Evaluation of Air Traffic Network Resilience: A UK Case Study

Abstract

With growing air travel demand, weather disruptions cost millions in flight delays and cancellations. Current resilience analysis research has been focused on airports and airlines, rather than the en-route waypoints, and has failed to consider the impact of disruption scenarios. This paper analyses the resilience of the United Kingdom (UK) air traffic network to weather events that disrupt the network’s high-traffic areas. A Demand and Capacity Balancing (DCB) model is used to simulate adverse weather and re-optimise the cancellation, delay, and rerouting of flights. The model’s feasibility and effectiveness were evaluated under 20 concentrated and randomly occurring extreme disruption scenarios, lasting 2 h and 4 h. The results show that the network is vulnerable to extended weather events that target the network’s most central waypoints. However, the network demonstrates resilience to weather disruptions lasting up to two hours, maintaining operational status without any flight cancellations. As the scale of disruption increases, the network’s resilience decreases. Notably, there exists a threshold beyond which further escalation in disruption scale does not significantly impair the network’s performance.

1. Introduction

Air travel demand has been growing rapidly since the 20th century and has continued into the 21st century. Between 2003 and 2019, the air traffic in Europe increased by 31%. Europe experienced continuous Air Traffic Flow Management (ATFM)-related delay degradation from 2013–2019, with 2018 and 2019 being exceptionally poor years in performance, mainly due to growing capacity issues. The COVID-19 pandemic led to a historic decrease of air travel demand; however, the traffic has been steadily returning to pre-pandemic levels. EUROCONTROL [1] reports that for the first week of March 2024, the traffic was at 92% of the equivalent week in 2019. Despite the lower demand, the network is performing worse compared to pre-pandemic numbers in terms of arrival and departure punctuality, with around 75% of all flights departing no later than 15 min after their scheduled time, and 80% of flights arriving no later than 15 min of their scheduled times [2]. This highlights the need for solutions to the demand–capacity balancing problem for the airspace to be utilised effectively at scale. And in Europe, the ATFM en-route capacity is an especially significant factor in tackling the DCB problem.

The UK airspace is one of the busiest in Europe in terms of the total number of departing and arriving flights, and also carried many transatlantic flights [3]. Therefore, any disruptions to the en-route capacities can result in major delays to both short-haul and long-haul flights. Delays caused by adverse weather have seen an increase in recent years. EUROCONTROL [4] report that in the summer of 2023, the UK airspace saw adverse convective weather in 20 out of the 92 days, increasing weather-related ATFM delays by 60% from the previous summer. With increasing disruptions, the question of how to change the scheduling of the system becomes important in minimizing the impact and all costs, as flights avoid stormy areas in the interest of safety [5].

This increasing traffic volume and poor delay performance suggests that there is an urgent need for research into air traffic resilience. Wang et al. [6] conducted a resilience analysis of the network within a sector. By considering potential horizontal and vertical conflicts between flights as a measure of complexity, the study employed a genetic algorithm to reallocate flight departure times. The results show the approach could reduce the sector’s traffic complexity and volatility, thereby enhancing resilience. However, this methodology operates under a significant assumption, namely that altering flight departure times does not impact flight trajectories before entering the sector or after leaving the sector. This means potential issues such as flight rerouting or handover complexity in adjacent sectors are ignored. Wang et al. [7] investigated the resilience of Chinese airport networks under disruptions. They defined the intensity of disruptive events through the following two metrics: the number of affected airports and the intervals between consecutive aircraft utilizing runways. The study observed system performance under various scenarios of reduced airport capacity resulting from these disruptions. The research encompassed a network of 251 operational airports and accounted for common delay propagation patterns. However, this methodology has a significant limitation: it does not consider en-route airspace conditions or potential flight plan alterations such as rerouting or air holding. The approach assumes that flight trajectories adhere strictly to their original flight plans. Sampaio et al. [8] used complexity network theory to perform resilience analysis on the Brazilian air traffic network. They found that the network was resilient to random disruptions, but sensitive to disruptions on central nodes. Lordan et al. [9] found the same phenomenon on the global airport network. Their findings revealed that the network follows a truncated power–law distribution. There was a significant decline in overall network performance when critical airports were targeted. Hossain and Alam [10] and Cong et al. [11] performed similar analyses on the Australian and Chinese airport networks, respectively. They identify key central airports that are most susceptible to failures. While these airport network-based studies effectively identify critical airports using various metrics and provide insights for system recovery under disruptive conditions, they invariably overlook the impact of en-route operations. From the reviewed works, only Du et al. [12] focused on the resilience of waypoint networks rather than airports. They used a memetic algorithm to remove edges randomly or directionally, then decide critical edges based on the robustness of the new network. However, this research still considers the problem from the perspective of network topological characteristics without accounting for the actual traffic patterns and flows.

While significant research has been conducted on the resilience of air traffic networks, current studies exhibit notable limitations. Primarily, these investigations have focused exclusively on network topology analysis, neglecting the crucial aspect of traffic flow redistribution. The prevailing assumption that air traffic will be naturally accommodated within the network overlooks critical operational factors such as takeoff and landing schedules and flight status. Furthermore, resilience studies based solely on topological characteristics tend to only consider connectivity between two airports, disregarding the substantial impact of airport schedule and capacity constraints. To address these shortcomings, this study employs a DCB model to re-optimize flight schedules when analyzing the UK waypoint network under various disruption scenarios. This model incorporates decision-making processes regarding flight status, including potential delays or cancellations, as well as trajectory allocation in response to mid-flight disruptions, thus providing a more comprehensive and operationally relevant assessment of network resilience. Moreover, we considered the impact of different scales of disruptions. This can be simulated through the use of a rolling horizon optimisation. The main contributions of this paper are as follows:

• Present a versatile DCB model with rolling horizons, capable of simulating various scale disruptions to the network.
• Simulate random and targeted disruptions on the UK air traffic network and analyse the network’s performance under disruptions to its high-traffic areas.

Section 2 introduces the DCB model, Section 3 presents the simulation scene setup, Section 4 discusses the results of the analysis and Section 5 presents the conclusions and potential further directions.

2. DCB Model

2.1. Introduction

The airspace can be represented in terms of a directed graph $\mathcal{G}=(\mathcal{N},\mathcal{A})$, where $\mathcal{N}$ is the set of nodes representing waypoints and airports and $\mathcal{A}$ is the set of arcs connecting the nodes. Flights are represented as flows in the network, starting at their origin node, traversing the arcs, and terminating at a destination node. The origin and destination nodes represent the airports or the boundaries of the UK airspace, if the flight is non-domestic.

The model is formulated as an integer mathematical optimisation problem, which strives to make decisions regarding which flights need to be delayed, which flights need to be rerouted, and which flights need to be cancelled. The main constraints for these decisions are the capacities at the airports, as well as the airspace along the route of the flights. The objective is to minimise the costs resulting from these decisions. The capacities of the airports and arcs are calibrated using the real flight data. It is assumed that the baseline capacities allow for the normal operation of all flights, without any delays.

A rolling horizon optimisation technique is adopted. The whole simulation time is split into two-hour windows, for which the optimisation problem is solved only for the flights that are scheduled to depart during that time window. The decision is made by taking into account the capacities taken up by flights from the previous simulation windows, as well as the predicted capacities based on the scheduled flights. This is adopted because of its ability to simulate sudden disruptions to the capacities.

Overall, the presented model allows for both ground-holding and air delay (through speed deviation). Rerouting is also considered. The presented formulation is adapted from Agustín et al. [13], working on links rather than nodes; hence, it is more flexible and provides a tighter formulation.

2.2. Definitions and Notation

Input sets

These are the sets of data that are initially supplied to the model. Further parameter explanations can be found in Appendix A.

$\mathcal{K}$	=	Set of airports. Can be further split into departure airports ${\mathcal{K}}^d$ and arrival airports ${\mathcal{K}}^a$, where $\mathcal{K}={\mathcal{K}}^d\cup {\mathcal{K}}^a$. It is likely that an airport is present in both sets.
$\mathcal{F}$	=	Set of flights. Can be further split into domestic flights ${\mathcal{F}}_d$, outgoing flights ${\mathcal{F}}_o$, incoming flights ${\mathcal{F}}_i$ and crossing flights ${\mathcal{F}}_c$. $\mathcal{F}={\mathcal{F}}_d\cup {\mathcal{F}}_o\cup {\mathcal{F}}_i\cup {\mathcal{F}}_c$.
$\mathcal{T}$	=	Set of time periods, where T is the last time period in the horizon. The adopted value for each time period is 5 min.
${\mathcal{N}}_f$	=	Set of nodes that define the possible routes flight f can take.
${\mathcal{N}}_f^{*}$	=	Set of nodes that define the scheduled route for flight f. ${\mathcal{N}}_f^{*}\subseteq {\mathcal{N}}_f$.
${\mathcal{A}}_f$	=	Set of arcs that define the possible routes flight f can take.
${\mathcal{A}}_f^{*}$	=	Set of arcs that define the scheduled route for flight f. ${\mathcal{A}}_f^{*}\subseteq {\mathcal{A}}_f$.

Incidence sets

These sets are derived from the input sets for clarity and ease of the subsequent formulation.

Incoming incidence set: ${\mathrm{\Gamma }}_f^{-}\left(n\right)=\{m\mid (m,n)\in {\mathcal{A}}_f\}$

These are nodes m that have an arc directed to node n.

Outgoing incidence set: ${\mathrm{\Gamma }}_f^{+}\left(n\right)=\{m^{\prime }\mid (n,m^{\prime })\in {\mathcal{A}}_f\}$

These are nodes $m^{\prime }$ that have an arc directed from node n.

Pre-calculated sets

These sets are derived from the input sets, in order to help with computation. The problem size is significantly reduced with the introduction of these sets.

${\mathcal{T}}_f^{m,n}$	=	Set of feasible time periods for flight f to arrive at node n through arc $(m,n)$. ${\underline{T}}_f^{m,n}$ and ${\overline{T}}_f^{m,n}$ are the earliest and latest feasible time periods, respectively.
${\mathcal{T}}_{*f}^{m,n}$	=	The interval containing all time periods between ${\underline{T}}_f^{m,n}$ and ${\overline{T}}_f^{m,n}$.
${\mathcal{T}}_f^n$	=	Set of feasible time periods for flight f to arrive at node n through any feasible arc. ${\underline{T}}_f^n$ and ${\overline{T}}_f^n$ are the earliest and latest feasible time periods, respectively.

Decision variables

$x_{f,m,n}^t$	=	$\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{flight}f\mathrm{arrives}\mathrm{at}\mathrm{node}n\mathrm{through}\mathrm{arc}(m,n)\mathrm{by}\mathrm{time}\mathrm{period}t\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$
${\alpha }_f$	=	Actual flight duration of flight f.
${\alpha }_f^{+}$	=	Positive time unit deviation from the scheduled duration for flight f.
${\alpha }_f^{-}$	=	Negative time unit deviation from the scheduled duration for flight f.

2.3. Objective Function

Here, we considered four kinds of costs to reveal the real world operation objectives:

$$\mathrm{Total}\mathrm{cancellation}\mathrm{cost}=TCC=\sum _{f\in \mathcal{F}}{c}_f^c\left(1-x_{f,k_f^d,q\left(k_f^d\right)}^{{\overline{T}}_f^{k_f^d,q\left(k_f^d\right)}}\right)$$

(1)

$$\mathrm{Total}\mathrm{rerouting}\mathrm{cost}=TRC=\sum _{f\in \mathcal{F}}\left[\sum _{(m,n)\in {\mathcal{A}}_f}{c}_f^{m^{\prime },n^{\prime }}{x}_{f,m,n}^{{\overline{T}}_f^{m,n}}-\sum _{(m^{\prime },n^{\prime })\in {\mathcal{A}}_f^{*}}{c}_f^{m^{\prime },n^{\prime }}{x}_{f,m^{\prime },n^{\prime }}^{{\overline{T}}_f^{m^{\prime },n^{\prime }}}\right]$$

(2)

$$\begin{array}{cc}\hfill \mathrm{Total}\mathrm{delay}\mathrm{cost}& =TDC=\sum _{f\in \mathcal{F}}\left[\sum _{t\in {\mathcal{T}}_f^{p\left(k_f^a\right),k_f^a}}{c}_{f,t}^{k_f^a}\left(x_{f,p\left(k_f^a\right),k_f^a}^t-x_{f,p\left(k_f^a\right),k_f^a}^{t-1}\right)\right.\hfill \\ \hfill & -\left.\sum _{t\in {\mathcal{T}}_f^{k_f^d,q\left(k_f^d\right)}}{c}_{f,t}^{k_f^d}\left(x_{f,k_f^d,q\left(k_f^d\right)}^t-x_{f,k_f^d,q\left(k_f^d\right)}^{t-1}\right)\right]\hfill \end{array}$$

(3)

$$\mathrm{Total}\mathrm{speed}\mathrm{deviation}\mathrm{cost}=TSDC=\sum _{f\in \mathcal{F}}{c}^a{\alpha }_f^{-}$$

(4)

Note that objective (3) is expressed in the form of total delay minus the cost reduction if the delay is a ground hold. As pointed out by Bertsimas et al. [14], this allows for a super-linear cost function. A linear cost function with air and ground delays separated can favor solutions in which a large delay is assigned to one flight and a small delay to another flight, which raises problems if considering equity amongst airlines. A super-linear cost function will favor solutions in which delays are more equally distributed among flights. However, because the costs are expressed in this way, the model may favour solutions in which a flight is purposely ground-held and then sped up en-route to obtain a negative delay cost. To deal with this, the speed-deviation cost objective is added. The four objectives can be combined into a single-objective function:

$$min\sum \left(w_1\times TCC+w_2\times TRC+w_3\times TDC+w_4\times TSDC\right),$$

(5)

where $w_1$, $w_2$, $w_3$, $w_4$ are the chosen weights for each of the individual objectives.

2.4. Constraints

Capacity constraints

$$\begin{array}{cc}\hfill & \sum _{f\in \mathcal{F}}\left(x_{f,k_f^d,q\left(k_f^d\right)}^t-x_{f,k_f^d,q\left(k_f^d\right)}^{t-1}\right)\hfill \\ & +\sum _{f\in \mathcal{F}}\left(x_{f,p\left(k_f^a\right),k_f^a}^t-x_{f,p\left(k_f^a\right),k_f^a}^{t-1}\right)\le D_k^t-\left(B_{k_f^d,q\left(k_f^d\right)}^t+B_{p\left(k_f^a\right),k_f^a}^t\right)\forall t\in \mathcal{T},k\in \mathcal{K},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \sum _{f\in \mathcal{F}}\left(x_{f,m,n}^t-x_{f,m,n}^{t-1}\right)\le Q_{m,n}^t-B_{m,n}^t\hfill \\ & \forall t\in \mathcal{T},(m,n)\in \mathcal{A}\setminus \{(k^d,q\left(k^d\right)),(p\left(k^a\right),k^a)\},\hfill \end{array}$$

(7)

Connectivity constraints

$$\begin{array}{cc}\hfill & \sum _{m\in {\mathrm{\Gamma }}_f^{+}\left(n\right)}{x}_{f,n,m}^{t+{\underline{l}}_{f,n,m}}\le \sum _{m\in {\mathrm{\Gamma }}_f^{-}\left(n\right)}{x}_{f,m,n}^t\le \sum _{m\in {\mathrm{\Gamma }}_f^{+}\left(n\right)}{x}_{f,n,m}^{t+{\overline{l}}_{f,n,m}}\hfill \\ & \forall f\in \mathcal{F},n\in {\mathcal{N}}_f\setminus \{k_f^d,k_f^a\},t\in {\mathcal{T}}_f^n,\hfill \end{array}$$

(8)

$$x_{f,k_f^d,q\left(k_f^d\right)}^{{\overline{T}}_f^{k_f^d,q\left(k_f^d\right)}}-x_{f,p\left(k_f^a\right),k_f^a}^{{\overline{T}}_f^{p\left(k_f^a\right),k_f^a}}=0\forall f\in \mathcal{F},$$

(9)

$$x_{f,k_f^d,q\left(k_f^d\right)}^{{\overline{T}}_f^{k_f^d,q\left(k_f^d\right)}}=1\forall f\in {\mathcal{F}}_i\cup {\mathcal{F}}_c,$$

(10)

$$x_{f,m,n}^t-x_{f,m,n}^{t-1}\le 0\forall f\in \mathcal{F},(m,n)\in {\mathcal{A}}_f,t\in {\mathcal{T}}_f^{m,n},$$

(11)

$$x_{f,m,n}^t-x_{f,m,n}^{t-1}=0\forall f\in \mathcal{F},(m,n)\in {\mathcal{A}}_f,t\in {\mathcal{T}}_{*f}^{m,n}\setminus T_f^{m,n},$$

(12)

Delay constraints

$$x_{f,k_f^d,q\left(k_f^d\right)}^t-x_{f,p\left(k_f^a\right),k_f^a}^{t+r_f-d_f+A_f}\le 0\forall f\in \mathcal{F},t\in {\mathcal{T}}_f^{k_f^d,q\left(k_f^d\right)},$$

(13)

$$\begin{array}{cc}\hfill & \sum _{t\in {\mathcal{T}}_f^{p\left(k_f^a\right),k_f^a}}t\left(x_{f,p\left(k_f^a\right),k_f^a)}^t-x_{f,p\left(k_f^a\right),k_f^a)}^{t-1}\right)\hfill \\ & -\sum _{t\in {\mathcal{T}}_f^{k_f^d,q\left(k_f^d\right)}}t\left(x_{f,k_f^d,q\left(k_f^d\right))}^t-x_{f,k_f^d,q\left(k_f^d\right))}^{t-1}\right)={\alpha }_f,\forall f\in \mathcal{F},\hfill \end{array}$$

(14)

$$x_{f,k_f^d,q\left(k_f^d\right)}^{{\overline{T}}_f^{k_f^d,q\left(k_f^d\right)}}(a_f-d_f)+{\alpha }_f^{+}-{\alpha }_f^{-}={\alpha }_f,\forall f\in \mathcal{F}$$

(15)

Variable bounds

$$x_{f,m,n}^t\in \{0,1\}\forall f\in \mathcal{F},(m,n)\in {\mathcal{A}}_f,t\in {\mathcal{T}}_{*f}^{m,n},$$

(16)

$${\alpha }_f,{\alpha }_f^{+},{\alpha }_f^{-}\ge 0\forall f\in \mathcal{F},$$

(17)

Constraint (6) enforces the combined departure and arrival capacity at airports at all time periods. Constraint (7) ensures that the link capacity at each time is not exceeded. Both of the capacity constraints include the in-use capacity resulting from flights in a previous time window and the booked capacity based on the schedule of the future flights in the subsequent time windows. Because scheduled capacities are considered, the model prioritises future flights over current ones. Constraint (8) ensures that the flight reaches the next node in the given time frame for each arc. Note that $n\in {\mathcal{N}}_f\setminus \{k_f^d,k_f^a\}$ denotes that this is not applicable once the aircraft is within the bounds of its origin or destination airport. This is important, as including the airport nodes in this constraint can lead to erroneous results where delay is assigned during takeoff, which is prohibited in practice [15]. Constraint (9) ensures that if a flight takes off from the origin airport, it will arrive at its destination airport, otherwise it is cancelled. Constraint (10) enforces incoming and crossing flights to not be cancelled. This model is concerned with optimising the traffic over the UK airspace; therefore, flights originating outside the airspace are assumed to take place regardless of the decisions provided by this model. Constraint (11) enforces the connectivity in time. If a flight f reaches node n through arc $(m,n)$ by time t, $x_{f,m,n}^{t^{\prime }}$ is set to 1 for $t^{\prime }>t$. Similarly, Constraint (12) handles cases where t is not present in ${\mathcal{T}}_f^{m,n}$, but is present in ${\mathcal{T}}_{*f}^{m,n}$. In these cases, $x_{f,m,n}^t$ takes the value of $x_{f,m,n}^{t-1}$. Constraint (13) ensures that a flight does not exceed its maximum permissible duration. Constraints (14) and (15) ensure that the actual flight duration is equal to the scheduled duration plus/minus any positive/negative deviations. Constraints (16) and (17) specify the variable bounds.

2.5. Cost Coefficients

As mentioned in the formulation, the delay costs are expressed in the form of super-linear functions of the total delay and ground delay. More specifically, the total delay is expressed in the following form:

$$c_{f,t}^{k_f^a}\left(t\right)={(t-a_f)}^{1+{ϵ}_2}$$

Conversely, the ground-holding cost is expressed as:

$$c_{f,t}^{k_f^d}\left(t\right)={(t-d_f)}^{1+{ϵ}_2}-{(t-d_f)}^{1+{ϵ}_1},$$

where ${ϵ}_1<{ϵ}_2$ are user define constants. The values of ${ϵ}_1$ and ${ϵ}_2$ are found by fitting the super-linear function to the piece-wise linear function provided for the cost of delays in EUROCONTROL [16]. This stems from the fact that the cost per minute of delay increases as the delay gets larger. Cancellation, rerouting, and speed-deviation costs are derived from the operational costs provided in EUROCONTROL [16], as well as aviation fuel prices.

3. Simulation Setup

3.1. Air Traffic Network

The relevant air traffic network is a simplified UK air traffic network, with a reduced number of nodes, covering two of the three of UK’s Flight Information Regions (FIRs), the London FIR and the Scottish FIR. The network is shown in Figure 1, where green boundaries are the FIRs and deep blue dots represent the major airports.

The data consists of flights traversing the airspace on 1 June 2015. To further reduce the data size and hence the ATFM problem size, only flights from 06:00 to 18:00 are selected. In total, the flight data consists of 3861 flights. It is made up of four types of flights:

• Domestic flights: These are flights that originate and terminate within the UK airspace. They account for 230 of all flights.
• Outgoing flights: These are flights that depart a UK airport, however; their destination is outside the UK airspace. They account for 1654 of all flights.
• Incoming flights: Conversely, these flights arrive to a UK airport from outside the airspace. They account for 1469 of all flights.
• Crossing flights: Both the departure and arrival airports of these flights are outside the UK. They simply traverse the UK airspace. They account for 508 of all flights.

To ensure the route-generation algorithm functions effectively, entry and exit points on FIR boundaries are clustered using the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm [17].

3.2. Rolling Horizon

The simulation duration is set to 12 h, during which disruptions are introduced at a specific time, lasting either two or four hours. The network’s performance is then tracked through Key Performance Indicators (KPIs).

There are two types of disruptions, namely targeted ones and random ones. A targeted disruption is used to evaluate the significance of a weather disruption to key connectivity nodes, which are chosen using a node centrality metric. For the chosen nodes, the capacity of every arc connected to the node is set to zero, making those parts of the network unusable during the disruption interval, as is the case when an airspace experiences a significant weather event. A random disruption disables a number of arbitrary nodes in the network. The simulation process is shown in Figure 2. The simulation duration is set from 06:00 to 18:00 on the experiment day, with a step size of two-hour time windows. During each time window, when a disruption occurs, the DCB model determines the status of each affected flight, implementing measures such as delays, cancellations, or reroute to minimize overall costs. Following the disruption, air traffic gradually recovers to normal operation levels. All relevant data, including the implemented measures and recovery dynamics, are recorded and analyzed until the simulation concludes.

3.3. Route Generation

The main priorities for airlines when planning flight trajectories are fuel consumption and minimum travel times. Since the problem is modelled in 2D, it is assumed that airlines prioritise the horizontally shortest paths. To generate the possible routes for each origin–destination pair, a modification of the [18] algorithm is used. The algorithm can efficiently generate several potential routes. To further diversify the possible paths that each flight may take, the algorithm can remove the commonly used nodes, see Algorithm 1.

Algorithm 1 Flight route generation.

1: for all origin–destination pairs $(O,D)$ do   2:  $I\leftarrow \mathrm{level}\mathrm{of}\mathrm{diversification}$   3:  $N\leftarrow \mathrm{all}\mathrm{possible}\mathrm{nodes}$   4:  $R\leftarrow \mathrm{all}\mathrm{routes}\mathrm{for}(\mathrm{O},\mathrm{D})$   5:  for $i\leftarrow 1$ to I do   6:    Generate K shortest paths from N using Yen’s algorithm   7:    Add top 3 paths to R   8:    Compute the most used nodes in the shortest paths   9:    Remove up to 5 of the most common nodes from N 10:  end for 11: end for

The algorithm presents each flight with a wide selection of routes, with the scheduled route following the shortest one, which have been presented in Figure 3. The diversification ensures that all flights will have potential routes to follow, even if in disruption scenarios.

3.4. Centrality Metrics

For targeted disruptions on the network, the nodes are chosen based on a measure of centrality. A survey on a resilience analysis of air traffic networks by Wang et al. [6] shows that degree and betweeness centralities are the most commonly used metrics. This paper adopts a weighted betweeness centrality measure, which can be formally expressed as:

$$s_i=\sum _{p=1}^P{w}_p\sum _{r=1}^R{a}_{ir}\forall i\in \mathcal{N},$$

(18)

where $\mathcal{N}$ is the set of all nodes, P is the set of origin–destination airport pairs, $w_p$ is the number of flights in each pair, R is the set of all routes for the pair, and $a_{ir}$ is an adjacency matrix, taking a value of 1 if node i is present in route r and 0 if it is not. The heatmap of the computed node centrality is plotted on Figure 4. As is to be expected, the most active parts of the network are in the southeastern part of the airspace near the major hubs, namely London Heathrow Airport and London Gatwick Airport. Additionally, the most connected nodes also lie in the corridor between these airports and mainland Europe, where a large part of the flight traffic is focused.

Additionally, to measure the scale of each disruption, a normalised centrality sum is computed as:

$$C_d={\displaystyle \frac{{\sum }_{(i,j)\in A_d}(s_i+s_j)}{{\sum }_{(i,j)\in \mathcal{A}}(s_i+s_j)}}$$

(19)

where $A_d$ is the set of arcs that are disabled due to the weather event, $\mathcal{A}$ is the set of all arcs in the network. The normalised centrality sum essentially expresses what fraction of the total centrality in the network is disrupted.

3.5. Key Performance Indicators

The KPIs are selected from both the ATFM and resilience perspective. For the ATFM KPIs:

• Number of cancelled flights and the associated costs.
• Number of rerouted flights and the associated costs.
• Number of flights that depart more than 15 min late and the associated ground-holding costs.
• Number of flights that arrive more than 15 min late and the associated air delay costs.

For the network resilience KPIs, three metrics are chosen to analyze the performance and resilience of an air traffic network: System Wide Delay (SWD), punctuality, and the General Resilience Index (GRI).

3.5.1. System Wide Delay

To represent the demand–supply relationship of the network, a queuing diagram can be constructed by considering the scheduled arrival times and actual arrival times, as shown in Figure 5. The blue line represents the cumulative scheduled demands (arrivals) as a function of time, $S\left(t\right)$. The red line represents the cumulative actual arrivals (indicating the supply) as a function of time, $A\left(t\right)$. Up until time $T_1$, the two functions overlap, indicating a balance between the supply and demand of the system; however, then a divergence occurs due to a disruption. The actual arrivals now lag behind the scheduled ones, showing an imbalance between supply and demand. After $T_2$, the system begins to recover; however, it takes up until time $T_3$ for the system to fully recover from the disruption.

The SWD can be obtained by considering the striped area between the two functions:

$$SWD={\int }_{T_s}^{T_e}\left[S\left(t\right)-A\left(t\right)\right]dt$$

(20)

where $S\left(t\right)$ is the cumulative scheduled arrivals, $A\left(t\right)$ is the cumulative actual arrivals, $T_s$ is the start of the flight schedule and $T_e$ is the end of the flight schedule. This can also be applied to departures in the same manner.

3.5.2. Punctuality

Punctuality is a direct measure of system performance and it can be expressed as:

$$P_{t_1,t_2}={\displaystyle \frac{N_{ontime,t_1,t_2}}{N_{scheduled,t_1,t_2}}}$$

(21)

where $P_{t_1,t_2}$ represents the arrival punctuality for the time period $[t_1,t_2]$, $N_{ontime,t_1,t_2}$ is the total number of flights in the time period that arrived no later than 15 min from their scheduled time and $N_{scheduled,t_1,t_2}$ is the total number of flights scheduled to arrive in the time period. Once again, this can be applied to departures in the same manner to obtain the departure punctuality.

3.5.3. General Resilience Index

The GRI captures the absorptive, adaptive, and restorative capability of the network [7]. It can be defined for the network as:

$$GRI=R\times \left({\displaystyle \frac{RAP{I}_{RP}}{RAP{I}_{DP}}}\right)\times {\left(TAPL\right)}^{-1}\times RA$$

(22)

where R is the minimum performance (punctuality will be used) of the system during the disruption period $[t_d,t_{ns}]$, where $t_d$ indicates the time when the effects of the disruption occurs ($T_1$ in Figure 5) and $t_{ns}$ is the time when a new steady state is reached ($T_3$ in Figure 5). The minimum performance occurs at time $t_r$ ($T_2$ in Figure 5), after which the performance starts to improve again. $RAP{I}_{DP}$ is the rate of performance decline from the period when the first negative effects occur to the start of the recovery time (i.e., from $t_d$ to $t_r$). Similarly, $RAP{I}_{RP}$ is the rate of performance recovery from $t_r$ to $t_{ns}$. $TAPL$, or the Time-Averaged Performance Loss, is calculated as:

$$TAPL={\displaystyle \frac{{\int }_{t_d}^{t_r}\left(MOP\left(t_0\right)-MOP\left(t\right)\right)dt}{t_r-t_d}}$$

(23)

where $MOP$ is the Measure of Performance, which again is the punctuality. $t_0$ indicates the time at which the disruption occurs (note that this is different from $t_d$, since there can be a lag between the disruption and when the negative effects start to occur). Finally the Recovery Ability (RA) is defined as:

$$RA=\left|{\displaystyle \frac{MOP\left(t_{ns}\right)-MOP\left(t_r\right)}{MOP\left(t_0\right)-MOP\left(t_r\right)}}\right|$$

(24)

4. Results

The following two kinds of disruptions occurred in the experiment: the two-hour disruptions performed from 08:00 to 10:00, and the four-hour weather event disruptions performed from 08:00 to 12:00. These intervals were selected due to their high traffic density. Observations from the Rapid Developing Thunderstorms satellite product, provided by the Nowcasting and Very Short Range Forecasting Satellite Application Facility, indicate that convective weather can occur at any time during the summer. The chosen time window allowed us to focus on a critical operational period, where demand frequently exceeds capacity, creating a relevant scenario for assessing the impact of disruptions. Six instances of the targeted weather events are performed:

a. 

The top 5 most connected nodes are disabled.

b. 

The top 10 most connected nodes are disabled.

c. 

The top 20 most connected nodes are disabled.

d. 

The top 30 most connected nodes are disabled.

e. 

The top 40 most connected nodes are disabled.

f. 

The top 50 most connected nodes are disabled.

Figure 6 shows the affected waypoints for each of these disruptions along with the computed normalised centrality sum, indicating the scale of each disruption. In the first two smallest instances, the affected waypoints are entirely in the southeastern part of the network, between London and mainland Europe. However, once 20 and more central nodes are disabled, the waypoints north of London are also affected. By the last two instances, almost all waypoints in the southeastern part of the network are affected, representing around 44% of the total centrality of the network.

The random weather event disruptions are performed by disturbing 50 arbitrary nodes in the network. For data variation, four of these instances are performed. Figure 7 shows the affected waypoints for the random disruptions.

Unlike the targeted disruptions, the random disruptions are not concentrated in particular locations, and instead affect many small parts of the network; thus, the $C_d$ values for the random disruptions are significantly smaller than for all the targeted ones. This indicates that while the random disruptions affect a similar amount of arcs, the affected arcs may not be in high-traffic areas.

4.1. Two-Hour Disruptions

The results from the two-hour weather event disruption are displayed in

Table 1. Results from simulations with two-hour disruptions.

Disruption	${\mathit{C}}_{\mathit{d}}$	Cancelled Flights	Rerouted Flights	Late-Departing Flights *	Late-Arriving Flights *	Total Cost (€)
t-a-2	0.11	0	137	69	73	602,575 €
t-b-2	0.17	0	169	89	95	750,845 €
t-c-2	0.27	0	171	190	196	1,464,968 €
t-d-2	0.34	0	184	203	207	1,585,642 €
t-e-2	0.40	0	178	224	229	1,761,545 €
t-f-2	0.44	0	183	233	238	1,823,808 €
r-a-2	0.044	0	72	6	6	40,645 €
r-b-2	0.074	0	77	14	19	124,003 €
r-c-2	0.065	0	40	7	9	48,552 €
r-d-2	0.048	0	50	4	3	55,930 €

* Only includes flights with delays exceeding 15 min.

. The first column indicates the type of disruption. The first character indicates whether the disruption is targeted, ‘t’, or random, ‘r’. The second character refers to the disruptions as indicated in Figure 6 and Figure 7. The last character indicates that it is a two-hour disruption. Therefore, for example, disruption “t-d-2” refers to the targeted two-hour disruption where the top 30 most central nodes are disabled.

There are no cancelled flights under the two-hour disruption events. This means that the model provides sufficient rerouting capabilities to manage a quite large weather event, even in the areas with the highest traffic. To analyse the impact of the scale of the disruption, $C_d$, on the ATFM-related KPIs, the relations are plotted on Figure 8.

The plots show that with the increasing size of the disruption, the number of rerouted, ground-held and late-arriving flights increases as well. The number of rerouted flights increases very steeply initially; however, at larger $C_d$ values, the increase is very marginal. On the other hand, the number of delayed flights rises more steadily and does not stagnate as much, even at a large $C_d$. This suggests that the model’s first priority is to reroute a flight and if that is not possible, the flight is delayed. Additionally, the number of flights experiencing departure and arrival delays is very similar, suggesting that ground holds, rather than air delays, dominate the total delay. This is made further evident by the cost breakdown, as departure delays account for over 95% of the total cost. Therefore, even though the number of rerouted flights is large, the rerouting is mostly insignificant in terms of the distance and flight duration change.

The largest cost increase is seen between disruptions “t-b-2” and “t-c-2”, which also marks the largest increase in $C_d$ between the disruptions. Based on Figure 6, “t-c-2” is the first instance where the disruption spreads to the north of the London area, hence impacting more flight routes.

Arrival punctuality is used to monitor the performance of the network and assess its resilience. Figure 9 compares the punctuality and average arrival delays of flights for the different disruptions. The punctuality performance is directly related to the scale of the disruption, $C_d$, with the trough of the punctuality decreasing as disruption gets more significant. Interestingly, the time at which the performance reaches its low point and starts to recover is very consistent across all targeted disruptions. The rate of the decline of the performance is also consistent once the disruption reaches a scale of $C_d=0.27$. Overall, even in the worst case two-hour weather event, the punctuality stays above 80% and recovers to around 90% by the end of the day. For reference, from 1 October 2023 to 1 June 2024, the average daily arrival punctuality in the UK was at around 70%. Therefore, despite the two-hour disruption to the highest-traffic areas, the network is able to perform quite well in terms of arrival punctuality of the whole day of operations. The average arrival delay is also under 15 min for the whole day of operations. However, if only considering the delayed flights, the average arrival delay is around 58 min with a standard deviation of 28 min. This is also consistent across all scales of targeted disruptions. In contrast, for the random disruptions, the punctuality rate is barely affected and the average delay for the whole day of operations is very small. This can be attributed to several factors. A primary reason is the network’s inherent design, characterized by multiple interconnected nodes and flexible routing options. This interconnected structure facilitates rerouting and traffic redistribution when certain sectors face disruptions, effectively mitigating their impact. Additionally, the relatively short duration of these disruptions allows the network to maintain its capacity to absorb and adapt effectively. Together, these factors contribute to the network’s ability to maintain efficiency despite disruptions.

The punctuality curves can be used to compute the GRI as described in Section 3. The obtained values are plotted with a curve of best fit on Figure 10. The SWD is plotted in the same manner on Figure 11. These plots show that there is a strong negative relationship between the GRI and the scale of the disruption. Conversely, there is a strong positive relationship between the SWD and the scale of the disruption.

Based on these metrics, the model can help recover from random disruptions, but targeted disruptions have a much greater impact on system performance. At high $C_d$ values, there is no significant change in GRI. This indicates that when the scale of disruption exceeds a certain threshold, the available airspace is limited and cannot accommodate a large volume of traffic. Due to this inherent constraint, the model’s ability to provide recovery based on the disrupted airspace is limited.

4.2. Four-Hour Disruptions

The results from the four-hour weather event disruption are displayed in

Table 2. Results from simulations with four-hour disruptions.

Disruption	${\mathit{C}}_{\mathit{d}}$	Cancelled Flights	Rerouted Flights	Late-Departing Flights *	Late-Arriving Flights *	Total Cost (€)
t-a-4	0.11	53	208	74	81	5,129,274
t-b-4	0.17	68	274	85	97	6,631,913
t-c-4	0.27	153	244	201	214	14,667,322
t-d-4	0.34	169	272	216	230	16,151,495
t-e-4	0.40	187	278	251	267	17,969,697
t-f-4	0.44	195	269	266	279	18,771,137
r-a-4	0.044	1	185	14	17	275,606
r-b-4	0.074	7	170	40	51	1,080,992
r-c-4	0.065	5	104	31	37	724,319
r-d-4	0.048	2	116	27	34	368,965

* Only includes flights with delays exceeding 15 min.

.

The main difference with the two-hour disruption is that, even at the smallest scale of disruption and even for random disruptions, the model can only maintain operations by cancelling flights. The results show that all flight cancellations occur within the first two hours of the four-hour storm. Considering the largest disruption, “t-f-4”, out of the 604 flights between 08:00 and 10:00, 195 get cancelled. This is a very large portion of all flights; hence, it can be concluded that the network is very susceptible to extended weather events in the high-traffic areas. Based on the model’s constraints, flights have a maximum ground delay period of two hours, after which flights are cancelled. This could explain why no cancellations occur for the two-hour disruption and why all the cancellations occur in the first two hours of the four-hour event. During the shorter disruption, a lot of the flights are able to essentially wait out the storm, and depart once the capacities have returned to normal. However, with the extended weather event, these flights would exceed their maximum ground-holding time and be cancelled instead.

The punctuality rates and the average arrival delays are plotted on Figure 12.

The observed patterns are similar to the two-hour event. The punctuality rates hit the low point around 80 min after the beginning of the disruption, just as with the two-hour disruption. Again, the performance is closely correlated to the scale of the disruption. The punctuality rates observed are slightly higher than in the two-hour event; however, this is because cancelled flights are excluded from the calculations. Therefore, in terms of the punctuality and delay, the network performance is similar to that with the shorter disruption. In fact, when considering the average delay of delayed flights, the average is lower than for the two-hour disruptions. The arrival delay average is 56 min across all disruption scales, which is 6 min less than observed in the two-hour disruptions. The standard deviation is similar, at 31 min. However, for the random disruptions, the average arrival delay of delayed flights is 45 min, which is 20 min higher than the observation seen during the two-hour disruptions.

Overall, as with the two-hour event, the random disruptions have a significantly lower impact than the targeted disruptions. The relationship between the scale of the disruption and the GRI, as well as the SWD, is plotted on Figure 13 and Figure 14. Once again, there is a strong correlation between the scale of the disruption and the performance indicators. However, for the GRI, the difference between the random and targeted disruptions is not as large as shown in the two-hour disruption results. Nonetheless, the aforementioned phase transition is still visible. Note in Figure 13 that as $C_d$ increases, GRI generally exhibits a declining trend, but there are instances where it rises. This increase in GRI with higher $C_d$ values is primarily due to the nature of random disruptions. Unlike targeted disruptions, which focus on high-traffic areas or critical nodes, random disruptions are dispersed throughout the network. Consequently, although a comparable number of arcs with specified $C_d$ values are impacted, these arcs are not necessarily associated with high-traffic or strategically critical areas. This dispersion can lead to an increase in GRI even as $C_d$ rises.

Considering the total cost, the four-hour weather events are significantly more disruptive to the system than the two-hour events. Even certain four-hour random disruptions can be more detrimental than targeted two-hour disruptions. The results indicate that even for a two-hour disruption targeting high-traffic areas, the model can effectively assist the system in gradually resuming normal level operations. Even at the maximum disruption scale, the model can optimize the schedule without any flight cancellations. However, for weather events lasting four hours, the model’s optimization capacity significantly decreases, leading to a substantial increase in the total number of flight cancellations and associated costs. Flight cancellations may occur even in areas that are not high-traffic zones.

5. Conclusions

Weather-induced delay is an important problem in the aviation industry. With the ever-growing demand, the air traffic networks’ ability to stay operational during weather events is crucial and can result in the savings of millions due to prevented delays and cancellations. The resilience of a network becomes an important factor in determining how well it will perform when subjected to weather disruptions. Current resilience analysis research focuses on the importance of particular airports in an air traffic network; however, there are limited studies on the role of en-route waypoints in facilitating the network’s performance during weather events. This paper conducts a resilience analysis of the UK air traffic network under weather disruptions to its high-traffic areas. An ATFM model is used to optimise the cancellation, rerouting, and delay of flights.

Disruptions of different scales (measured by the fraction of total node centrality that is disabled) and durations are explored. Additionally, we compared the model’s capabilities to support system recovery under targeted and random disruptions. The simulation results indicate that the model can assist the system in gradually resuming normal operation levels during two-hour disruption events, even when these events occur in the busiest parts of the network. The system can maintain operations with sufficient capacity and rerouting ability to prevent any flight cancellations. The number of rerouted and delayed flights is significant; however, these occurrences are contained in the disruption window. Post-disruption, the network returns to optimal performance. When the weather events extend to four hours, it becomes more challenging for operations to return to normal levels. Prolonged disruptions lead to flight cancellations, even if the scale of disruption is small or located in lower traffic areas. Overall, the model exhibits a much stronger recovery capability under random disruptions compared to targeted disruptions. The recovery capability of the model is negatively correlated with the scale of disruption. However, beyond a certain disruption scale, further increases in disruption scale do not significantly reduce the model’s recovery capability.

The main limitation of the study is that the ATFM model does not consider flight continuation. This limits the ability to model the propagative effects of the delays induced by weather disruptions. Additionally, the en-route capacity is modelled using a aircraft separation constraint, rather than considering sector capacity or complexity, which could be improved further. In reality, airspace resilience is influenced by a range of factors, including sector capacity limits, human factors, as well as technological constraints; these could all lead to an unexpected slow recovery from disruptions. Further research should aim to expand the modeling approach to incorporate these additional dimensions, in order to provide a more comprehensive assessment of the airspace system resilience performance.