ij}^{(ent)}

, and

s_{ij}^{(0)}

, as shown in Figure 7.

In Figure 7, the left axis represents the flight altitude h; the bottom represents the continuous descent operation time

t

of the aircraft; the top axis represents the horizontal distance

s

of the aircraft from point AG;

t_{ij}^{(p)}

represents the flight time of aircraft

i

from the handover point to the end of the

p

-phase;

t_{ij}^{(4)}

represents the flight time of aircraft

i

from the handover point

P_{in}

to the approach gate

P_{AG}

on alternative route

j

;

s_{ij}^{(p)}

represents the horizontal distance of aircraft

i

from point AG at the end of

p

-phase on alternative route

j

;

s_{ij}^{(con)}

and

s_{ij}^{(ent)}

represent the horizontal distances of aircraft

i

from point AG at key waypoints

P_{con}

and

P_{ent}

on alternative route

j

, respectively; and

t_{ij}^{(con)}

and

t_{ij}^{(ent)}

represent the flight times from the handover point to the key waypoints

P_{con}

and

P_{ent}

on alternative route

j

, respectively.

2.2.3. Multi-Aircraft 4D Trajectory MIP Planning Model

Assuming that

I

is the set of aircraft entering the terminal area,

i \in I = {1, 2, \dots, N_{1}, N_{1} + 1, \dots, N_{1} + N_{2}, N_{1} + N_{2} + 1, \dots, N_{1} + N_{2} + N_{3}, N_{1} + N_{2} + N_{3} + 1, \dots, N_{1} + N_{2} + N_{3} + N_{4}}

. The set

I

is divided into four categories—

{I_{1}, I_{2}, I_{3}, I_{4}} \subseteq I

—based on the aircraft’s entry position into the terminal area. Specifically, I₁ = {1, 2, …, N₁}; I₂ = {N₁ + 1, N₁ + 2, …, N₁ + N₂}; I₃ = {N₁ + N₂ + 1, N₁ + N₂ + 2,…, N₁ + N_{2 +} N₃}; and I₄ = {N₁ + N₂ + N₃ + 1, N₁ + N₂ + N₃ + 2, …, N₁ + N₂ + N₃ + N₄}. When

i \in I_{1}

, it represents aircraft entering the left downwind leg via their arrival routes from the corresponding handover point (corresponding to STAR-01 in Figure 3), with a total of

N_{1}

aircraft. When

i \in I_{2}

, it represents aircraft entering the right downwind leg via their arrival routes from the corresponding handover point (corresponding to STAR-02 in Figure 3), with a total of

N_{2}

aircraft. When

i \in {I_{3}, I_{4}}

, it represents aircraft flying directly to the final approach via their arrival routes from the corresponding handover points (corresponding to STAR-03 and STAR-04 in Figure 3), with a total of

{(N}_{3} + N_{4})

aircraft.

We defined {

r_{i}, c_{i}

} as the decision variables of the MIP planning model, where

r_{i} = j

represents the alternative route number

j

assigned to aircraft

i

. The assignment rules follow that aircraft entering the left or right downwind legs can only choose their respective alternative routes, while aircraft flying directly to the final approach can only fly along the preset final approach routes.

c_{i}

represents the number of standard holding patterns assigned to the aircraft. The coding mechanism for decision variables adopts decimal encoding, where each chromosome consists of

2 (N_{1} + N_{2} + N_{3} + N_{4})

genes. The first

(N_{1} + N_{2} + N_{3} + N_{4})

genes of the chromosome represent the alternative route numbers assigned to the aircraft, and the latter

(N_{1} + N_{2} + N_{3} + N_{4})

genes represent the number of standard holding patterns assigned to the aircraft. The chromosome model is shown in Figure 8.

In accordance with the specified chromosome model, the trajectory planning model, which is based on a genetic algorithm, has been constructed as illustrated in Equation (9).

\begin{array}{l} \min f * (R, C) \\ s . t \{\begin{array}{l} 1 \leq r_{i} \leq n, i \in {1, 2, \dots, N_{1}} \\ n + 1 \leq r_{i} \leq n + n^{'}, i \in {N_{1} + 1, N_{1} + 2, \dots, N_{1} + N_{2}} \\ r_{i} = n + n^{'} + 1, i \in i \in {N_{1} + N_{2} + 1, \dots, N_{1} + N_{2} + N_{3}} \\ r_{i} = n + n^{'} + 2, i \in {N_{1} + N_{2} + N_{3} + 1, \dots, N_{1} + N_{2} + N_{3} + N_{4}} \\ 0 \leq c_{i} \leq 7, i \in {1, 2, \dots, N_{1} + N_{2} + N_{3} + N_{4}} \\ r_{i} \in Z \\ c_{i} \in Z \end{array} \end{array}

(9)

where

f * (R, C), R = [r_{1}, r_{2}, \dots, r_{N_{1} + N_{2} + N_{3} + N_{4}}], C = [c_{1}, c, \dots, c_{N_{1} + N_{2} + N_{3} + N_{4}}]

is the objective function aimed at minimizing the total flight time of arriving aircraft within the terminal area;

r_{1} ~ r_{N_{1}}

represents the alternative route numbers assigned to aircraft entering the left downwind leg;

r_{N_{1} + 1} ~ r_{N_{1} + N_{2}}

represents the alternative route numbers assigned to aircraft entering the right downwind leg;

r_{N_{1} + N_{2} + 1} ~ r_{N_{1} + N_{2} + N_{3}}

represents the alternative route numbers assigned to aircraft flying directly to the final approach via STAR-03;

r_{N_{1} + N_{2} + N_{3} + 1} ~ r_{N_{1} + N_{2} + N_{3} + N_{4}}

represents the alternative route numbers assigned to aircraft flying directly to the final approach via STAR-04; and

c_{1} ~ c_{N_{1} + N_{2} + N_{3} + N_{4}}

represents the number of standard holding patterns assigned to each aircraft. According to the airline’s operations manual, each aircraft must reserve 30 min of holding fuel, and therefore the maximum number of holding patterns for each aircraft is limited to 7. Variables

r_{1} ~ r_{N_{1} + N_{2} + N_{3} + N_{4}}

and

c_{1} ~ c_{N_{1} + N_{2} + N_{3} + N_{4}}

are set as integer values.

In order to maximize the arrival efficiency of the traffic flow and minimize the total flight time of the arriving aircraft within the terminal area while satisfying the constraints, the objective function is shown in Equation (10).

\min Z = \sum_{i \in I} \sum_{r_{i} \in J} (240 c_{i} + t_{i r_{i}}^{(4)})

(10)

where 240

c_{i}

denotes the waiting time of aircraft

i

at the terminal area handover point, with a standard waiting time of 1 circle of 240s;

t_{i r_{i}}^{(4)}

denotes the flight time of aircraft

i

on the alternative route

r_{i}

, which can be derived from the calculations in [34].

In the multi-aircraft 4D trajectory cooperative planning problem, it is crucial to ensure the flight safety of each aircraft within the terminal zone. The main conflict constraints in the terminal area include the trailing interval constraint of the same traffic stream and the convergence conflict constraint of different traffic streams. The time for aircraft

i

to reach each key waypoint can be calculated from the decision variable

{r_{i,} c_{i}}

, which is used to determine whether there is a conflict between aircraft, as shown in Equation (11).

\{\begin{array}{l} a_{i}^{(0)} = t_{i r_{i}}^{in} + 240 c_{i} \\ a_{i}^{con} = t_{i r_{i}}^{in} + 240 c_{i} + t_{i r_{i}}^{(con)} \\ a_{i}^{ent} = t_{i r_{i}}^{in} + 240 c_{i} + t_{i r_{i}}^{(ent)} \\ a_{i}^{AG} = t_{i r_{i}}^{in} + 240 c_{i} + t_{i r_{i}}^{(4)} \end{array}

(11)

where

t_{i r_{i}}^{in}

represents the time aircraft

i

arrives at the terminal area handover point;

a_{i}^{(0)}

represents the time aircraft

i

begins flying along the arrival route;

a_{i}^{con}

represents the time aircraft

i

arrives at the convergence point

P_{con}

with other traffic flows;

a_{i}^{ent}

represents the time aircraft

i

arrives at the entry point

P_{ent}

of the downwind or final approach leg in the T-shaped structure;

a_{i}^{AG}

represents the time aircraft

i

arrives at the approach gate

P_{AG}

; and

t_{i r_{i}}^{(con)}

t_{i r_{i}}^{(ent)}, t_{i r_{i}}^{(4)}

, respectively, represent the flight times of aircraft

i

on alternative route

j

from the handover point to each key waypoint.

In order to guarantee the security of aircraft flight operations, the regulations mandate the implementation of aircraft trailing interval constraints within the terminal area. Consequently, the introduction of the same route non-surpassing constraint function

α_{1} (i)

, the public route non-surpassing constraint function

α_{2} (i)

, and the entrance interval constraint function

α_{3} (i)

is warranted, as illustrated in Equation (12).

\{\begin{cases} α_{1} (i) = \{\begin{array}{l} 0, (a_{i}^{(0)} - a_{w}^{(0)}) (a_{i}^{ent} - a_{w}^{ent}) > 0, \forall i \in I_{q}, w \in I_{q} \ {i}, q \in {1, 2} \\ 0, (a_{i}^{(0)} - a_{w}^{(0)}) (a_{i}^{con} - a_{w}^{con}) > 0, \forall i \in I_{q},, w \in I_{q} \ {i}, q \in {3, 4} \\ 1, else \end{array} \\ α_{2} (i) = \{\begin{array}{l} 0, (a_{i}^{con} - a_{w}^{con}) (a_{i}^{ent} - a_{w}^{ent}) > 0 \\ 1, else \end{array}, \forall i \in I_{3} \cup I_{4}, w = (I_{3} \cup I_{4}) \ {i} \\ α_{3} (i) = \{\begin{array}{l} 0, |a_{i}^{ent} - a_{w}^{ent}| > S_{i w}, \forall i \in I_{q}, w \in I_{q} \ {i}, q \in {1, 2} \\ 0, |a_{i}^{ent} - a_{w}^{ent}| > S_{i w}, \forall i \in I_{3} \cup I_{4}, w = (I_{3} \cup I_{4}) \ {i} \\ 1, else \end{array} \end{cases}

(12)

where

S_{iw}

is used to denote the safe separation standard that is to be followed between aircraft

i

and aircraft

w

There is a possibility of conflicts when aircraft from different traffic flows converge, and, similarly, there is a possibility of conflicts at the approach gate between aircraft arriving from the left and right extended downwind legs and those flying directly to the final approach. To address these two potential conflicts, we introduce traffic flow convergence conflict constraint functions

ψ_{1} (i)

and

ψ_{2} (i)

, as shown in Equation (13).

\{\begin{cases} ψ_{1} (i) = \{\begin{array}{l} 0, |a_{i}^{con} - a_{w}^{con}| \geq S_{i w} \\ 1, else \end{array}, \forall i \in I_{3} \cup I_{4}, w = (I_{3} \cup I_{4}) \ {i} \\ ψ_{2} (i) = \{\begin{array}{l} 0, |a_{i}^{AG} - a_{w}^{AG}| \geq S_{i w} \\ 1, else \end{array}, \forall i \in I, w = I \ {i} \end{cases}

(13)

In the T-shaped structure, aircraft flying directly to the final approach have landing priority. If an aircraft

i

entering the downwind leg arrives at the approach gate before an aircraft

w

flying directly to the final approach, and the time interval between their arrivals at the approach gate is less than the safety separation standard, the flight time of aircraft

i

on the downwind leg should be extended to ensure that aircraft

w

flying directly to the final approach has landing priority, as shown in Figure 9.

Unlike traditional scheduling models based on sequential optimization, the MIP planning model proposed in this study is based on trajectory planning. It selects the optimal trajectory from the set of alternative trajectories for each aircraft that meets safety constraints and optimization objectives, thereby forming the convergence scheduling result. Since the flight time for aircraft entering the downwind leg, even when extended to the longest alternative route, is still less than the time it takes for an aircraft flying directly to the final approach to complete one standard holding pattern, the algorithm prioritizes solutions that adopt the extended downwind leg strategy to meet the optimization goal of minimizing the total flight time of arriving aircraft within the terminal area. Thus, the MIP model automatically implements the final approach priority strategy at the algorithmic level, without the need for additional constraint conditions.

To ensure that the genetic algorithm can effectively search for solutions that meet safety constraints, the constraints of the MIP planning model are incorporated as penalty terms into the objective function. This constructs a fitness function aimed at minimizing the total flight time; here, only the fitness function with the objective of minimizing total flight time is provided. The fitness functions with the objectives of minimizing fuel consumption and minimizing overall total cost are similar to Equation (14).

F (R, C) = \sum_{i \in I} (t_{i r_{i}}^{(4)} + 240 c_{i}) + M \sum_{i \in I} (α_{1} (i) + α_{2} (i) + α_{3} (i) + ψ_{1} (i) + ψ_{2} (i))

(14)

where

M

is a sufficiently large positive number.

3. Experimental Results and Discussion

3.1. Arrival Routes and Available Data

Taking the terminal area of ZSQD as the background for the case analysis, the standard instrument arrival procedure is shown in Figure 10. There are four handover points within the terminal area of ZSQD—WFC, HCH, P74, and VEVED. The traffic flow arriving from WFC enters the left downwind leg along the arrival route WFG-91A; the traffic flow arriving from HCH enters the right downwind leg along the arrival route HCH-91A; and the traffic flow arriving from P74 and VEVED flies directly to the final approach.

In conjunction with the existing standard instrument arrival procedures for the ZSQD terminal area, and to ensure the safe separation of aircraft arriving from all directions, the extended downwind leg structure is designed as follows: the length of the downwind leg is 15 km, the length of the left base leg is 10 km, and the length of the right base leg is 12 km, as shown in Figure 11.

In this study, the downwind leg is divided into equal segments to provide sufficient trajectory options for flexible route adjustments, while avoiding excessive route complexity. The interval between equidistant division points is set to 1 km, as shown in Figure 11.

We use the Aviation Environmental Design Tool (AEDT 3d) developed by the Federal Aviation Administration to calculate the fuel consumption and gas emissions of actual flight trajectories. The emission indices for CO₂ and SO₂ are referenced from [37], while the NO_x emission index is calculated using the Boeing Fuel Flow Method 2 (BFFM2), as detailed in [38]. Additionally, to calculate emissions, we also conducted a reference study based on [39]. The historical trajectories of 22 arriving aircraft at ZSQD between 17:00 and 18:00 on 6 December 2023 were selected for analysis. The specific data are presented in Table 1. The statistical analysis revealed that the actual total flight time of the 22 aircraft in the terminal area was 416.2 min, the total fuel consumption was 9071.44 kg, the total CO₂ emission was 28,575.04 kg, the total SO₂ emission was 7257.15 g, and the total NO_x emission was 110,094.30 g.

3.2. Alternative Trajectory Analysis

In this study, experiments were conducted in the MATLAB 2022b environment, and the flight times of 22 aircraft traversing different alternative routes were utilized as inputs to the trajectory planning model, which was then solved using the GA algorithm. In this instance, the population size is set to 200, the maximum number of evolutionary generations is set to 500, the crossover operator employs a single-point crossover method with a crossover probability of 0.8, and the variational operator utilizes a single-point variational method with a variational probability of 0.3.

(1): 4D Trajectory Alternative Set Analysis

In this example, we consider a flight arriving from the WFG direction with a B737-800 aircraft. We present the results of optimizing the vertical profile of this flight under 16 alternative routes, with the objective of minimizing the integrated total cost (the 16 alternative routes for aircraft arriving from the WFG direction are labeled as P₁ to P₁₆ in Figure 11). At the terminal handover point, the altitude of the flight is 4300 m, the speed is 260 knots, and the flight downturn angle is

γ \in [2, 6]

. The fuel consumption and gas emissions of the optimized trajectory were calculated by solving the optimal control model. A comparison of the actual and optimized vertical profiles for the B737-800 is presented in Figure 12.

Table 2 presents a comparison of fuel consumption and flight time between the actual trajectory and the optimized one under different alternative routes (16 alternative routes for the B737-800 arriving from the WFG direction). As evidenced in Table 2, the CDO time and fuel consumption of all alternative arrival routes are superior to the actual situation, exhibiting a reduction of 16.8–31.3% in flight time and 3.6–27.3% in fuel consumption and gas emission.

(2): Analysis of the trajectory planning results of the 4D trajectory alternative set for different optimization objectives

Figure 13 illustrates the flight time and fuel consumption distribution of 22 aircraft under different optimization objectives. In comparison to the actual flights, the trajectory planning method has been demonstrated to effectively reduce fuel consumption among arriving aircraft, while also significantly reducing the total flight time. This has the effect of improving overall flight operation efficiency.

The results of the experimental trials conducted with varying optimization objectives are presented in Table 3. When the minimum time is the optimization objective, a reduction of 27.3% in the total flight time is observed, accompanied by a reduction of 19.9% in fuel consumption, CO₂ emission, and SO₂ emission, and a reduction of 20.4% in NO_x emission. When the integrated total cost is minimized as the optimization objective, the total operating time is reduced by 22.4%; fuel consumption, CO₂ emission, and SO₂ emission are reduced by 22.9%; and NO_x emission is reduced by 23.7%. When minimizing fuel consumption was the optimization objective, a reduction of 19.3% in total operating time was observed, accompanied by a reduction of 23.4% in fuel consumption, CO₂ emissions, and SO₂ emissions, and a reduction of 24.2% in NO_x emissions. While the objective of minimizing flight time resulted in a notable reduction in total flight time, it proved ineffective in enhancing fuel efficiency and mitigating environmental impacts. Conversely, the strategy that targeted minimum fuel consumption demonstrated a superior performance in reducing fuel and gas emissions, albeit at the cost of time efficiency. In contrast, the objective of minimum total integrated cost achieved a balance between time efficiency and fuel efficiency, effectively reducing flight time, fuel consumption, and environmental impact.

In conclusion, in order to achieve an equilibrium between flight efficiency, energy saving, and emission reduction, this paper has elected to minimize the comprehensive total cost as the optimization objective in order to generate an alternative set of 4D trajectories. Furthermore, the cost index recommended by the airlines was adopted in order to set the CI [40,41], and the vertical profile optimization of the alternative routes was carried out based on the optimal control model of the vertical profiles proposed in Section 2.

3.3. Convergence Trajectory Scheduling Results and Horizontal Trajectory Analysis

A multi-aircraft trajectory planning experiment was conducted using the 4D trajectory alternative set generated with the objective of minimizing the overall total cost. Based on the experimental results, a space–time diagram for multi-aircraft trajectory planning was plotted, showing the waiting times and selected alternative routes for each aircraft, as illustrated in Figure 14. The number above the “○” indicates the route number chosen by the aircraft; the red solid line indicates that the aircraft had no waiting time at the terminal area handover point; and the red dashed line indicates that the aircraft had a waiting time at the terminal area handover point.

From Figure 14, it can be seen that, to ensure safe separation between aircraft, the algorithm flexibly selected the most suitable alternative route for each aircraft, effectively enhancing scheduling efficiency while ensuring no conflicts within the traffic flow and at convergence points, thereby guaranteeing scheduling safety. The red box in Figure 14 highlights two aircraft arriving simultaneously from different directions, with one entering the downwind leg and the other flying directly to the final approach. The aircraft entering the downwind leg from the WFG direction adopts the extended downwind leg strategy, giving landing priority to the aircraft flying directly to the final approach from the VEVED direction. This demonstrates the effectiveness of the algorithm in implementing the final approach priority principle, reducing unnecessary holding patterns in the terminal area, and effectively improving airspace utilization.

Figure 15 compares the actual horizontal trajectories of 22 aircraft with the optimized horizontal trajectories. In actual operations, air traffic controllers were unable to fully utilize the sequencing potential of the downwind leg, resulting in actual trajectories that exhibit lower orderliness and higher uncertainty. In contrast, the optimized routes effectively coordinate the arrival of arriving flights, leading to more structured and orderly flight trajectories, and significantly reducing the complexity of arrival operations in the terminal airspace.

3.4. Vertical Profile and Fuel Consumption Comparison

Figure 16 presents a comparison between the actual vertical profiles of the 22 aircraft and the optimized vertical profiles. The actual trajectory descent profile exhibits a considerable number of level flight segments within the low-altitude layer, necessitating frequent adjustments to flight speed to guarantee the safety interval and landing sequence. Following optimization, the descent apex of the aircraft is situated at a greater altitude, thereby converting the low-altitude level flight into a high-altitude level flight. This avoids the need for a midway thrust reduction descent and level flight operation, and results in a reduction in fuel consumption and CO₂ emissions.

Figure 17 compares the actual fuel profile of Aircraft 11 with the optimized fuel profile. It can be seen that in the actual descent profile, the fuel flow rate remains high during low-altitude level flight, while the trajectory obtained by the planning algorithm achieves continuous descent, maintaining a lower fuel flow rate. In actual operations, the fuel consumption for this flight was 391.11 kg, whereas the fuel consumption under the algorithmic planning was 303.22 kg—a reduction of 22.5%.

Figure 18 presents the fuel consumption comparison curves for 22 flights, with the exception of Aircraft 20. The remaining 21 aircraft demonstrate a greater degree of fuel efficiency under the trajectory planning algorithm in comparison to the actual flights.

3.5. Analysis of High-Traffic Scenarios

To comprehensively evaluate the performance of the trajectory planning algorithm in a high-density terminal area, this study increased the number of simulated flights based on the actual number of arriving flights within one hour. By analyzing the data of actual arriving flights, we calculated the altitude levels at which flights crossed the terminal area boundary and the proportion of different aircraft types, and we also recorded the hourly flight traffic over a 24 h period. Based on these statistical data, we used random sampling to construct a high-traffic simulation flight plan, generating simulated flight schedules for high-traffic scenarios within 1 h, simulating scenarios with 30, 40, and 45 arriving flights per hour.

When the number of arriving aircraft exceeds 45, the result of the planning algorithm indicates that a significant number of flights are circling and waiting at the handover point. This is accompanied by a notable increase in the value of the fitness function, reaching the upper limit of the penalty term. This suggests that the airspace has reached its maximum capacity. The maximum landing capacity of ZSQD is 24 flights per hour. The trajectory planning algorithm results in a maximum landing capacity of 45 flights per hour, which is significantly higher than the maximum landing capacity under the current operational mode. This demonstrates the efficacy of the algorithm in enhancing the terminal area capacity and operational efficiency.

Figure 19 shows the distribution of flight times under different numbers of arriving flights, with the actual trajectory data derived from the historical trajectories of 22 arriving aircraft. It can be seen that despite the increase in the number of arriving flights, the average flight time has hardly increased. The results indicate that the arriving flight trajectories have been effectively planned, and the traffic congestion in the busy terminal area, caused by air traffic controllers’ instructions, has been significantly improved.

Table 4 presents a comparative analysis of the total flight time, fuel consumption, and gas emissions between the actual trajectory and the planned trajectory for various arrival flights. In comparison to the air traffic controller’s scheduling scheme, the planned convergence scheduling scheme demonstrates a notable reduction in both time and fuel consumption. The average reduction in flight time is 21.7%, while the average reductions in fuel consumption, CO₂ emissions, SO₂ emissions, and NO_x emissions are 23.2%, 23.7%, 23.7%, and 23.7%, respectively. Meanwhile, Table 4 also presents the benchmarks of carbon emissions when the number of arriving aircraft is 30, 40, and 45, which are 29,508.32 kg, 40,254.80 kg, and 46,221.12 kg, respectively.

4. Conclusions

This study aimed to enhance the emission reduction potential of CDO by focusing on cooperative low-carbon trajectory planning for multiple aircraft in busy terminal areas. Two major innovations are presented. First, a T-shaped arrival route structure is proposed to address the limitations of traditional extended downwind approaches in high-traffic terminals. This structure integrates arrival flows from various directions and generates horizontal alternative routes, allowing for optimized sequencing and conflict-free 4D trajectory planning. Second, we propose a cooperative low-carbon trajectory planning method for multiple arriving aircraft in CDO, which simultaneously achieves aircraft sequencing and trajectory planning. The method not only provides the optimal aircraft sequencing for maximum operational efficiency, but also generates the specific low-carbon flight trajectories to achieve that sequencing.

The cooperative low-carbon trajectory planning model for multiple aircraft estimates the carbon emission baseline for arriving aircraft operations and reveals the significant potential for further emission reductions. In this study, the proposed cooperative low-carbon trajectory planning method for CDO of multiple arriving aircraft was applied to a real terminal airspace environment, and its applicability was validated through experiments. The experimental results indicate that strategies focusing solely on minimizing flight time and fuel consumption have limitations in reducing both fuel consumption and total flight time. In contrast, when the objective was total cost minimization, the total operation time was reduced by 22.4%; fuel consumption, CO₂ emissions, and SO₂ emissions were reduced by 22.9%; and NO_x emissions were reduced by 23.7%. This method effectively balances time efficiency and fuel efficiency. Finally, the experiments also revealed the baseline minimum carbon emissions when the numbers of arriving aircraft were 30, 40, and 45, which were 29,508.32 kg, 40,254.80 kg, and 46,221.12 kg, respectively.

The vertical profile optimal control model is based on the ideal assumption of no wind conditions. To improve the practical accuracy of the optimization method, future research could incorporate wind field models to simulate actual flight environments, achieving more precise trajectory optimization results. In addition, the impact of the temporal distribution of arrivals on carbon emissions will be investigated.

Author Contributions

Conceptualization, C.F. and C.W.; methodology, C.W.; software, C.F. and H.C.; validation, C.F. and H.C.; formal analysis, C.F and H.C.; investigation, C.W. and C.F.; resources, C.W.; data curation, C.F. and C.X.; writing—original draft preparation, C.F. and C.W.; writing—review and editing, C.F. and J.W.; visualization, C.F. and H.C.; supervision, C.W.; project administration, C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the key project of applied basic research multi-investment fund of Tianjin municipal (No. 21JCZDJC00780).

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors upon request.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. A typical CDO process of an arrival aircraft.

Figure 2. The explicit guidance for aircraft speed control.

Figure 3. A simplified standard terminal arrival route for busy terminal areas.

Figure 4. (a) Traditional open path arrival route structure to downwind leg; (b) T-shaped arrival route structure.

Figure 5. Alternative route assembly schematic.

Figure 6. Alternative set of 4D trajectories based on downwind leg segmentation.

Figure 7. Correspondence between flight distance and time of critical waypoint.

Figure 8. The chromosome model of decision variables in the MIP planning model.

Figure 9. Diagram illustrating priority landing for aircraft on a direct final approach.

Figure 10. Standard arrival flight procedures of ZSQD TMA.

Figure 11. Alternative routes of T-shaped arrival route structure (schematic diagram not to scale).

Figure 12. Actual and optimized vertical profile of B737-800.

Figure 13. Variation in flight time and fuel consumption with different optimization objectives. (a) Flight time distribution; (b) fuel consumption distribution.

Figure 14. Space–time diagram of multi-aircraft trajectory planning. Analysis of selected alternative routes and waiting times with the objective of minimizing total cost.

Figure 15. Horizontal trajectory comparison. (a) Actual horizontal trajectories; (b) optimized horizontal trajectories.

Figure 16. Vertical profile comparison. (a) Actual trajectory vertical profile; (b) optimized altitude profile.

Figure 17. Fuel flow comparison of Aircraft 11. (a) Actual trajectory fuel profile; (b) optimized trajectory fuel profile.

Figure 18. Comparison of fuel consumption of the 22 aircraft.

Figure 19. The distribution of flight times under different numbers of arrival flights.

Table 1. Initial conditions of 22 arrival aircraft.

Aircraft ID	Type	Entering Waypoint	Entering Time	Entering Altitude/m	Calibrated Airspeed/kn
1	B737	P74	17:00:00	6900	310
2	B738	HCH	17:00:35	4400	240
3	B737	VEVED	17:05:00	6300	320
4	B738	WFG	17:05:05	5100	280
5	B738	P74	17:07:00	6600	320
6	A320	WFG	17:07:00	5100	280
7	B738	HCH	17:08:00	4500	250
8	B738	WFG	17:10:00	5100	270
9	A320	HCH	17:11:00	4100	240
10	B738	VEVED	17:11:30	6400	300
11	A320	WFG	17:16:00	5100	260
12	A320	WFG	17:17:30	5100	270
13	A319	VEVED	17:19:30	6900	320
14	A320	VEVED	17:28:00	6900	300
15	A320	HCH	17:29:00	4400	250
16	B738	HCH	17:30:30	5000	270
17	B738	WFG	17:34:30	4300	260
18	A333	VEVED	17:41:00	6800	320
19	A320	WFG	17:48:00	4700	250
20	B738	HCH	17:49:10	4700	260
21	A332	P74	17:55:00	7500	320
22	A320	WFG	17:56:00	4800	250

Table 2. Comparison of actual and optimized fuel consumption and flight time.

Air Route	Flight Time/s	Fuel Consumption/kg	CO₂ Emission/kg	SO₂ Emission/g	NO_x Emission/g
Actual route	1240	482.10	1518.62	385.68	4387.11
j = 1	852	350.36	1103.63	280.29	3188.28
j = 2	864	354.39	1116.33	283.51	3224.95
j = 3	875	361.49	1138.69	289.19	3289.56
j = 4	886	368.66	1161.28	294.93	3354.81
j = 5	897	378.8	1193.22	303.04	3447.08
j = 6	910	383.18	1207.02	306.54	3486.94
j = 7	926	396.42	1248.72	317.14	3607.42
j = 8	934	400.48	1261.51	320.38	3644.37
j = 9	946	413.17	1301.49	330.54	3759.85
j = 10	958	413.70	1303.16	330.96	3764.67
j = 11	974	420.77	1325.43	336.62	3829.01
j = 12	986	431.65	1359.70	345.32	3928.02
j = 13	999	443.35	1396.55	354.68	4034.49
j = 14	1010	450.42	1418.82	360.34	4098.82
j = 15	1021	457.55	1441.28	366.04	4163.71
j = 16	1032	464.71	1463.84	371.77	4228.86

Table 3. Flight time, fuel consumption, and gas emission of planning trajectories with different optimization objectives.

Optimization Objective	Total Time/min	Fuel Consumption/kg	CO₂ Emission/kg	SO₂ Emission/g	NO_x Emission/g
Minimum flight time	302.7	7217.66	22,735.63	5774.13	87,607.01
The combined total cost is minimal	322.6	6956.82	21,913.983	5565.46	84,031.04
Minimum fuel consumption	335.5	6949.90	21,892.19	5559.92	834,03.09

Table 4. The benchmarks of fuel consumption and gas emission for different traffic flows.

Number of Arriving Aircraft	Scheduling Method	Total Time/min	Fuel Consumption/kg	CO₂ Emission/kg	SO₂ Emission/g	NO_x Emission/kg
30	Air traffic controller direction	559.5	12,315.65	38,794.30	9852.52	143.16
30	Planned trajectories	436.8	9367.72	29,508.32	7494.18	109.89
40	Air traffic controller direction	751.8	16,939.51	53,359.46	13,551.61	200.35
40	Planned trajectories	583.5	12,879.30	40,254.80	10,303.44	149.59
45	Air traffic controller direction	833.0	18,778.03	59,150.79	15,022.42	220.00
45	Planned trajectories	658.8	14,673.37	46,221.12	11,738.70	170.45

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Feng, C.; Wang, C.; Chen, H.; Xu, C.; Wang, J. Cooperative Low-Carbon Trajectory Planning of Multi-Arrival Aircraft for Continuous Descent Operation. Aerospace 2024, 11, 1024. https://doi.org/10.3390/aerospace11121024

AMA Style

Feng C, Wang C, Chen H, Xu C, Wang J. Cooperative Low-Carbon Trajectory Planning of Multi-Arrival Aircraft for Continuous Descent Operation. Aerospace. 2024; 11(12):1024. https://doi.org/10.3390/aerospace11121024

Chicago/Turabian Style

Feng, Cun, Chao Wang, Hanlu Chen, Chenyang Xu, and Jinpeng Wang. 2024. "Cooperative Low-Carbon Trajectory Planning of Multi-Arrival Aircraft for Continuous Descent Operation" Aerospace 11, no. 12: 1024. https://doi.org/10.3390/aerospace11121024

APA Style

Feng, C., Wang, C., Chen, H., Xu, C., & Wang, J. (2024). Cooperative Low-Carbon Trajectory Planning of Multi-Arrival Aircraft for Continuous Descent Operation. Aerospace, 11(12), 1024. https://doi.org/10.3390/aerospace11121024

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Cooperative Low-Carbon Trajectory Planning of Multi-Arrival Aircraft for Continuous Descent Operation

Abstract

1. Introduction

2. Methodology

2.1. Optimal Control Model for CDO

2.1.1. CDO Phase

2.1.2. Mathematical Model

2.1.3. CDO Profile Solving Model

2.2. Low-Carbon Multi-Arrival Aircraft 4D Trajectory Cooperative Planning

2.2.1. T-Shaped Arrival Routes

2.2.2. Alternative Route Generation

2.2.3. Multi-Aircraft 4D Trajectory MIP Planning Model

3. Experimental Results and Discussion

3.1. Arrival Routes and Available Data

3.2. Alternative Trajectory Analysis

3.3. Convergence Trajectory Scheduling Results and Horizontal Trajectory Analysis

3.4. Vertical Profile and Fuel Consumption Comparison

3.5. Analysis of High-Traffic Scenarios

4. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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