Generalized Method of Modeling Minute-in-Trail Strategy for Air Traffic Flow Management

Huang, Jinglei; Xu, Qiucheng; Yan, Yongjie; Ding, Hui; Tian, Jing

doi:https://doi.org/10.1155/2019/6059608

Mathematical Problems in Engineering

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

Research Article | Open Access

Volume 2019 | Article ID 6059608 | https://doi.org/10.1155/2019/6059608

Generalized Method of Modeling Minute-in-Trail Strategy for Air Traffic Flow Management

Jinglei Huang,¹Qiucheng Xu,¹Yongjie Yan,¹Hui Ding,¹and Jing Tian¹

Academic Editor: Mahmoud Mesbah

Received11 Aug 2019

Revised19 Nov 2019

Accepted10 Dec 2019

Published26 Dec 2019

Abstract

With the rapidly increasing air traffic demand, the demand-capacity imbalance problem of sector is surfaced gradually. And, minute-in-trail/miles-in-trail (MIT) is an effective strategy to balance the traffic demands and capacity. In this work, we consider the MIT strategy generation problem for the situation that a sector with corridors is affected by convection weather for time periods. Given the sector capacity , , under convection weather, we propose a three-phase optimization framework to generate E- strategy to achieve the demand-capacity balance. First, we take the sector capacity of time periods under convection weather as a whole, that is, , and then a dynamical programming-based method is proposed to allocate for corridors such that the capacity resources of each corridor , , can be determined. Second, a 0-1 combination algorithm is used to allocate the capacity resources into time periods for each corridor such that the candidate strategies set of each corridor can be determined, where a strategy is an array with numbers and each number represents the maximum allowed number of flights entering into sector from in one time period. Finally, a modified shortest path algorithm based on the backtracking method is taken to select the optimal strategy from for corridors such that the total delay cost and air traffic control load are minimized. Additionally, a dynamical programming-based method is proposed to generate E- strategy for the special case that the sector capacities of different time periods under convection weather are the same, that is, , and the generated strategies of time periods for a corridor are also the same. Experimental results show that compared with the proposed three-phase optimization method, rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively. When considering the special case, the experimental results show that compared with the proposed dynamical programming-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively.

1. Introduction

With the rapidly growing air traffic demands, air traffic control load [1] is increased, and the air traffic problems [2–4], such as air traffic congestion, the demand-capacity imbalance problem [5], and flight delays [6], are surfaced gradually. And, the demand-capacity imbalance problem is often occurred, particularly when capacity has been reduced due to the convection weather.

Generally, the demand-capacity imbalance problem can be dealt with in two ways [7]. One is the long-term strategy through the construction of infrastructure, such as new airports and runways to enlarge capacity. However, this procedure would take a long time and high cost.

The second way is to regulate the traffic flow by using the traffic management initiatives (TMIs) [8], such that the limited capacity can be used efficiently and the impact of unavoidable delays would be reduced [9]. And, the TMIs can be classified into two categories: (1) strategic actions, which are the strategies that taken before the aircraft has been taken off, including ground delay program [10], ground stop [11], airspace flow program [12], minute-in-trail or miles-in-trail [13], and Collaborative Trajectory Options Program [14] and (2) tactical actions, which are the strategies that taken after the aircraft is airborne, consisting of rerouting [15], speed adjustment, airborne holding [16], and fix balancing.

Among these strategies, the minute-in-trail/miles-in-trail is the most frequently used TMI because of its simplicity and ease of implementation [17], which is the strategy that imposes the time/distance spacing restrictions between every two adjacent aircraft flying along a routing path from the same corridor. Because the minute-in-trail strategy and miles-in-trail strategy can be interconverted, this paper focuses on the minute-in-trail strategy, hereafter referred to as MIT.

However, most of traffic managers rely largely on experience to determine the MIT restrictions, and no tool is available to support the traffic managers to balance the interests of all stakeholders, airlines, and passengers, resulting in the larger delay cost. Therefore, it is necessary to study the MIT strategy generation problem.

Many studies have focused on modeling the strategy of MIT. In [17], the authors presented a perspective on the MIT strategy, where the strengths and shortcomings of MIT were discussed in details and proved that MIT is an effective traffic management strategy for high-density sectors. Ostwald et al. [18] proposed an operational concept for arrival MIT restrictions using the MIA capability, where MIA is the tool used to evaluate the impacts of the proposed MIT restrictions on resources and flights before implementing them. Sheth et al. [19, 20] developed a model to compute MIT and pass back restrictions in the NAS for current traffic conditions, and the maximum ground delay and absorbable airborne delay are incorporated in the model.

In [21], the authors proposed a method based on the genetic algorithm for generating the MIT strategy to make full use of the storage capacities of the sector. Unfortunately, the stochastic optimization algorithm would lead to the different MIT strategies in different runs for the same scene; thus, the method cannot be used in the actual control practice. To achieve the goal of demand-capacity balance, Yuanhua and Zhang [22] proposed a method to restrict the interval of every two flights from surrounding areas and the departure time of flights of airports in this area through the strategy of GDP so that the total delay is minimized.

Machine learning has been applied to solve air traffic management problems. In [23], four machine learning algorithms, including support vector machine, random forest, decision tree algorithm, and softmax regression algorithm, are used to evaluate the miles-in-trail. Wang and Grabbe [24] offered an update analysis of the cause, frequency, and duration of historical MIT restrictions and subsequently using machine learning techniques to predict the occurrence of MIT restrictions to manage arrivals into the ATL airport.

However, most of the previous works mainly focused on generating a control strategy to restrict the flights into a sector with minimization of the total delay while the types of aircraft and passengers are not considered, resulting in the larger delay cost. In addition, the generated MIT strategy of the previous work is difficult to operate for air traffic managers because of the frequent change of traffic flow management strategy would lead to higher air traffic control load [25].

In [26], the authors proposed an evolutionary algorithm to generate the MIT strategy, where a traffic flow-capacity matching model based on workload restriction is presented and the stability function for evaluating the strategy in each slot time is defined, and an evolutionary algorithm is proposed to generate the MIT strategy. Unfortunately, the stochastic optimization algorithm would lead to the different strategies in different runs, resulting in the larger air traffic control load.

Motivated by these arguments, we propose a method for generating the E- strategy to control the aircraft from different directions heading towards a sector whose capacity is decreased from the normal operation capacity due to the convective weather. The main contributions of this paper are outlined as follows:(1)Given the sector capacity , , under convection weather, we propose a three-phase optimization framework to generate the E- strategy to achieve the demand-capacity balance. A dynamical programming-based method is proposed to allocate the total sector capacity of time periods under convection weather, , for corridors, and a 0-1 combination algorithm is used to determine the candidate strategies set for each corridor. Finally, a modified shortest path algorithm based on the backtracking method is taken to select the optimal strategy from for all corridors with minimization of the total delay cost and air traffic control load.(2)Additionally, a dynamical programming-based method is proposed to solve the E- strategy generation problem for the special case that sector capacities of different time periods under convection weather are the same, that is, , and the generated strategies of time periods for a corridor are also the same.

Experimental results show that compared with the rate-based method and need-based method, the proposed generalized method can reduce the average cost by 9.1% and 5.2%, respectively. When considering the special case, the experimental results show that compared with the rate-based method and need-based method, the proposed dynamical programming-based method can reduce the average cost by 9.2% and 7.0%, respectively.

The remainder of the paper is organized as follows: Section 2 describes the problem definition. Section 3 gives the details of computing delay cost and air traffic control load. The generalized E- strategy generation method is discussed in Section 4. Section 5 discusses the generation method for a special case. Experimental results and conclusions are shown and discussed in Sections 6 and 7, respectively.

2. Problem Description

2.1. Extended MIT Strategy

Minute-in-trail/miles-in-trail (MIT) is the strategy that requires the flights in a flow of air traffic crossing a certain corridor of sector must be separated by a certain number of minutes or miles. Through this strategy, we can control the volume of air traffic into sectors and airports at a safe level.

In the actual control practice, MIT, along with the maximum allowed number of flights, is the frequently used traffic management initiatives to balance the traffic flows and sector capacity under converse weather. That is, the generated strategy includes two restrictions for flights across one transfer-of-control point (or, a corridor) of sector:(i)The maximum allowed number of flights in a time period(ii)The minimal time interval between every two adjacent flights from the same corridor

In this paper, we focus on the MIT strategy, along with the maximum allowed number of flights, hereafter referred to as the extended MIT strategy (E-).

2.2. Flow Control Time

Table 1 shows some notations used in this paper.

Once the E- strategy is issued, the duration time of strategy, , the flow control time , must be attached to the E- strategy so that the air traffic managers can control the traffic flows more effectively.

Flow control time is the total time needed to balance the sector capacity and traffic flows using the E- strategy, which is defined as follows:where is the duration time periods of convection weather on sector, that is, the time periods of demand-capacity imbalance. Because the sector capacity is decreased from the normal operation capacity due to the convection weather, some planned flights are delayed to the subsequent time periods. And, is the time period to process the delayed flights such that the balance of demand-capacity can be recovered, which is defined as follows:where is the time period needed to process the delayed flights and recover the demand-capacity imbalance for corridor .

Assume that the sector is affected by convection weather during , and let the flights set across the corridor } be the flights scheduled to arrive at this sector during . The time axis can be discretized by subdividing into a set of consecutive time periods, and the time span τ of each time period can be set any value. However, because of the accuracy of weather forecast, τ is set to 15 minutes, 30 minutes, or 1 hour, practically. For convenience, we define to be the number of flights across the corridor during t-th time period.

Thus, the imbalance time periods can be calculated as follows:where τ is the time span of a time period, and in this work, we set one hour.

If the maximum allowed number of flights across the corridor in a time period is under convection weather and the corresponding normal level is in one time period, the demand-capacity recovery time periods is the least value that meets the following in-equation:

Figure 1 shows an example of calculating the recovery time for , where the convection weather occurs from 20:00 to 22:00, and we set each time period at one hour; thus, . In these two time periods, the maximum allowed number of flights across the corridor is flights per hour and flights per hour, respectively. And, the maximum allowed number of flights across under normal condition is 16 flights per hour.

From Figure 1, we can get , , , and . Thus, according to in-equation (4), we can calculate the recovery time periods to recover the left 12 flights of the demand-capacity imbalance periods.

2.3. Problem Definition

Based on the above definitions, the problem definition for generating the E- strategy is as follows.

Given the following inputs,(1)The number of corridors of a sector, that is (2)The sector capacity under convection weather condition, , (3)The sector capacity of a time period under normal condition(4)The maximum allowed number of flights (or, corridor capacity) of a time period for corridor , , under normal condition

and the following constraints:(1)The flights are not allowed to take off in advance, that is, (2)The maximum allowed number of flights across is recovered to a normal level once the convection weather is over(3)The maximum allowed number of flights across a corridor in one time period under convection weather conditions cannot exceed the corresponding normal level, that is,

We try to generate an optimal E- strategy to control the air traffic flows from different corridors that enter into a sector whose capacity is decreased from the normal operation capacity due to the convective weather. And, the E- strategy generally includes two restrictions on flights:(1)The maximum allowed number of flights across a corridor in a time period , i.e., (2)the minimal time interval between every two adjacent flights across the same corridor in one time period

Thus, the total flight delay cost and air traffic control load are minimized.

Figure 2 gives an example of the E- control strategy for a sector in one hour, where is 31 flights/h, and the maximum allowed number of flights for each corridor in one hour is 8 flights/h, 8 flights/h, 12 flights/h, and 3 flights/h, respectively.

However, once the maximum allowed number of flights of corridor in time period is determined, the minimal time interval can be calculated as follows:where τ is the duration time of a time period and is the minimal separation time between aircraft i and j to ensure safety, which can be calculated according to Table 2. In Table 2, three types of aircraft, , , and , are considered [27]. For example, if a heavy aircraft follows a large aircraft, then their minimal separation time must be at least 61 seconds.

Thus, the E- strategy generation problem is to determine of corridor in time period , , .

Let be the total delay cost of corridor when of all time periods are determined simultaneously, and let be the air traffic control load of corridor , which is used to approximate the strategy stability. Therefore, the strategy generation problem can be formulated as a resource allocation problem as follows:subject towhere α and β can make trade-off between the contributing factors.

The first sets of constraints define a limited capacity resource in each time period for sector, and the second sets of constraints show the limited value of the maximum allowed number of flights for each corridor in a time period .

To explain the strategy generation problem more clearly, an example is given in Figure 3, where the sector has four corridors, and the sector capacities of three time periods under convection weather are , , and , respectively. The strategy generation problem is to determine the values of , , , under the constraints that such that the total delay cost and air traffic control load are minimized. Unfortunately, the delay cost and air traffic control load of corridor can be calculated (discussed in the next section) only if the values of , , and are given simultaneously.

Therefore, it is very difficult to directly solve this resource allocation problem since , , , cannot be determined simultaneously. And, in this work, a three-phase method is proposed to generate the E- strategy.

3. Computation of Objection Function

3.1. Computation of Delay Cost

Once the E- strategy is generated, it will be feed back to the collaborative decision-making system of local airport and regenerate the calculated time of takeoff (CTOT) for flights such that the number of flights across a corridor in a time period and the minimal time interval between every two adjacent flights could meet the constraints of E- strategy.

Here, we take a corridor, for example, as shown in Figure 3, given the maximum allowed number of flights across in one time period under convection weather conditions, that is, , , and , and the CTOT of flight , , can be calculated via the ration-by-schedule algorithm [28], which is based on the first-scheduled-first-served principle. The main steps are listed as follows:(a)Determine the number of time slots and the time span of each slot (b)Sort the flights according to their estimated time of takeoff (c)Allocate the time slots for the sorted flights in order, and the allocated slot is at or later than the flight’s estimated time of takeoff (d)The beginning time of slot is taken as

The procedure is repeated until each flight is allocated a time slot. At last, the delay time of flight can be determined.

In Step (a), the number of time slots is the maximum allowed number of flights across in one time period, and the time span of each slot can be defined as follows:where τ is the time span of a time period.

Figure 4 gives an example of calculating for flight , where , , , and flights per hour. Because the time span of one time period is one hour, the time span of each slot are 60/4 = 15 minutes, 60/3 = 20 minutes, 60/4 = 15 minutes, and 60/8 = 7 minutes, respectively. Based on the first-scheduled-first-served principle, we allocate these slot times to the flights according to their estimated time of takeoff (which is given as input) as shown in Steps (b) and (c).

As shown in Figure 4, for flight , whose estimated takeoff time is 20:07 and through the procedure of allocating time slots, its calculated takeoff time is 20:15; thus, the delay time is 8 minutes.

Once the delay time of flights are obtained, the delay cost for corridor can be calculated as follows:where represents the delay cost of flights and is the delay cost of passengers.

The delay cost of flights depends on the delay time of flights , which can be calculated as follows:where is the set of affected flights, which can be determined according to the flow control time . represents the delay cost per hour for flight , which depends on the type of aircraft as shown in Table 3, where three types of aircraft are given.

And, the delay cost of passengers can be calculated as follows:where represents the number of passengers of flight and represents the delay cost of one people of flight . In this work, we take the cost analysis model [29] to estimate the delay cost of passengers, where the unit delay cost of ordinary passengers is set 50 per hour and the important passenger is 100 per hour.

3.2. Computation of Air Traffic Control Load

In this work, a method similar to [26] is taken to calculate the air traffic control load for corridor , which is defined as follows:

What is more, the air traffic control load can also represent the strategy stability, where the bigger value of will cause the less strategy stability.

4. E-MIT

Given the sector capacity of time period () under the convection weather, a three-phase method is proposed to generate the E- strategy for flights with minimization of delay cost and air traffic control load.

4.1. Allocation of Sector Capacity

In this work, we first take the sector capacity of time periods under convection weather as a whole, that is, ; thus, the strategy generation problem can be regarded as a classic resource allocation problem that allocates for corridors with minimization of total delay cost. And, in this work, a dynamical programming-based method is proposed to solve this resource allocation problem such that the capacity resources of each corridor , , can be determined.

Figure 5 shows an example of allocating sector capacity, where the sector has four corridors and . Our goal is to determine the capacity resources , , , and .

If corridor is allocated to j resources, we can calculate the delay cost for corridor as follows:(i)Firstly, according to the equation (8), the time span of each slot under convection weather and normal condition can be calculated as and , respectively(ii)Then, through the procedures of allocating time slots in Section 3.1, the delay time of each flight can be calculated(iii)At last, the delay cost for corridor can be calculated according to equation (9)

Let be the minimum total delay cost when the first i corridors, , are allocated to the total j resources, which can be calculated according to the following two steps:(1)The first i-1 corridors, , are allocated to the total m resources(2)The remaining resources are allocated to corridor

Thus, the recursive relation formula of dynamical programming can be formulated as follows:

Algorithm 1 gives a detailed description of the dynamical programming-based method to allocate the total sector capacity for corridors.

(1)	Input the total sector capacity and the number of corridors ;
(2)	Initialize the values , ;
(3)	for to do
(4)	;
(5)	end for
(6)	;
(7)	for to do
(8)	;
(9)	for to do
(10)	= INF;
(11)	= ;
(12)	for to do
(13)	if then
(14)	Continue;
(15)	end if
(16)	if then
(17)	= ;
(18)	= m;
(19)	end if
(20)	end for
(21)	= ;
(22)	end for
(23)	end for
(24)	//backtrack output results
(25)	;
(26)	for to 1 do
(27)	;
(28)	;
(29)	end for

In lines 2–5 in Algorithm 1, we initialize the values of , , and . From lines 13 to line 15, we check whether allocating flights of time periods to corridor will violate the constraint (3). If the allocated flights meet the constraints and have less delay cost than the current cost recorded by , we will update the minimum delay cost and record the allocated capacity resource for in lines 17-18.

Once Algorithm 1 is finished, the total allocated capacity resources of time periods for each corridor , , can be determined as shown in line 27.

4.2. Candidate Strategies for a Corridor

Once the total capacity resource of time periods for each corridor () is determined, we will use 0-1 combination algorithm to allocate into time periods for each corridor such that the candidate strategies set of each corridor can be determined, where a strategy is an array with numbers, and each number represents the maximum allowed number of flights entering into sector from in one time period.

To simplify the discussion, the total allocated capacity resource of corridor is relabeled as N; thus, the corridor capacity resource allocation problem can be formulated as the problem of placing −1 baffles between N numbers, which is the combination optimization problem of selecting −1 from N + 1, and, there has solutions. In this work, we take the 0-1 combination algorithm to search those solutions, and Algorithm 2 shows the overall flow of the 0-1 combination algorithm.

(1)	Input the capacity resource N and time periods ;
(2)	Initialize the array with the size of N + 1;
(3)	//set the first −1 of array to 1, which corresponds to a solution;
(4)	for to −1 do
(5)	index[i] = 1;
(6)	end for
(7)	Store the solution according to the values of array ;
(8)	while has Done (, N, ) do
(9)	for to N + 1 do
(10)	//find the first 1-0 combination and change it to 0-1 combination;
(11)	if index[i] = = 1 and index[i] = = 0 then
(12)	index[i] = 0;
(13)	index[i] = 1;
(14)	Store the solution from ;
(15)	//move 1 of the left of 0-1 combination to the left of array;
(16)	int count = 0;
(17)	for to i do
(18)	if index[j] = = 1 then
(19)	index[j] = 0;
(20)	index[count++] = 1;
(21)	end if
(22)	end for
(23)	end if
(24)	break;
(25)	end for
(26)	end while

In line 2 in Algorithm 2, we initialize the array with the size of N + 1, and is the flag that represents whether the corresponding baffle is placed or not. If , there is a baffle between the i-th number and the (i + 1)-th number. Otherwise, the i-th number and the (i + 1)-th number are allocated to the same time period.

The combination optimization problem is solved in lines 8–26, and the main steps of 0-1 combination algorithm are listed as follows:(1)Set the first −1 numbers of array to 1 as shown in lines 4–6, which corresponds to a solution(2)Find the first 1-0 combination of array and change it to 0-1 combination as shown in line 11–13(3)Move all 1 of the left of 0-1 combination to the left of array

Repeat Steps 2 and 3 until the last −1 numbers of array are 1.

Once the 0-1 combination algorithm is finished, we can get the candidate strategies set for each corridor () according to the values of array .

Figure 6 shows an example of determining candidate strategies for corridor using the 0-1 combination algorithm, where = 3 and = 3. Firstly, we define an array with size of 4, and according to Steps 4–25 of Algorithm 2, we can find possible solutions for the combination problem of selecting −1 from , as shown in Figure 6(a). Secondly, for each solution, according to the values of array , we can get the positions of baffles placed between numbers. Here, we take the third solution (“0110”), for example, because = 1, and we place a baffle between the first number and the second number. Similarly, a baffle is placed between the second number and the third number as shown in Figure 6(b). At last, we calculate the numbers between adjacent baffles, and each number represents the maximum allowed number of flights entering into sector from in one time period.

4.3. Strategy Generation Algorithm

For each corridor (), there are (=) solutions of allocating resources into time periods, and different solutions in will have different delay costs and air traffic control load. In this section, we will introduce a modified shortest path algorithm based on the backtracking method to select the optimal strategy from for each corridor such that the total delay cost and the air traffic control load can be minimized.

To represent all the possible candidate strategies of all corridors, a directed search graph is constructed, which contains three sets of vertices , where s is the start vertex, is the candidate strategies set of corridor , and a strategy is an array with numbers, and t is the end vertex. Besides, edge set . In addition, the edge costs are defined as follows:where and are the delay cost and air traffic control load if corridor uses the strategy that node u represents.

Figure 7 shows an example of a directed search graph .

Thus, the optimal strategies for each corridor can be determined by finding the shortest path from s to t on , and each node represents a strategy.

To speed up the search procedure of the strategies for all corridors with minimization of the total delay cost and air traffic control load, we propose a modified shortest path algorithm based on the backtracking method [30] to select the allocation solution for each corridor while meeting the constraint that the corridor capacity of each time period cannot exceed the corresponding normal level, .

Once the modified shortest path algorithm [30] is finished, each node in the returned shortest path represents the best solution, that is, , , . And, according to formula (5), we can determine the minimal time interval between every two adjacent flights from the same corridor and time period .

5. A Special Case

In this section, we consider a situation that the input sector capacities of the time periods are the same, that is, , and the generated strategies of different time periods for a corridor are also same (that is, for a corridor , ).

Figure 8 shows an example of a special case. For this case, if the strategy of a time period for each corridor, that is, , , is given, the delay cost for corridor , , can be determined.

Thus, this special case can be solved by allocating the sector capacity resource into corridors with minimization of total delay cost. And, a dynamical programming-based method similar to Section 4.1 is used to solve this special case.

6. Experiments

6.1. Experimental Setup

The proposed method has been implemented in C language on a Linux 64 bit workstation (Intel 2.4 GHz, 256 GB RAM).

To explore the effectiveness of the proposed algorithm, we construct a case benchmark by referring to the typical operating day of sector 06 in the terminal area of Beijing as shown in Table 4, which has four corridors, , , , and , and the sector capacity is 48 in one time period under normal condition.

In Table 4, the first row represents the maximum allowed number of flights across the corridor in a time period under normal conditions. The other rows list the traffic flows from each direction across a corridor in different time periods, where the used corridors of flights are predetermined and the sequence of flights are kept unchanged. In addition, the aircraft type, the estimated arrival time of aircraft and the type of passengers are also given.

6.2. Results and Analysis

To explore the effectiveness of the proposed generalized method, we perform two other methods, rate-based method and need-based method, which are often used in the actual control practice:(i)Rate-based method: based on the descending ratio of capacity in time period (), (), the maximum allowed number of flights of corridor in time period is set .(ii)Need-based method: let be the rate that the traffic flows of corridor to the total flows of corridors in time period . That is, for each time period , , the capacity is set .

As a baseline situation, we solve the resource allocation problem using an integer linear programming (“ILP”), and the manual in the study of Gurobi [31] is used as the ILP solver to find the optimal solution.

In experiments, we set ; that is, the sector in 20:00-21:00 and 21:00-22:00 are affected by the convective weather. And, the sector capacities of these two time periods, and , are assumed to be 24 and 28 flights per hour, respectively.

Table 5 shows the experimental results. represents the number of affected flights, whose estimated time of takeoff is not equal to the calculated time of takeoff . The total delay time is defined as follows:

And, the average delay time is defined as .

As shown in Table 5, compared with the proposed three-phase optimization method, the rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively, which shows the effectiveness of the proposed method. In addition, the proposed three-phase optimization method can reduce the affected flights of 3 aircraft and 3 aircraft on average when compared to the rate-based method and need-based method, respectively.

In Table 5, the delay cost obtained by the proposed method is a litter higher (1.5%) than the baseline situation of “ILP,” which demonstrates the effectiveness of the proposed method.

Table 5 also lists the strategy generated by the proposed three-phase method for each directions to control the maximum allowed number of flights and the minimal time interval between every two adjacent flights in different time periods, and the minimal interval between every adjacent flights is determined according to formula (5). Here, we take corridor 2, for example, whose flow control time is five hours and the strategies are listed as follows in detail:(i)20:00 to 21:00: the maximum allowed number of flights is 4 flights/h, and the minimal time interval between adjacent flights is 8 minutes(ii)21:00 to 22:00: the maximum allowed number of flights is 9 flights/h, and the minimal interval between adjacent flights is 3 minutes(iii)22:00 to 23:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes(iv)23:00 to 00:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes(v)00:00 to 01:00: the maximum allowed number of flights is 12 flights/h, and the minimal interval between adjacent flights is 3 minutes

6.3. Impacts of Sector Capacity on E-

Further to demonstrate the effectiveness of the proposed three-phase optimization method, we perform the comparison experiments on different sector capacities of these two time periods, where is set to 30 and is ranged from 25 to 35. Figure 9 shows the changing trend of the delay cost, the number of affected flights, air traffic control load, and average delay time with changing under the same . Form Figure 9, it can be seen that the proposed three-phase method can generate the control strategy with the least delay cost and the least air traffic load when compared to the rate-based method and need-based method. And, as shown in Figures 9(b) and 9(d), the proposed three-phase method can achieve the least average delay time with a little more of affected flights.

(a)

(b)

(c)

(d)

6.4. Impacts of on E-

To evaluate the impact of on the generated strategy E-, we perform the experiments on the test benchmark as shown in Figure 10, where the duration time periods of convection weather ranged from 2 to 5, and in each time period, the sector capacity , , is set to 31.

(a)

(b)

As shown in Figure 10, with the increasing , the delay cost and average delay time are increased, which is consistent with the actual situation. What is more, compared with the rate-based method and need-based method, the proposed three-phase optimization method can achieve the least delay cost and delay time.

6.5. Results of the Special Case

In this section, to demonstrate the effectiveness of the proposed dynamical programming-based method for the special case, we perform experiments with the rate-based method and need-based method as shown in Table 6, where the sectors in 20:00-21:00 and 21:00-22:00 are affected by the convective weather, and the corresponding sector capacity of these two time periods are both set to be 31 flights per hour.

From Table 6, it can be seen that, compared with the proposed DP-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively, which shows the effectiveness of the proposed method for the special case. In addition, the proposed DP-based method can save the average delay of 2.0 minutes and 1.6 minutes on average when compared to the rate-based method and need-based method, respectively.

In addition, we also compare the DP-based method with the proposed three-phase method as shown in Table 7. In this experiment, the sector capacities and are set to the same values, which are ranging from 15 to 35 flights per hour. As shown in Table 7, compared with the DP-based method, the proposed three-phase method can save 2% delay cost because it could explore larger solution space, while the DP-based method assumes that the generated strategies of different time periods for a corridor to be same; thus, the air traffic control load is zero as shown in Table 7.

7. Conclusion

In this work, we consider the MIT strategy generation problem for the situation that a sector with corridors is affected by convection weather for time periods. Given the sector capacity , , under convection weather, we propose a three-phase optimization framework to generate E- strategy to achieve the demand-capacity balance. Firstly, we take the sector capacity of time periods under convection weather as a whole, that is, , and then a dynamical programming-based method is proposed to allocate for corridors such that the capacity resources of each corridor , , can be determined. Secondly, a 0-1 combination algorithm is used to allocate the capacity resources into time periods for each corridor such that the candidate strategies set of each corridor can be determined, where a strategy is an array with numbers, and each number represents the maximum allowed number of flights entering into sector from in one time period. Finally, a modified shortest path algorithm based on the backtracking method is taken to select the optimal strategy from for corridors such that the total delay cost and air traffic control load are minimized. Additionally, a dynamical programming-based method is proposed to generate the E- strategy for the special case that the sector capacities of different time periods under convection weather are the same, that is, , and the generated strategies of time periods for a corridor are also the same. Experimental results show that compared with the proposed three-phase optimization method, rate-based method and need-based method will spend more 8.1% and 6.3% of delay cost, respectively. When considering the special case, the experimental results show that compared with the proposed dynamical programming-based method, the rate-based method and need-based method will spend more 10.2% and 7.5% of delay cost, respectively.

Data Availability

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

Disclosure

This manuscript is an extension of our previous publications [32, 33].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (NSFC) under grant no. 61903346 and Jiangsu Provincial Natural Science Foundation (BK20170157).

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Copyright © 2019 Jinglei Huang 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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