Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network

Jiang, Furong; Zhang, Zhaoning; Dai, Xiaoxu

doi:https://doi.org/10.1155/2023/4602148

Journal of Advanced Transportation

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 4602148 | https://doi.org/10.1155/2023/4602148

Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network

Furong Jiang,¹Zhaoning Zhang,¹and Xiaoxu Dai¹

Academic Editor: Tao Liu

Received06 Mar 2023

Revised11 Apr 2023

Accepted27 Apr 2023

Published05 May 2023

Abstract

A multilayer network approach to model and analyze air traffic networks is proposed. These networks are viewed as complex systems with interactions between airports, airspaces, procedures, and air traffic flows (ATFs). A topology-based airport-airspace network and a flight trajectory network are developed to represent critical physical and operational characteristics. A multilayer traffic flow network and an interrelated traffic congestion propagation network are also formulated to represent the ATF connection and congestion propagation dynamics, respectively. Furthermore, a set of analytical metrics, including those of airport surface (AS), terminal controlled airspace (TCA), and area-controlled airspace (ACA), is introduced and applied to a case study in central and south-eastern China. The empirical results show the existence of a fundamental diagram of the airport, terminal, and intersections of air routes. Moreover, the dynamics and underlying mechanisms of congestion propagation through the AS-TCA-ACA network are revealed and interpreted using the classical susceptible-infectious-removed model in a hierarchical network. Finally, a high propagation probability among adjacent terminals and a high recovery probability are identified at the network system level. This study provides analytical tools for comprehending the complex interactions among air traffic systems and identifies future developments and automation of layered coupled air traffic management systems.

1. Introduction

Traffic flow is a ubiquitous dynamic process in human society. Air traffic, which differs from most traffic variations, is strictly monitored by air traffic management (ATM). Air traffic congestion is a scientific problem restricting the development of civil aviation and flight safety.

The ATM system’s operational supervision initiatives are the determining factors for solving air traffic congestion problems. To cope with increasing air traffic demand and congestion, the global ATM data management system is being upgraded and transformed from subregional management to unified management at the national level. By 2021, China has implemented a national flow management system that plays the role of a brain center in the operation and management of civil aviation air traffic. The promotion and implementation of ATM system planning, such as the Single European Sky ATM Research (SESAR) and NextGen, are anticipated to improve air traffic system resilience and alleviate air traffic congestion to a considerable extent.

Different from urban traffic congestion, which is generated by the conflict between the vehicles driven by each independent decision-maker (i.e., driver) and the limited road traffic resources, air traffic flow (ATF) congestion is a combination of the air traffic control system (ATCS) and the airline operation system in a limited space environment. Therefore, as shown in Figure 1, ATF congestion generally occurs in a spatiotemporal environment with limited traffic operation resources (such as the airport surface (AS) (including aprons, taxiways, and runways), terminal controlled airspace (TCA), and area-controlled airspace (ACA) with route convergence).

In recent decades, several studies on ATF modeling have been conducted. As an important location for the outflow and inflow of air traffic, the AS operation has various constraints, such as release time, available apron, and runway assignment, which cause traffic congestion. Studies on the traffic congestion on ASs have mainly focused on the queuing problem connecting the apron, taxiway, and runway [1–3]; flight delays [4–7]; and taxiing management [8–10]. Yang et al. discussed the congestion of departing traffic flow but ignored the influence of arriving traffic flow [11]. Two distinct phases are involved, namely, free-flow and congested phases, which are represented by a density-speed-flow relationship [12]. When the AS congestion intensifies and continues for some time, traffic congestion spreads to the airspace upstream of the traffic flow, causing congestion in the terminal area. Few studies have systematically examined the characteristics of traffic flow congestion on ASs, especially concerning simultaneous taxiing after landing and before takeoff.

Congestion in the TCA is another key manifestation of air traffic dynamics. The traffic flow congestion propagation within the TCA has motivated the formulation of several ATF control models in recent years to predict the overall impact on air traffic delays and support flow management in the TCA of hub airports. Inspired by the fundamental diagram of traffic flow [13, 14], Zhang et al. proposed a cell transmission model-based terminal airspace flow model with an assumed “flow density-velocity” relationship [15]. More detailed traffic flow phase transitions (free-flow, smooth, semistable, and congested) were identified by Yang et al. [12]. In contrast to AS traffic congestion, which can be analyzed using a two-dimensional model, TCA congestion is a time-dependent change process in a three-dimensional space. Hence, a strong correlation between the TCA and AS congestions is observed. Accordingly, most studies do not distinguish the mechanism of congestion between these two. However, from the perspective of traffic flow, ground flow and ATF have different motion states, flow environments, and restrictive conditions. Therefore, the AS traffic flow and arrival/departure traffic flows in the TCA can be analyzed separately in terms of their motion states.

Another critical area of air traffic congestion is the ACA. In this space, aircraft operating in medium-altitude and high-altitude air routes must remain safely spaced and orderly while flying. Bilimoria and Lee proposed a spatial clustering method based on the relative distance of aircraft and identified the air traffic status by establishing the congestion index of “Gaggle Density” [16]. Lee et al. evaluated and identified the degree of congestion caused by aircraft entering the control sector with different headings using the minimum course change parameter of aircraft collision avoidance [17].

Various recent studies highlight the importance of considering network modeling to study the propagation of traffic congestion and flight delays [18–21]. Lin et al. propose a new flight delay model to capture the impact of en-route congestion on flight delays over an air traffic network [22]. Wang et al. provide an algorithm, which using sliding correlation windows to extract the airport pairs with stable delay lags, to estimate statistically significant time lags between airport delays from noisy, aggregate operational data [23]. Wu et al. developed an Airport-Sector Network Delays model which takes both airports and airspace capacities into account [24].

Other studies have focused on airspace sector capacity assessments [25–27] and aviation network modeling based on complex networks [28–34]. Most studies that reveal the topology and dynamic behavior of air traffic network congestion are based on scale-free small-world network modeling in which airports and flights are represented by nodes and edges, respectively. However, this modeling method only considers airport nodes; it ignores the root cause of congestion in actual air traffic network flows. The entire process of civil aviation flight operation is under the monitoring and command of the ATCS, which controls the AS, TCA, and ACA.

A review of literature indicates that existing studies focus on the modeling and evaluation of air traffic systems without fully comprehending the inherent airspace flow dynamics and rules that govern their coevolution. Moreover, reports on the congestion propagation of air traffic at the universal network system level are limited.

The air traffic network flow system refers to the formation of an air traffic network among airports through air routes. The network and movement of aircraft in this network generates ATF; the two comprises the air traffic network flow system. When flow input is absent, that is, no aircraft flies in the airspace, the air traffic network is a static traffic network with only spatial characteristics. In contrast, when aircraft is flying in the airport and air route, the non-negative flow endows the air traffic network with certain dynamic characteristics. With the support of the network transmission of flow, the operation of the air traffic network flow system also has certain dynamic characteristics. Moreover, the quantitative characterization of the complexity of air traffic systems from the perspective of traffic flow and airspace interactions is lacking. To conduct such characterization at both the macroscopic and microscopic levels, a novel multilayer network approach is proposed. The multilayer network consists of two physical networks, namely, a topology-based airport-airspace network (TAAN) and a flight trajectory network (FTN) and two hierarchical networks, namely, a multilayer traffic flow network (MTFN) and an interrelated traffic congestion propagation network (ITCPN). To support such a multilayer network framework and reveal the internal ATF node congestion propagation dynamics, different classes of analytical metrics are developed as follows: traffic flow metrics of (i) AS, (ii) TCA, (iii) ACA, (iv) traffic congestion metrics, and (v) hierarchical congestion propagation metrics. The main contribution of this study is the formulation of the multilayer coupled congestion propagation network model. It also contributes the following findings obtained through the verification of the theoretical and empirical data of a representative region in China. The content and flow of this study are shown in Figure 2.(1)Empirical flow diagram (FD) of AS, TCA, and ACA: At the level of the air traffic network based on MTFN, AS, TCA, and ACA are used as network nodes for the first time, and a bidirectional fundamental FD based on empirical data is created. Three distinct flow states were identified for the nodes of the three levels, namely, free, smooth, and congested.(2)Time slot-based congestion propagation network: At the regional level, the layered coupled congestion propagation network model is verified by analyzing the traffic flow data of the AS, TCA, and ACA on a typical busy day. The introduction should be succinct, with no subheadings. Limited figures may be included only if they are truly introductory and contain no new results.

2. Multilayer Network for Congestion Propagation Modeling

In region-based operations, the complex ATF dynamics between AS, TCA, and ACA and the operational status changes in AS, TCA, and ACA arise from the interactions between the regional airway distribution, air traffic, and ATCS. To analyze the traffic flow dynamics of AS-TCA comprehensively, a novel analytical framework based on a multilayer network is established. The framework consists of two physical networks, namely, TAAN and FTN, and two hierarchical networks, namely, MTFN and ITCPN. These networks are elaborated in the following sections.

2.1. TAAN

Conventionally, airspace networks are modeled as graphs with waypoints as nodes and route segments (links) as edges. However, as presented in this section, a topology-based airspace network is constructed, utilizing the dense interchanges of air routes in the airspace and surrounding airspace as airspace nodes to form the ACA network. As shown in Figure 3, the nodes in the ACA network represent major air route interchanges or air route interchanges formed by multiple interchanges. The connectivity in the ACA network reflects the adjacency of air routes between interchanges and dense air routes. High-altitude air traffic congestion typically occurs during route interchanges. Therefore, the topology-based airspace network modeling method can simplify the actual route network and abstract the complex route network distribution into a simple network with node and edge connections.

For the airspace around and above an airport, the traffic flow movement space is divided into AS, TCA, and ACA according to the traffic flow rule and control range of the ATCS. The adjacent nodes are also connected with edges, as shown in Figure 4.

2.2. FTN

Consider a Tianjin-Shanghai flight flow as an example. When an aircraft departs from Tianjin Airport and climbs to a certain altitude, the departure procedure ends, and the TCA ATC (Air Traffic Controller) transfers the flight to the ACC ATC. After climbing out of the ZBTJ (Tianjin) TCA, the aircraft passes through ZBAA (Beijing) ACC, ZSJN (Jinan) ACC, and ZSSS (Shanghai) ACC and finally approaches and lands through ZSSS (Shanghai) TCA. The route passing the ACCs encounters three route interchanges, which can be represented as ACA nodes, as shown in Figure 5. Accordingly, the airspace traversed by the entire flight process is represented in the form of nodes to simplify the movement path of ATF.

2.3. MTFN

To define an MTFN, TAAN and FTN as well as three types of nodes (AS, TCA, and ACA) are combined.

Considering the coupling network after the fusion of the TAAN and FTN, an MTFN was established according to the traffic flow motion connectivity. Subnetwork A represents the AS layer, and the AS nodes are not connected. Subnetwork B represents the TCA layer; if the terminals are close to or contain each other, the adjacent TCA nodes are connected by edges. Subnetwork C represents the ACA layer. According to the sector distribution of low/high-altitude airspaces, adjacent ACA nodes are connected by edges. As shown in Figure 6, the nodes among the layers are linked according to the connection relationship of traffic flow.

2.4. ITCPN

The nodes in Figure 6 correspond to ASs, TCAs, and ACAs, signifying the primary zones where air traffic converges and becomes concentrated. When the volume of air traffic within these zones reaches high levels, congestion emerges. Connected air traffic convergence areas tend to facilitate congestion spread across multiple areas in a cascade effect. This congestion propagation follows a pattern similar to the transmission of an epidemic among individuals residing in proximity. Consequently, the infectious disease model is an apt approach to illuminating the transmission of air traffic congestion.

In studies of traffic congestion propagation mechanisms, infectious disease models such as susceptible-infectious-susceptible (SIS), susceptible-infectious-removed (SIR), and susceptible-exposed-infectious-removed (SEIR) are commonly employed [5, 7, 35]. It is crucial to note that disease propagation is irreversible, i.e., epidemiology spread is a unidirectional diffusion that mirrors congestion propagation. For this reason, traditional infectious disease models are deemed appropriate for explaining air traffic congestion propagation caused by single source of congestion.

Nonetheless, congestion propagation between nodes is solely associated with node degree and congestion transmission rate within traditional infectious disease models, with no differentiation between node types or direct congestion transmission rates of various node types. The string-coupled three-layer network model [28] overcomes this shortcoming in the SIR model and can be implemented in MTFN to describe traffic congestion propagation both between nodes of each layer and nodes of different layers.

Furthermore, traffic node states can be classified into free, congested, and removed states which correspond to susceptible, infected, and removed states in SIR model, respectively. In Section 3, definitions of free, congested, and removed states will be investigated based on node type, connecting distribution patterns of nodes within, and among distinct layers.

The connection relationship between the nodes and layers in the subnetwork is constructed based on the TAAN and FTN. The definitions of parameters are shown in Table 1. To represent the probability that node A in subnet A is randomly selected to have j edges connected to B, P_A (j) is defined. Similarly, P_B (i, j, k) represents the probability that a node is randomly selected in subnet B with i edges connected to A, j edges to B, and k edges to C. The probability of randomly selecting a node in subnet C that has j edges connected to B and k edges connected to C is represented by P_C (j, k).

According to the foregoing parameter definitions, the total number of nodes in the three subnetworks and the number of nodes in the crowded, free, and removed states can be expressed as follows:

The joint degree distribution, marginal degree distribution, average degree values (φ = 1), and second moment of degree (φ = 2) are calculated; the results are summarized in Table 2.

The probability that the AS node of layer A is transformed into a congested state by the influence of the TCA node of layer B (connected layer) is λ₂₁. The probabilities that the TCA node of layer B is transformed into a congested state by the influence of the AS node of layer A, TCA node of layer B, and ACA node of layer C (connected layers) are λ₁₂, λ₂₂, and λ₃₂, respectively. The probabilities that the ACA node in layer C is affected by the TCA node in layer B and ACA node in layer C in the congested state are λ₂₃ and λ₃₃, respectively. In subnetworks A, B, and C, the transition probabilities of the congested state nodes to the removed state are μ₁, μ₂, and μ₃, respectively.

Based on the foregoing assumptions, the SIR density model of ITCPN is established. Let , , which represent the corresponding densities. The other densities , , , are similarly defined. According to the previous assumption, we build a networked mean-field spreading model, which is composed of -dimensional ordinary differential equations as follows:.

To simplify the model further, the coupled network is assumed to be degree-independent, that is, the degree of any node is independent of the degrees of its neighboring nodes. In the traditional SIR model with average degree value equals , Θ (τ) is the probability that any adjacent contact can be affected to transform into a crowded state. It can be expressed as the average joint probability of connecting from a node of degree α to a node of degree β, , which is abbreviated as .

In the string-coupled three-layer network model, the formulas are expressed as follows:

The classical SIR model was used to analyze the propagation of the crowded state of the node in the ITCPN. The basic regeneration number, R₀, is an important parameter for revealing the propagation characteristics. In this study, R₀ represents the number of nodes converted into a crowded state; the conversion is due to the influence of one congested node propagation in all the free-state network nodes. The fundamental reproduction number, R₀, can be calculated using the next generation matrix, Γ = ⁻¹ [35], where F is the rate of new occurring congestion, and is the rate of transferring congested nodes from the free nodes group. Matrix Γ can be rewritten using similar transformations as follows:

The fundamental reproduction number of the model is R₀ = ρ (Γ), where ρ (Γ) is the spectral radius of matrix Γ.

2.5. Summary

The primary aim of this study is to explore the complex dynamics of ATF and the operating states of airports and airspaces through which traffic flows; empirical and comprehensive approaches are used. To achieve this objective, the ATF trajectory, AS-TCA-ACA air traffic operating network, and region-level interrelated traffic congestion propagation characteristics are thoroughly investigated using different network representations of such complex systems.

The MTFN is a three-layer network representing the connection between the AS, TCA, and ACA. As described in Section 2.3, to model the MTFN, generating the cross-layer linkage between the TAAN and FTN is crucial. This identifies the aircraft flying along each airspace sector and derives the traffic flow metrics presented in Section 3. An ITCPN is developed based on the integrated TAAN-FTN and the propagation law of the node congestion state. The propagation parameters of the node congestion state in the three-layer AS-TCA-ACA network are represented based on the characteristics of the classical SIR model in the layered coupled network. This provides a mathematical model for simulating the parameters and verifying empirical data.

3. Analytical Metrics

3.1. AS Traffic Flow Metrics

An FD that showed the phase transition of traffic flow on a road section by describing the one-to-one relationship among the fundamental parameters of traffic flow was proposed by Daganzo [13]. Considering that the dynamic monitoring of AS traffic flow mainly includes the taxiing traffic flow of aircraft about to take-off and has landed, the following are defined:(1)Take-off Traffic Flow, Q_To. Number of aircraft taking off from the airport within a specified time.(2)Landing Traffic Flow, Q_Ld. Number of aircraft landing at airports within a specified time.(3)Pre-takeoff Taxiing Traffic Density (PretTTD), ρ_To-t. Total number of aircraft taxiing before take-off within a specified time.(4)Postlanding Taxiing Traffic Density (PoldTTD), ρ_Ld-t. Total number of aircraft taxiing after landing within a specified time.(5)Average Pre-takeoff Taxiing Speed (PretTS), V_To-t. The average taxiing speed of all aircraft ready to take-off at the airport in the taxiing state within a specified time; this can be regarded as the average velocity of the entire traffic flow in the pre-takeoff taxiing state.(6)Average Postlanding Taxiing Speed (PoldTS), V_Ld-t. The average taxiing speed of all aircraft in the taxiing state after landing at the airport within a specified time; this can be regarded as the average velocity of the entire traffic flow in the postlanding taxiing state.

Airports are a major source of initial delays in flight. The operation for taxiing on the ground and flying in air can be regarded as a complete closed-loop traffic flow. The traffic flow within the AS mainly includes flights taxiing on the ground after landing and flights preparing to take-off after leaving the parking stand.

The congestion of AS mainly occurs in the movement range of aircraft on the ground, such as aprons, taxis, and runways. The flight congestion caused by the limited resources of the airport is represented by flight departure delay. Considering that the taxiing route and taxiing time in different airports cannot be directly compared because of the different configurations of runways, taxiways, and parking stands, the taxiing speed must be normalized. If the total taxiing process is 1 and the taxiing time is x (in minutes), the average taxiing speed after normalization is 1/x. The length of the time slot is 30 min. Figure 7 shows ρ_To-t, ρ_Ld-t, and the nonlinear relationship between the normalized _To-t and _Ld-t.

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We set the length of the time window presented in Sections 3 to 30 min. The nonlinear relationships between PretTTD, PoldTTD, normalized PretTS, and normalized PoldTS are shown in Figure 7(a). For a detailed analysis of the AS taxiing traffic flow dynamics, the three phases are divided by dissecting the fundamental diagram (Figures 7(b) and 7(d)) and temporal-spatial diagrams (Figure 8).(1)Free phase corresponds to relatively low PretTTD and PoldTTD, adequate taxiway availability, and low interaction among aircraft. Generally, the entry of aircraft into the parking stand is not restricted if no flight follows after its landing. Consequently, the PoldTS in the free-state is typically higher than the PretTS in the same state. This is because departing flights have more limiting factors, such as flight time, runway, and airspace.(2)Smooth phase corresponds to relatively high PretTTD and PoldTTD with a distinct reduction in both PretTS and PoldTS compared with the free phase owing to the high flow density. At this stage, the traffic flow in the AS significantly increases, the average pre-takeoff taxiing speed of the aircraft considerably decreases, and the waiting time for takeoff increases, although the operation is within the normal range.(3)Congested phase corresponds to the drop in both the PretTTD and PretTS as PoldTTD and PoldTS continue to increase as a consequence of the higher priority of aircraft landing than of take-off. At this stage, the postlanding traffic flow moves to the AS at high density and velocity. Meanwhile, the pretake-off traffic flow is heavy because numerous aircraft are in the waiting state owing to resource limitations. Moreover, the density and velocity are both low.

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As for the ITCPN model presented in Section 2, both the free and smooth phases are defined as a free state (susceptible state), and the congested phase is defined as a congested state (infected state). When the AS changes from a relatively distinct congested phase to a smooth phase, both the PretTTD and PretTS rise back to a relatively high level, whereas PoldTTD and PoldTS drop to an average level. This can be considered as the end of the congested state and defined as the removed state.

3.2. TCA Traffic Flow Metrics

Similar to the approach of defining parameters in the AS node, the spatial range is defined from an altitude of 1000 m to the upper limit of the airspace of the terminal area (generally 6000 m). Moreover, the latitude and longitude boundaries of the horizontal range of the terminal area are considered as nodes. To study the motion state of traffic flow in TCA nodes, the following are defined.(1)Departure Traffic Density (DTD), ρ_Dpt. Number of aircraft departing from the terminal area within a specified time.(2)Approach Traffic Density (ATD), ρ_Arv. Number of aircraft approaching the terminal area within a specified time.(3)Average Departure Speed (ADS), V_Dpt. The average departure speed of all departure flights in the terminal area within a specified time can be regarded as the average departure speed of the entire departure traffic flow.(4)Average Approach Speed (AAS), V_Arv. The average approach speed of all departure flights in the terminal area within a specified time can be regarded as the average velocity of the entire approach traffic flow.

The terminal area is the main section in which increased flight congestion occurs. Owing to the reduction in airspace capacity caused by the weather, convergence of waypoints, traffic congestion in the air section, and other reasons, the traffic flow in this area may turn into a crowded state. Previous studies have shown that terminal traffic flow has the basic properties of a fluid, and the traffic capacity between approach and departure has the property of a convex function [36].

Owing to the different flight paths of the approach and departure procedures in the terminal area, the approach and departure processes cannot be directly compared. Therefore, the traffic flow speed in the terminal area is assumed to be normalized regardless of the difference in the basic time of approach and departure caused by flight procedure differences. In the terminal area, if the total flight process is 1 and the approach (departure) time is x (in minutes), the average normalized approach (departure) speed is 1/x. The length of the time slot is 30 min. Figure 9 shows ρ_Dpt, ρ_Arv, and the nonlinear relationship between the normalized V_Dpt and V_Arv.

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Figure 9(a) shows the nonlinear relationship between the DTD, ATD, normalized ADS, and normalized AAS. For a detailed analysis of TCA traffic flow dynamics, similar to that in Section 3.1, the three phases were divided by scrutinizing the fundamental diagram (Figures 9(b)– 9(d)) and temporal-spatial diagrams (Figure 10).(1)Free Phase corresponds to relatively low DTD and ATD, large available space, and negligible interaction among aircraft. The departure speed of an aircraft is generally extremely high if no congestion occurs in the subsequent route after takeoff; hence, the ADS in the free-state is generally higher than the AAS in the same state. This is because the approach procedure has more restrictive conditions than the departure procedure in the terminal area.(2)Smooth Phase corresponds to a relatively high DTD and ATD with a distinct reduction in ADS compared with the free phase owing to the high flow density. At this stage, the traffic flow of the terminal airspace significantly increases, and the AAS of the aircraft considerably decreases. Moreover, the ADS fluctuates within a certain range, although it is within the normal operation range.(3)Congested Phase corresponds to a drop in the AAS as the ATD continues to increase. At this stage, the departing traffic flow moves at a relatively high density and speed in the terminal area. Furthermore, the approach traffic flow has numerous aircraft in waiting states at low speed owing to resource limitations.

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For the ITCPN model presented in Section 2, the free and smooth phases are defined as a free state (susceptible state), and the congested phase as a congested state (infected state). The TCA may change from a relatively distinct congested phase to a smooth phase in which the ATD drops to a normal level and the AAS rises back to a relatively average level, whereas the DTD and ADS return to a smooth phase. This can be considered as the end of the congested state; this is defined as the removed state.

3.3. ACA Traffic Flow Metrics

The studies that have been conducted on congestion propagation in the ACA are limited. In general, if congestion occurs at the TCA and lasts for a long time, congestion also occurs at the upstream ACA. This can be manifested by the decrease in the horizontal or vertical interval of the aircraft to a minimum, adjustment in the altitude or speed of the aircraft, aircraft circling, and waiting at the designated waypoint, etc. In the high-altitude airspace, ATF congestion mainly occurs at the air route interchange. The area where multiple air routes cross is prone to a shortage of airspace resources, leading to congestion. To simplify the representation of the ACA nodes, as shown in Figure 11, the intersection where the main route crosses is selected as the ACA node.

To study the motion state of the traffic flow in ACA nodes, the following are defined:(1)Cruising Traffic Density (CTD), ρ_CR_us. Number of aircraft cruising across the ACA node area within a specified of time.(2)Average Cruising Speed (ACS), V_CR_us. The average cruising speed of all cruising flights across the ACA node area within a specified time can be regarded as the average cruising speed of the entire cruising traffic flow.(3)Average Angular Velocity (AAV), ω_CR_us. The average angular velocity of all cruising flights across the ACA node area within a specified time can be regarded as the average angular velocity of the entire cruising traffic flow.

In high-altitude airspaces, the air route intersection is the main area of increasing flight congestion. Owing to the reduction in airspace capacity caused by public navigation stations and the limited range of available airspace, the traffic flow in this area may enter a crowded state.

The motion state of traffic flow at the ACA node is studied using the analysis method of FD in the AS and TCA. Because the cruising traffic flows flying into and out of ACA nodes have varying angles, flight distances, and flight paths, the status of all cruising flights passing through ACA nodes cannot be directly compared. Therefore, the geometric center where the ACA node traffic flow converges is selected as the center of the circle. The length that can contain the main convergence area is considered as the radius. The circular area and its center (orange), which is set as the ACA node, are shown in Figure 11. The cruise traffic flow velocity of ACA nodes is assumed to be normalized ignoring the time difference between entering and leaving the ACA node area owing to flight path variations. In the ACA node area, if the total flight process is 1, and the time from entering the node airspace range to leaving this range is x min, the average normalized cruise speed is 1/x. If the angle of the total flight process is α and the time from entering the node airspace range to leaving this range is x (in minutes), the average normalized angular velocity is α/x. The length of the time slot is 15 min. The nonlinear relationship between ρ_Crus, V_Crus, and ω_Crus are shown in Figure 12.

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Figure 12(a) shows the relationship between the CTD, ACS, and AAV. The figure indicates that the AAV is fundamentally proportional to the ACS, whereas the CTD is negatively correlated with the relationship between the two. Sections 3.1 and 3.2 elaborate the analysis of ACA traffic flow dynamics. The three phases are divided by scrutinizing the fundamental diagram (Figures 12(b) and 12(d)) and temporal-spatial diagram (Figure 13).(1)Free Phase corresponds to a relatively low CTD, large available cruise space, and low interaction among aircraft. When the cruise flight route is not congested, the aircraft speed is generally overly high. Moreover, the ACS and AAV in the free-state are considerably high.(2)Smooth Phase corresponds to a relatively high CTD with considerable reductions in ACS and AAV compared with the free phase owing to high flow density. At this stage, the traffic flow at the air route interchange significantly increases, and the ACS and AAV of the aircraft decrease to a certain extent. As shown in Figure 13, the CTD, ACS, and AAV all fluctuate within a certain range over time but within the normal operating range.(3)Congested Phase in this case differs from that in the AS and TCA. Here, the CTD does not rise but falls when the ACA is crowded. This is because if the intersection of air routes is congested, the aircraft is directed to circle around or wait; consequently, both the ACS and AAV drop to a low value. In the congested phase, owing to the large airspace occupied by circling or waiting aircraft, the number of aircraft cruising across the intersection of air routes decreases. Furthermore, the CTD decreases compared with that in the smooth phase.

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Regarding the ITCPN model presented in Section 2, the free and smooth phases are defined as free states (susceptible states), and the congested phase is defined as a congested state (infected state). The ACA can change from a relatively distinct congested phase to a smooth phase in which the ACS and AAV rise back to a relatively average level, whereas the CTD returns to a smooth phase. This may be considered as the end of the congested state, which is defined as the removed state.

4. Numerical Simulation Analysis

A congestion propagation simulation analysis is implemented based on the ITCPN model presented in Section 2 and the analysis of the phase transition of the AS-TCA-ACA node discussed in Section 3. The following equation represents the average crowded state density of subnetworks A, B, and C at time t:

4.1. Fundamental Parameter Characteristics of ITCPN Model

Set N_A = N_B = 80. This indicates that the number of airports is equal to the number of terminal areas, ignoring the case in which adjacent airports share terminal areas. The status of the airport node has no relationship with those of other airport nodes; hence, the node degree in layer A is zero. Some nodes in the terminal area are distant from adjacent nodes according to the distribution law of traffic flow in the terminal area in an actual airspace. Their mutual influence can be ignored and represented as isolated nodes in this layer. In a part of the terminal area, mainly the terminal area corresponding to the airport group, nodes are clustered in a small range; they are represented as circular nodes connected to each other in pairs. Accordingly, the following can be assumed: n₁₂ = <k>₁₂ = 1, n₂₁ = <k>₂₁ = 1, n₂₂ = 3, and <k>₂₂ = 2.

A route node network of a layer is formed by connecting adjacent nodes. If N_C = 30, a random route node network in which n₃₃ = 6 and <k>₃₃ = 4.47 is formed, as shown in Figure 14.

One-to-one correspondence does not exist between the nodes of the route and terminal area. A terminal area node is connected to at least one route node, and a route node can be connected to multiple terminal area nodes. In the simulated random network, n₂₃ = 1, n₃₂ = 4, <k>₂₃ = 1, and <k>₃₂ = 2.67.

4.2. Node State Density (i, s, r)-Time (T)

Let’s assume that the initial delay occurs in the terminal area (i.e., i_B (0) = 0.01) and that the propagation probability of crowded nodes (λ) is 0.5. The changes in the crowded, free, and removed state densities of the nodes at each layer over time are shown in Figure 15.

Figure 15 indicates that if the propagation probability of a crowded state among nodes is the same, the ACA node of layer C is closely related to each node owing to its high average degree of nodes. The congested state density of the ACA node increases with time at the highest rate, followed by the TCA node; the AS node had the lowest propagation rate. Under this condition, the basic regeneration number, R₀, is 3.21.

In a real aviation network, the congestion propagation rate among ACA nodes is considerably less than that of TCA nodes. This is because the traffic flow in the route can be alleviated by speed regulation and the adjustment of the horizontal and longitudinal spacings among aircraft.

4.3. Infection Rate (λ) and Crowded State Density (i)

All infection rates in the network are set as 0.5, 0.4, and 0.3, and the recovery rate is assumed to be 1. With a decrease in infection rate, as shown in Figure 16, the peak value of the congestion rate decreases, and the time point of the peak value subsequently shifts.

4.4. Infection Rate (λ) and Fundamental Reproduction Number (R₀)

The effect of infection rate on the fundamental reproduction number (R₀) considering different network structures is investigated. As presented in Figure 17, when one of the infection rates changes, the other rates are fixed at 0.1.

As shown above, the cross-infection rates (λ₁₂, λ₂₁ and λ₂₃, λ₃₂) have the same influence on R₀. This shows that the degree of influence of congestion propagation between the AS and TCA is the same in both directions. This means that the probability of congestion in the TCA resulting from the congestion in the AS and the probability of flight congestion in the AS resulting from the congestion in the TCA have the same influence on the large-area traffic flow congestion throughout the system. In this network model, an increase in λ₃₃ can considerably increase R₀ because of the large average degree of layer C. This can be explained by the complex ACA distribution; the traffic flow congestion propagation rate in the ACA has a more significant impact on the large-scale congestion of the entire system. However, traffic congestion in actual operations can be alleviated though various means, such as rerouting, regulating speed, and converting the altitude level. This can lead to a low numerical value of the system (λ₃₃) such that the congestion propagation in the network decreases.

5. Empirical Results: Case Study in Central and South-Eastern China

5.1. Data Description

The central and south-eastern parts of China have 12 airports. Because the air networks are densely distributed, the traffic operational scope of the 12 airports, including 6 of the top 15 airports in China, is selected in terms of flight volume. These airports include the Shanghai Hongqiao, Shanghai Pudong, Hangzhou Xiaoshan, Nanjing Lukou, Wuhan Tianhe, and Zhengzhou Xinzheng airports and five medium-sized airports. The nodes number and corresponding names are listed in Table 3. To explore the congestion propagation process of air traffic in a large space based on the multilayer network framework and analytical metrics, the flight trajectory data on a typical busy day in December 2019 were collected. The time resolution of the trajectory data is 15 s. The sample trajectory data for the TCA and ACA are shown in Figure 18. An MTFN integrated with TAAN and FTN is established according to the data analysis method presented in Section 3. The node distribution and MTFN are shown in Figures 19 and 20, respectively.

(a)

(b)

(c)

(a)

(b)

(c)

5.2. Model Verification

Using the data analytical metrics described in Section 3, aircraft taxiing, takeoff, and landing as well as the flight trajectory data for a day are analyzed. Moreover, the set time slot is every 30 min. The states of each node in the ITCPN are established according to real data changes over time, as shown in Figure 21. The light, dark, and gray grids represent the free, congested, and removed states, respectively.

A comparison of the crowded node density between the empirical and simulation data is shown in Figure 22. In the simulation, the TCA2 node (i.e., the node whose degree values in layer B connecting A, B, and C are 1, 3, and 1, respectively) is initially in a crowded state. We assume that other nodes are in free-state at initial and there is only one traffic congestion cause. The propagation probability values of the crowded state among the nodes are set as λ₁₂ = 0.2, λ₂₁ = 0.1, λ₂₂ = 0.8, λ₂₃ = 0.2, λ₃₂ = 0.4, and λ₃₃ = 0.4; the recovery rates are set as μ₁ = 0.3, μ₂ = 0.3, and μ₃ = 0.7. The maximum value of λ₂₂ in the simulation indicates that the influence of the congestion propagation between two adjacent terminals is the highest. In contrast, λ₁₂, λ₂₁, and λ₂₃ are relatively low, indicating that the possibility of traffic flow congestion at the AS and intersection traffic congestion in air routes caused by terminal area congestion is low. In the ACA layer network, the nodes are connected by air routes, including long air routes that can buffer and absorb the congested traffic flow at the intersection. Accordingly, the recovery rate of ACA nodes, μ₃, considerably exceeds those of μ₁ and μ₂.

6. Conclusions

The study of the state of complex ATF in the range of ground and air operations is essential to understand the nature of the interaction between the ATF, airport, and airspace and to reveal the technical potential of advancing ATM. Among the various analytical approaches, network analysis is an effective and intuitive method for examining the behavior of complex systems with varying granularities. The objective is to depict the characteristics of the operational status of ATF and explain the internal mechanism of traffic flow congestion state propagation in different ground and air operation spaces. Accordingly, a comprehensive multilayer network is constructed to integrate and present factors, such as airports, terminal areas, air route networks, aircraft trajectories, traffic flow space states, and congestion state propagation laws. To characterize the congestion propagation among different ATF regions further, an ITCFN is established. The following are implemented:(1)The node state parameters and crowded state propagation parameters in the ITCPN network are modeled and analyzed.(2)The AS and terminal area of Shanghai Pudong Airport and the typical air route crossing nodes in central China are considered as examples. The proposed network node state and analysis index, including traffic flow dynamics and three flow stages (free, congested, and removed), are analyzed.(3)Empirical analysis is performed on the flight data of 12 airports and 7 terminal areas and route networks in central and south-eastern China on a typical busy day to verify the accuracy of the ITCPN network model parameters.

This study uses general methods to study the flight data in some regions of China empirically. The effectiveness of using string-coupled three-layer network model to describe congestion at the convergence of air traffic flows both on ground and in the airspace is verified.

In the numerical simulation of Section 4, the study further reveals that cross-infection rates have the same influence on R₀, suggesting that the degree of influence of congestion propagation between the AS and TCA is the same in both directions. An increase in λ₃₃ has a considerable impact on R₀ due to the complex ACA distribution, specifically on the traffic flow congestion propagation rate in the ACA, which has a more significant impact on the large-scale congestion of the entire system. Thus, it is imperative to devise a more coherent and effective layout of the air traffic network, while implementing more efficient methods of traffic planning and management, so as to curtail the widespread ramifications of air traffic congestion within the network.

This study presents a comparison of the crowded node density based on empirical and simulation data in Section 5. Results show that the congestion propagation between two adjacent terminals has the highest influence, while the possibility of congestion in the AS and intersection traffic congestion in air routes caused by terminal area congestion is low. This underscores the pivotal role played by the interplay of traffic management within airport terminal areas in instigating and exacerbating air traffic congestion. By prioritizing the management of traffic flow within airport terminal areas, it is possible to mitigate the adverse consequences of large-scale traffic congestion.

In future research, testing the generality of the findings and verifying the key characteristics of more complex aviation networks and airport layouts are necessary. This model is useful for aviation network layout design, congestion prediction, and stability evaluation.

Nomenclature

ACA:	Area-controlled airspace
ACC:	Area control center
AS:	Airport surface
ATC:	Air traffic controller
ATCS:	Air traffic control system
ATF:	Air traffic flow
ATM:	Air traffic management
FTN:	Flight trajectory network
ITCPN:	Interrelated traffic congestion propagation network
MTFN:	Multilayer traffic flow network
SIR:	Susceptible-infectious-removed infectious disease models
TAAN:	Topology-based airport-airspace network
TCA:	Terminal controlled airspace.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. U2233209) and the Fundamental Research Funds for the Central Universities (Grant no. 3122022053).

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Copyright

Copyright © 2023 Furong Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Journal of Advanced Transportation

Ground-Air Traffic Congestion Propagation Model Based on Hierarchical Control Interdependent Network

Abstract

1. Introduction

2. Multilayer Network for Congestion Propagation Modeling

2.1. TAAN

2.2. FTN

2.3. MTFN

2.4. ITCPN

2.5. Summary

3. Analytical Metrics

3.1. AS Traffic Flow Metrics

3.2. TCA Traffic Flow Metrics

3.3. ACA Traffic Flow Metrics

4. Numerical Simulation Analysis

4.1. Fundamental Parameter Characteristics of ITCPN Model

4.2. Node State Density (i, s, r)-Time (T)

4.3. Infection Rate (λ) and Crowded State Density (i)

4.4. Infection Rate (λ) and Fundamental Reproduction Number (R0)

5. Empirical Results: Case Study in Central and South-Eastern China

5.1. Data Description

5.2. Model Verification

6. Conclusions

Nomenclature

Data Availability

Conflicts of Interest

Acknowledgments

References

Copyright

4.4. Infection Rate (λ) and Fundamental Reproduction Number (R₀)