Abstract
With the rapid growth of civil aviation, the increasing expansion of air traffic flow has brought serious challenges to the service capacity of the current airspace system, making the operation of the control sector increasingly complex. The accurate quantification of sector situational complexity is the basis for improving the service capability of airspace systems. The existing research on complexity ignores the resilience of the air traffic system in case of flight change, which cannot fully describe the dynamic characteristics of the air traffic situation. For this reason, a new air traffic complexity evaluation algorithm based on system resilience is proposed. Firstly, an air traffic situation network based on between-flight interaction is established. Then an overall sector complexity index based on network efficiency, average network failure rate, and average network recovery rate is built. Then, the complexity index is verified by analyzing the real radar number of ZSSSAR01 (sector 1 of Shanghai). By establishing a sector complexity optimization model, the complexity of sector air traffic and its volatility can be greatly reduced by changing the departure time of some flights. Finally, by optimizing the complexity of the sector, the workload of controllers is reduced, and the safety and efficiency of air traffic operations are improved.
1. Introduction
Along with the stable development of the global economy, the demand and flow of air traffic are continually increasing. However, it is encountered by airspace congestion, delayed flight, and workload increment of controllers. Moreover, the operation situations of air sectors, as the basic airspace unit, become more complex. The complexity of air traffic operation situations is the core influencing factor of workload for controllers. Thus, research on the evaluation and control strategies of air traffic complexity in sectors will significantly contribute to reducing the workload of controllers and improving operational safety and efficiency.
The existing studies generally agree that the air operation should have set force to avoid air traffic interaction. Based on this, a number of methods are proposed to reduce the air traffic interaction in terms of complexity. To avoid air traffic interaction, the existing research on the evaluation of air traffic complexity is focused on aircraft density, dynamic density, traffic flow disturbance, and complexity measurement based on between-flight relationships. For instance, Wang et al. regarded potential interactions as an influential factor in the occurrence of flight conflicts and built an air traffic complexity model with the dynamic weighted network by considering the effects of both airspace and traffic flow [1, 2]. Rhonda et al. analyzed air traffic complexity according to between-flight relationships based on flight paths to avoid between-flight interaction [3–5]. To reduce conflict in the terminal area, Dong and Wen analyzed the complexity of terminal traffic organization relationships using theories of nonlinear mechanics and approximate entropy [6]. To control excessive interaction during aircraft approach, Wang et al. studied the approaching effect of aircraft and conflict disengagement index, built a complexity calculation model of sector between-flight influence relationships, and proposed the algorithms for the sector complexity and the sector distributing complexity under single aircraft disturbance [7]. Hirabayashi et al. proposed a “difficulty index” of air traffic control to measure airspace complexity, conducted simulation and visualization according to real data of flight plans, and attempted the future application based on track movement [8]. Xi proposed a semi supervised learning model for air traffic complexity evaluation. The experimental results show that this model can effectively utilize the complexity evaluation information contained in unlabeled samples, and achieve better performance than the traditional supervised evaluator when sacrificing the same amount of labeled samples [9]. Hua and Chen put forward an airspace complexity measuring method based on en-route construct constraints and an “airplane pair” model, which modified an airspace coupling complexity model [10]. Experimental data of practical airspace motion validate that this model can reflect the space coupling situations and airspace complexity between aircraft. However, the indices used in the above studies cannot fully describe air traffic complexity and excessively focus on the current air traffic situation complexity in sectors while ignoring the air traffic complexity in sectors at the strategic stage. Other researchers have studied complexity from the perspectives of air traffic density and dynamic density. Prandini et al. proposed a new air traffic complexity measuring method based on-air traffic density [11], but this method is only applicable to independent separation control among aircraft for future movement. Zhu et al. presented a small-sample-based integrated learning model to measure the air traffic complexity in sectors [12]. These studies focus on the exploration of air traffic complexity, but rarely evaluate the future air traffic complexity in sectors since the strategic stage of flight departure. The complex network theory is a major tool for research on complexity science and complex systems. Sebastian et al. systematically evaluated the robustness of transportation networks by leveraging the community structure and proposed two effective and scalable network dismantling algorithms by separating communities, considering both node removal and edge removal on transportation networks. The results provide a scalable method for network dismantling, shed light on the important roles of intercommunity nodes/edges for robustness improvement, and deepen our understanding of the modularity in transportation networks [13]. Then, Wandelt et al. proposed a novel perspective to solve the network dismantling problem. The efficacy of the novel method for network dismantling provides an excellent trade-off between attack quality and scalability [14]. Especially, the complex network theory has been introduced into research on-air traffic control systems and air traffic complexity. Wang and Song used complex networks to study airway networks and route networks in China [15]. Later, based on the community structure of complex networks, they described the structure of air traffic situations, established an airplane clustering model, and proposed an airplane cluster discovery algorithm based on depth-first traversal [16]. In all, the existing studies overly focus on tactical operations, but ignore the strategy control of air traffic complexity in sectors. Moreover, the dynamic and elastic changes in air traffic situations under the continual variation of flight paths are ignored. At the same time, other studies did not consider the conceptual requirements of Trajectory Based Operation in the future air traffic management system. In this study based on the idea of system resilience, the evaluation index of air traffic complexity in sectors was designed to respond to the Trajectory Based Operation requirements in the air traffic management system. The air traffic complexity in sectors at the strategic stage is evaluated and optimized to decrease the workload of controllers and improve the operational efficiency of air traffic control systems because the controller’s workload is affected by factors such as air traffic flow, airspace structure, air traffic control equipment quality differences, individual differences, and cognitive strategies. Air traffic complexity is to study the air traffic flow and airspace characteristics to evaluate the complexity of the control sector. Therefore, reducing the complexity of air traffic through optimization will reduce the workload of the controller to a certain extent, thereby improving the efficiency of air traffic operation to a certain extent.
2. Air Traffic Situation Network Analysis
2.1. Description of Problems
The operation situation of an air traffic control sector consists of flights from different airports (Figure 1). This study is aimed to predict the air traffic complexity of the target sector before flight takeoff. Based on the system resilience theory, the evaluation index to measure air traffic complexity in the target sector would be identified, and the air traffic complexity in the sector would be reasonably controlled. The problem studied in this paper is based on the following assumptions: the concept of trajectory-based operations (TBO) is correct, the trajectory prediction accuracy is high, and the impact of special weather and military activities on the trajectory is not considered. In this paper, three questions are examined. The first is the evaluation of air traffic complexity, the second is to verify the air traffic complexity evaluation model proposed in this paper, and the third is the optimization problem of sector air traffic complexity.
2.2. Between-Flight Interaction
Based on the above assumptions, this paper proposes a new method to measure the between-flight interaction.
Before takeoff, a flight plan usually is delivered to the en-route control sector. The plan covers the flight number, takeoff airport, target airport, planned takeoff time, and a set of en-route waypoints. By screening the flights’ related waypoints in the target sector, it is valuable to find out all flights to enter the target sector in the future.
Firstly, based on the historical data of the flight, the flight duration and flight altitude from the departure point to waypoints are calculated for the flight determination. The mean and standard deviation of the flight duration and altitude at every waypoint are calculated from the data in the sample. Then, a 95% confidence interval is further calculated for the flight duration and altitude sample. This interval is used as a range of flight times and altitudes that can be used for subsequent studies, which will improve the validity of the data for model construction. Finally, the degree of interaction is visualized in the form of a graph (Figure 2).
Figure 2 shows how to determine the interaction of flight time and altitude between the two interacting flights. Flight and Flight take off from different airports, denoted as and . And both of Flight and Flight fly over the same waypoint, denoted as . Then, focus on the set of flights that pass through the same waypoint, , to determine the degree of interaction.
If the predicted passing-point time ranges of flights and meet any of equation (1), the passing-point time ranges of two flights are considered as “interacting,” and the longitudinal interaction between two flights, , is the ratio of overlapping time to the smaller value of the flight time of two flights.otherwise .
If the predicted passing-point altitude ranges of flights and meet any of equation (3), the passing-point altitude ranges of two flights are considered as “intersecting,” and the vertical interaction between two flights, is the ratio of the overlapping altitudes to the smaller value of the altitude range of two flights.otherwise .
The sum of the longitudinal interaction and vertical interaction of two flights is determined in equation(5).
2.3. Construction of Air Traffic Situation Network
Figure 3 illustrates a sector air traffic situation network model. According to the complex network theory, the flights to pass the waypoint are considered as nodes. If longitudinal interaction or vertical interaction occurs between two flights, these two nodes share an edge, and the weight of the edge is the interactive value of the two flights. The connection relationship between the nodes is expressed by an adjacent matrix .
Next, a sector air traffic situation network model based on the simulated flight data will be built. Table 1 lists the sets of flights predicted to pass waypoints LJG, DO, and FQG in the sector. According to the between-flight interaction judging method, the adjacent matrix of between-flight interaction is determined (Table 2).
2.4. Evaluation of Air Traffic Situation Complexity
The air traffic situation complexity of a sector is essentially the interactions among all elements in an air traffic system [15]. The network model of the sector air traffic situation described above clearly represents a network of situations consisting of interactions between flights, and the air traffic situation becomes more complex when the sector involves more interactions between flights.
Latora and Marchiori studied the small-world properties of complex networks and proposed an overall network efficacy index to measure network damage resistance [17]. This index measures the shortest path length between nodes in a network. In equation (6), is the overall network efficacy and is the between-node distance. When no path connects two nodes and , is .
This index reflects the network efficiency only from the aspect of structural characteristics, but the air traffic situation of a sector is a weighting network, in which flights are considered as nodes, the between-flight interactions as edges, and the values of interactions as weights. Hence, the average distance is unsuitable for this network, and the weights of all edges in the network must be considered. The network efficiency of the air traffic situation, , in a sector is calculated as follows:
In the above equation, the is the weight between nodes and . The value of is determined from equations (1)–(5). The air traffic situation network efficiency of a sector reflects the interactions among the flights predicted to enter the sector. Since the air traffic complexity of a sector is essentially the interactions among flights in the sector, network efficiency can be used as an index to measure the air traffic complexity of this sector. The larger is, the more potential interactions between flights within the sector, and the greater complexity of air traffic in the sector.
However, the use of network efficiency alone is inadequate to reflect the air traffic complexity of this sector. The network efficiency values of two air traffic situations are both 8, but the air traffic complexity reflected by these two situations is different (Figure 4), and the change of air traffic situation after the deployment of a flight is also different. The air traffic situation that dynamically changes with flight adjustment reflects the system’s resilience. Thus, the interaction between elements of an air traffic situation network is measured by the idea of system resilience. The system performances change differently after the disturbance (Figure 5). Before, the system under the original state is unaffected by disturbance. After disturbance occurs, the system performance starts to decline over time, but the system may develop along three different routes. On route A, when the negative effect of a destructive event dissipates, the system performance will gradually recover in the case of no external intervention until it returns to the original stable status. On route B, the system finally reaches a new stable status. On route C, the destruction degree exceeds the tolerable limit of the system, so the system completely collapses in the end. Nan and Sansavini proposed a system resilience quantitative assessment method and validated its effectiveness in electrical power systems [18]. Cui et al. built a command information system supernetwork model and proposed a command information system supernetwork elasticity measuring method based on the probability attack and elastic strategy. Then in the case of air defense command information systems of a certain region, they measured the system plasticity in response to different attacking ways and recovery strategies [19]. Wang et al. established an airport network model and analyzed the operation performance and resilience of air traffic systems based on network topology structures and dynamic properties [20]. Wang and Miao established a Mainland China air sector network model and the index of sector network resilience and compared this index under different recovery strategies [21]. Moreover, subgraph structural resilience was proposed to characterize the dynamic evolving laws of network topology structures. It was found that subgraph structural resilience and network macroscopic structural changes were very consistent [22].
In the air traffic situation network, the disturbance refers to the adjustment of the trajectory by air traffic controllers. In the strategy phase, some of the flight conflicts can be resolved by adjusting parameters such as expected departure time and flight altitude. After the adjustment, the interaction between flights changes, and both the complexity of the intrasectoral situation and the structure of the intrasectoral air traffic situation network also change. Assuming that the smaller the perturbation of the flight track adjustment, but the greater the damage to the structure of the air traffic posture network, it is suggested that a large number of potential interactions between flights can be weakened by a smaller number of flight track adjustments. It reflects the less complexity in the original situational network. In other words, a larger complexity in the original air traffic posture would require a large number of flight trajectories to be adjusted to effectively reduce the potential interaction relationships between flights. Therefore, it is valuable to reflect the strength of the complexity of the traffic dynamics in the sector through the resilience of the system.
When controllers adjust relevant flights in the air traffic situation network of a sector, the air traffic situation network may change along three directions. In the first direction, the efficiency of the air traffic situation network in the sector is gradually weakened, but is slowly enlarged with time, which is consistent with route A in Figure 5. In the second direction, which is basically consistent with the first direction, the network is more severely destroyed, and the variation trend is basically consistent with route B in Figure 5. In the third direction, when the controller adjusts the flight trajectories, the air traffic situation network is gradually destroyed and finally collapses.
When system resilience is adopted to reflect the degree of air traffic situation complexity, the network efficiency cannot comprehensively measure the air traffic complexity in the sector. Nan and Sansavini proposed a quantitative assessment method of system resilience [18], in which the system destruction rate is defined as the rate of performance loss after a system is affected by external disturbance. The system destruction rate upon disturbance can be defined by the average system destruction rate:
In the above equation, is the average system damage rate; is the number of lines at the destruction stage; is the system performance at the -th line.
Inspired by the use of resilience to express the average system destruction rate, we proposed the average network destruction speed :
In the above equation, is the total number of regulated flights; and are the total between-flight interactions in the sector before and after flight adjustment, respectively; represents the -th flight adjustment.
If the controllers adjust only a few flights and the between-flight interactions change little, the complexity of air traffic situation in the sector is low. If the controllers adjust many flights and the between-flight interactions change nonobviously, the complexity of air traffic situation in the sector is high. Hence, when at a larger average network destruction rate, the air traffic situation complexity in the sector is low. And when at a smaller average network destruction rate, the complexity is high.
The recovery rate refers to the performance recovery rate when the system is disturbed to the lowest performance level:
In the above equation, is the average system recovery rate, is the number of lines at the recovery stage. Inspired by the system recovery rate, we proposed the average network recovery rate :
In the above equation, is the total number of recovered flights; represents the -th randomly recovered flight; represents the sum of between-flight interactions in the sector after the flight is recovered; represents the sum of between-flight interactions in the sector before the flight is recovered.
According to the elements in the air traffic situation network of the sector, an interaction related to flight recovery is randomly selected. Then for each flight, the related interactions are independently recovered, and finally the values are averaged to get the average network recovery rate. After flight adjustment, the air traffic complexity of a sector can be lowered to a certain level. However, random factors such as weather may lead to an increase in the number of between-flight interactions in the target sector. For this reason, the average network random recovery rate is introduced to measure the variation of air traffic complexity. A larger average network random recovery rate indicates the air traffic in the target sector is more complex, and vice versa.
According to the above evaluation indices of sector air traffic complexity, an overall complexity index based on comprehensive consideration is proposed:
In the above equation, represents the overall complexity of the air traffic situation. represents the network efficiency of the sector air traffic situation network. represents the maximum network efficiency under the available network. represents the average network disruption rate. represents the average network recovery rate.
3. Validation with Simulations
3.1. Examples
This study will give an example to illustrate the calculation of various indicators to measure the complexity of air traffic situation. Given a simple scenario in which there are three waypoints, the calculation process is explained through the air traffic situation of three sectors in three periods (Figure 6). The flight simulation data for the three periods are shown in Table 3.
In the first period, there are four flights. The air traffic situation network is established according to equations (1)–(5) above, and then the network efficiency is calculated according to equation (7). The average network destruction rate and the average random recovery rate can be calculated (Table 4).
When calculating the average damage rate and average random recovery rate, the air traffic situation network is divided into 15 cases. The destruction rate and recovery rate under different degrees of network damage and different degrees of recovery. This study is based on the assumption that after the network is damaged, the recovery is as large as before the network damaged. In real air traffic control, the adjustment of flight trajectory may be affected by atmospheric conditions, aircraft performance and weight, efficiency standards used by airlines, and controllers’ perceptions, resulting in different network damage rates and random recovery levels. So here, the average network damage rate and network recovery rate under different conditions are averaged, and finally, the average network damage rate and the average random recovery rate are obtained (Table 5).
4. Validation with Real Data
To validate the new complexity metrics, this section will use real historical data from the Shanghai 01 sector. Since the air traffic volume in East China is relatively large around New Year’s Day in January each year, the data used in this study are the flight data of Shanghai Sector 01 from January 1 to January 7, 2019. Figure 7 shows the spatial structure of ZSSSAR01.
According to the data of these 7 days, the flight flying duration and flying altitude from various airports across the country to the Shanghai sector 01 waypoints are calculated (Figures 8 and 9). The data are used as a sample to estimate the flight flying duration and flight flying altitude range of the flight passing point in the real situation. Calculate the mean and standard deviation from each airport to each waypoint in the sector in the sample, and further calculate that the 95% confidence interval of the corresponding sample is the range of flight time and flight height from the corresponding airport to the corresponding waypoint. Then get the flying duration range and point-passing height range at each waypoint in each airport of this sector.
Figures 10 and 11 demonstrate the flying duration and flying height from ZSSS, ZSPD, ZSNB, ZSHX, ZSNJ, ZGGG, or ZGSZ to waypoints ABVIL or ELNEX in ZSSSAR01. Clearly, the flying duration and flying height from each airport to the waypoints in the sector change within stable ranges, and the flying duration is positively correlated with the distance from the takeoff point to this sector.
Statistical analysis of the above data indicates the flying duration and flying height passing the same waypoint change within certain ranges. Then the reasons for such changes were analyzed. During performance-based navigation (PBN), the overall system errors can cause deviation in the real route from the calibrated route, leading to a certain time deviation in the flight to pass the waypoints. The uncertainty in takeoff time also affects the passing time of the flight. Due to the effect of high-altitude winds, the flight rapidly changes, which results in a variation in the point passing the time of the flight. The change of flight flying height is affected by many factors, such as air status, aircraft performance and weight, efficiency standards used by the airline, and requirements for controllers.
According to historical data, the minimum flying duration, maximum flying duration, minimum point-passing height, and maximum point-passing height at each waypoint in each airport of ZSSSAR01 were statistically analyzed. At each segment from 08:00 to 20:59, relevant aircraft in the sector constituted an air traffic situation network. The air traffic complexity at each time segment from 08:00 to 20:59 was calculated. The number of aircraft passing this sector at each segment was counted, and the network connection density in the air traffic situation network constituted at each segment was computed. Since the air traffic situation network is a weighted network, network connection density represents the connection between nodes in the network, but cannot reflect the importance of the connection between nodes. Network efficiency can reflect not only the connection situation but also the importance of the connection.
The network connection density at each segment is not significantly correlated to complexity or aircraft number, but the complexity and aircraft number are significantly consistent (Figure 12). Hence, network connection density alone does not reflect the air traffic complexity of a sector, but the overall complexity index proposed here can reveal the air traffic complexity. However, the aircraft number in the duration 08:00–08:59 is the same as that in 09:00–09:59, but the complexity is different. Then the causes for such difference will be explained.
As shown in Figures 13 and 14, the total area of green dots in a waypoint represents the total number of interactions of relevant flights at this waypoint. In 09:00–09:59, the value of interactions among waypoints ABVIL, TOL, MULOV, and MOLGU in the sector is larger than that in 08:00–08:59. For instance, there are horizontal interactions in 4 flights and vertical interactions in 5 flights at waypoint MULOV in 08:00–08:59. In 09:00–09:59, there are horizontal interactions in 8 flights and vertical interactions in 9 flights. There is no horizontal interaction or vertical interaction at waypoint MOLGU in 08:00–08:59, but there are horizontal interactions in 3 flights and vertical interactions in 6 flights in 09:00–09:59. These results indicate the overall sector complexity can more accurately reflect the air traffic complexity of a sector.
5. Optimization of Air Traffic Complexity
5.1. Description of Problems
Given the air traffic complexity, the estimated time of arriving in the target sector can be controlled and the between-flight interactions can be reduced by changing estimated time of departure, which will lower air traffic complexity in the sector. Five hypotheses based on mathematical models are proposed.(1)Deterministic demand. The total number of flights is known.(2)No on-air waiting. Since on-air waiting will raise the fuel costs and safety costs, the cost of on-air waiting is higher than that of on-ground delay.(3)No flight cancellation. Flight cancellation may occur due to weather, flow control, a machinery breakdown, or other reasons. For simplification of mathematical models, flight cancellation is not considered here. It is assumed no flight will cancel flight plans.(4)Certain advanced quantity of flights. The variable under control here is the takeoff time alteration of flights. The consideration of certain flight advance quantity can further relieve the air traffic complexity in the target sector.(5)The capacities of other sectors passed on the flying routes and the air traffic complexity of other sectors are not taken into account.
5.2. Modeling
There are flights , a flight set , and waypoints in the target sector. The takeoff time of the flight is , the takeoff time variation is , the minimum flying duration passing waypoint is , the maximum flying duration is , the minimum flying height is , and the maximum flying height is . The duration 08:00–20:59 is separated into 13 segments, which each lasts one hour, .
Decision variable. Set the takeoff time changing amount of each flight as the decision variable.
Optimization objective 1: Let the sum of air traffic complexity from the 13 segments in the sector be the smallest. The air traffic complexity at time segment can be computed from equations (1)–(5), (7), (9), (11), and (12).
Optimization objective 2: Let the sum of flight takeoff time variations be the minimum.
Optimization objective 3: Let the total number of adjusted flights be the minimum.
If the takeoff time change of flight is 0, this flight is not adjusted; if the takeoff time change is not 0, this flight is adjusted.
The constraint is the range of takeoff time variation. According to the regulations of the Civil Aviation Administration of China, a delay within 15 minutes is regarded as a normal flight. Therefore, the amount of time adjustment is limited to 15 minutes.
Air traffic complexity optimization is a multiobjective problem, and different objectives are mutually related. Optimizing all the objectives of a multiobjective optimization problem is very difficult. For this reason, the objectives are assigned with different weights, and the multiobjective problem is transformed via linear weighting into single-objective optimization problems that can be solved. However, due to the lack of common measuring standards among different objectives, the important degrees among objectives cannot be quantified, making it difficult to reach final objective satisfaction. Moreover, the weight coefficients of different objectives should be determined by repeated experiments, which take much time.
Messac proposed a physical programming algorithm to solve multiobjective optimization design problems [23] that do not require the determination of weights and offer a new clue for solving multiobjective optimization problems. During physical programming, a dimensionless preference function reflecting the preference over each objective is designed. Then the preference functions are synthesized into a comprehensive preference function, which is regarded as the final optimization objective of physical programming.
In physical programming, four types of indices can be designed according to different preferences: (1) smaller indices correspond to better preference, (2) indices at certain values correspond to the optimal preference, (3) larger indices correspond to better preference, and (4) indices within certain range correspond to the optimal preference. This study only focuses on the first type and explicates the significance of physical programming with the first type as example (Figure 15).(1)much expected domain much expected target value is within this range;(2)expected domain expected target value is within this range;(3)acceptable domain acceptable target value is within this range;(4)unexpected domain unexpected target value is within this range;(5)much unexpected domain much unexpected target value is within this range.
Such design preference function unifies the indices with different physical meanings into the same order of magnitude and facilitates computation and comparison. The optimization based on physical programming undergoes three steps. First, a preference structure is designed, and the boundary values of the preference function are determined. Second, the preference function is solved, and according to its boundary values, the parameters of each preference function are solved. Third, objective functions are solved and thereby, the preference function values are reverse-calculated. After that, the overall preference of the physical programming problem is clarified, and further optimized algorithmically. According to the characteristics of the three objective functions, the preference functions are set as piecewise functions with values within [0, 1]. The preference functions are expressed as follows:
In the above equation, is the target value, is the boundary point of preference ranges and is the preferred value at each boundary point. Here the following equation is referred to
In this way, a preference function is constructed, and for any target value, the corresponding value of the preference function can be determined from equation (18). For 3 objectives, let be the -th objective. Then the overall preference of physical programming or namely the optimized target function can be determined from equation (19). At this moment, the multiobjective optimization problem is transformed to single-objective optimization problems.
5.3. Algorithms
Since linear algorithms lack the global searching ability and cannot find the optimal solutions to optimization problems, an intelligent algorithm is utilized here. The genetic algorithm is an intelligent algorithm simulating organic evolution and is widely used in combination optimization, machinery learning, machine learning, and signal processing. The combinational optimization problem in this study can be solved by the genetic algorithm. The genetic algorithm undergoes six steps [24].(1)Initialization of populations. The chromosomes of each individual are randomly assigned, and the values are set within the given range under the constraints.(2)Computation of fitness. The fitness function requires that the function value should be greater than or equal to 0. The values of the preference functions here are greater than or equal to 0, which meets the above requirement. The maximal fitness obtained by the genetic algorithm is exactly the optimal value.(3)The selecting operation. The individuals are selected by the roulette method. The selection operation is a process of “survival of the fittest” with fitness as the standard. The probability of selecting an individual is proportional to its fitness.(4)The crossover operation. Individuals are randomly selected from a population, and a random number is generated before the crossover operation of each pair of individuals. When the random number is smaller than the crossover probability, the starting and ending gene points of a fragment to be crossed over are randomly generated, and the fragments of two individuals to be crossed over are exchanged, forming two new individuals.(5)The mutation operation. A random number is generated and compared with the mutation probability. If the random number is smaller, two different gene points are randomly selected and their positions are exchanged, which are one mutation operation. For instance, the gene points at sites 1 and 4 before and after random mutations are and , respectively.(6)Determination of optimal chromosome and optimal preference function value. Based on comprehensive consideration, the parameters of the genetic algorithm are set as: population number of 80, gene generation number of 400, crossover probability of 0.9, and mutation probability of 0.8.
5.4. Optimization Results and Analysis
The complexity is optimized in the duration of 20:00–20:59. As the number of iterations rises, the air traffic complexity of the sector is increasingly closer to the optimal value (Figure 16), thus indicating that the algorithm has high convergence.
The complexity of each period is filtered. And then when the complexity of exceeds the threshold value, the period is optimized. The optimization results are shown in Figure 17. The threshold value depends on the tolerance of the workload by the controller. In this paper, the threshold value is set to 80% of the maximum value of historical complexity. After the optimization by genetic algorithm, the complexity of all four peak periods decreases to below the average level. And, by adjusting the flights for the four peak periods, the average complexity is reduced from 0.71 to 0.66. This shows that the optimization is efficient and meaningful.
From Figures 18–20, it is found that in the postoptimization period, , , and , all three indicators change significantly after optimization. After optimization, has been a more obvious drop than before optimization. The average network destruction rate has increased significantly after optimization than before optimization. The average random recovery rate decreased significantly after optimization than before optimization. Under the combined effect of these three, the complexity of air traffic has decreased significantly.
The standard deviation of complexity changes at smaller amplitude after the optimization (Figure 21). The complexity at each period after the optimization fluctuates at smaller level than before optimization, and will not cause severe fluctuation in the workload of controllers and is favourable for the works of controllers.
Figure 22 shows the aircraft volume, the adjusted aircraft volume, and the share of adjusted aircraft. The percentage of adjusted aircraft during the optimization period ranged from 18% to 23%. The change in departure time per segment ranged from 40 to 55 minutes, with an average change in departure time from 3.6 to 6.3 minutes per flight (Figure 23). According to the definition of flight delay and the results of the analysis, it can be seen that the small cost of adjusting the flight departure time will not cause a large delay to the flight, and at the same time, it will bring a relatively large reduction in the complexity of the sector’s air traffic and has a certain positive significance for reducing the workload of controllers.
6. Conclusion
A new air traffic complexity evaluation algorithm based on system resilience is proposed. Firstly, an air traffic situation network based on between-flight interaction is established. Then an overall sector complexity index based on network efficiency, average network failure rate, and average network recovery rate is built. Then the complexity index is verified by analyzing the real radar number of ZSSSAR01 (sector 1 of Shanghai). By establishing a sector complexity optimization model, the complexity of sector air traffic and its volatility can be greatly reduced by changing the departure time of some flights. Thereby, it has a certain positive significance for reducing the workload of controllers and improving the operational efficiency of air traffic.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Key Program of Tianjin Science and Technology Plan (no. 21JCZDJC00840) and the National Natural Science Foundation of China (no. U1833103).