Trajectory Planning Method in Time-Variant Wind Considering Heterogeneity of Segment Flight Time Distribution

Abstract

The application of Trajectory-Based Operation (TBO) and Free-Route Airspace (FRA) can relieve air traffic congestion and reduce flight delays. However, this new operational framework has higher requirements for the reliability and efficiency of the trajectory, which will be significantly influenced if the analysis of wind uncertainty during trajectory planning is insufficient. In the literature, trajectory planning models considering wind uncertainty are developed based on the time-invariant condition (i.e., three-dimensional), which may potentially lead to a significant discrepancy between the predicted flight time and the real flight time. To address this problem, this study proposes a trajectory planning model considering time-variant wind uncertainty (i.e., four-dimensional). This study aims to optimize a reliable and efficient trajectory by minimizing the Mean-Excess Flight Time (MEFT). This model formulates wind as a discrete variable, forming the foundation of the proposed time-variant predicted method that can calculate the segment flight time accurately. To avoid the homogeneous assumption of distributions, we specifically apply the first four moments (i.e., expectation, variance, skewness, and kurtosis) to describe the stochasticity of the distributions, rather than using the probability distribution function. We apply a two-stage algorithm to solve this problem and demonstrate its convergence in the time-variant network. The simulation results show that the optimal trajectory has 99.2% reliability and reduces flight time by approximately 9.2% compared to the current structured airspace trajectory. In addition, the solution time is only 2.3 min, which can satisfy the requirement of trajectory planning.

1. Introduction

Trajectory-Based Operation (TBO) [1] and Free-Route Airspace (FRA) [2], as suggested by the International Civil Aviation Organization, are two innovative conceptions in today’s Air Traffic Management (ATM). They have enormous potential to relieve air traffic congestion and reduce flight delays due to their characteristics. First, TBO aims to satisfy the requests of airspace users to the maximum extent possible by requiring aircraft to follow a 4-dimensional trajectory (latitude, longitude, altitude, and time) [3]. Second, FRA can improve efficiency by allowing aircraft to choose their preferred trajectory, which is more flexible than the current structured airspace. Therefore, this new operational framework has higher requirements for the reliability and efficiency of the trajectory.

An effective way to satisfy these requirements is to consider the influence of uncertainty in the planning stage. According to studies, wind uncertainty is one of the most important factors that greatly affect flight time [4]. Since flights have a large spatial and temporal horizon, incomplete knowledge of the current and future wind conditions may result in flight delays [5]. Ultimately, this will translate into additional expenses for travelers, airlines, and navigation service providers [6]. The trend in existing studies to qualify the effect of wind uncertainty is to use the Ensemble Prediction System (EPS) [7,8]. To plan reliable and effective trajectories to improve the performance of ATM under this new operational framework, trajectory planning models considering wind uncertainty have received much attention in recent research.

Legrand et al. proposed a trajectory planning model in which segment flight time is determined by the ratio of distance to wind-influencing flight velocity. They used the minimum shortest path algorithm for each wind condition in the EPS to calculate a trajectory ensemble, from which the optimal trajectory can be clustered [9]. However, the clustering algorithm may necessitate a significant number of computational resources. Franco et al. addressed this gap by proposing a stochastic optimization model in which the objective is to minimize fuel consumption (or segment flight time) [10]. Franco et al. further extended this model by applying the deviation and expectation of the flight time to describe reliability and efficiency, respectively [5]. Therefore, this problem can be converted into a deterministic trajectory planning problem. Under this circumstance, common algorithms, including the geometric algorithm, path planning algorithm (such as Dijkstra and A*), heuristic algorithm, and random tree algorithm [11,12], can effectively solve this problem. However, on the one hand, neither the deviation nor the variance intuitively reflects the reliability of the aircraft arrival time; on the other hand, this kind of model implicitly assumes that the distributions of the flight time on different segments are homogeneous (i.e., obeying the same distribution, such as the uniform or beta distribution) [13]. According to the analysis by Franco, for the same aircraft with different true air speeds, the fitting performances of the same distribution vary significantly [8]. Unfortunately, existing studies do not appear to provide any relevant evidence, such as R-Squared values, to substantiate the appropriateness of the fitting effect.

Note that all the studies mentioned above are all focused on time-invariant wind uncertainty. Nevertheless, wind uncertainty is four-dimensional rather than three-dimensional (i.e., the uncertainty changes with time) due to the large horizon of the flight time (especially for a long trajectory). As far as the authors are aware, there are only two methods published in the literature that take into account time-variant weather conditions. The first model analyzed the effect of time-variant adverse weather by predicting future obstacle locations (such as those of thunderstorms) at regular time interval t [14]. However, the focus of this article is on avoiding adverse weather, not the reliability and efficiency of the trajectory. In the second model proposed by González-Arribas in 2023, time is also discrete, and the distributions of segment flight time are derived from the initial condition distribution, which is also assumed to follow a homogeneous Gaussian distribution [15].

The deviation and variance in flight time commonly used in existing studies cannot qualify reliability directly because they are indicators that reflect dispersion. Therefore, it is necessary to provide a precise definition of reliability to expand the trajectory planning model. Furthermore, due to the fact that the air traffic network is a complex system and it is hard for researchers to analyze the probability distribution function (PDF) of the flight time for each segment, assuming that the distributions are homogenous may be far from reality. Finally, due to time-variant wind uncertainty, which may lead to the optimal trajectory in a three-dimension condition becoming suboptimal during operation, the reliability and efficiency of the trajectory cannot be ensured. Therefore, the trajectory planning models designed based on three-dimensional wind may be unreliable, so they cannot actually reduce flight delays.

This paper proposes a trajectory planning model that accounts for time-variant wind uncertainty to solve the problems mentioned above. We adopt the Mean-Excess Flight Time (MEFT) as the objective function, which has been proven effective in representing the reliability of travel time in the domain of road transportation [16,17,18]. Furthermore, we avoid the assumption that the distributions of segment flight time are homogeneous by using the first four moments to describe their stochasticity instead of the PDF. Due to the complexity of the problem, a two-stage algorithm is used to solve this model. First, we calculate the ensemble of feasible trajectories containing the optimal one by the Time-Variant Shortest Path (TVSP) algorithm and the K-Time-Variant Shortest Path (K-TVSP) algorithm. Second, the optimal trajectory is obtained by the exhaustive method. Compared with enumerating all feasible trajectories, this approach can significantly decrease the solution time.

The structure of this paper is as follows: Section 2 presents the trajectory prediction model, considering time-variant wind forecast uncertainty. Section 3 describes the proposed reliable trajectory planning method dependent on the MEFT. Section 4 focuses on the design of the two-stage algorithm for solving the proposed problem. A case study is shown in Section 5 to demonstrate the efficiency of the model and algorithm. Finally, we provide some conclusions and discussions in Section 6.

2. Time-Variant Ensemble Trajectory Prediction Model

Existing ensemble trajectory prediction methods based on the EPS can be divided into the ensemble method and the probabilistic transformation method (see Figure 1).

(1)

Ensemble method: A deterministic trajectory prediction model is applied for each member of the EPS. In this case, the output is a trajectory ensemble, from which the probability distribution of trajectory parameters (e.g., flight time, fuel consumption, etc.) can be derived [5,9,10,19,20].

(2)

Probabilistic transformation method: The probability distributions of wind are evolved along the trajectory from the EPS by a probabilistic trajectory prediction model, resulting in the probability distributions of trajectory parameters [8,21].

In contrast, the probabilistic transformation method implicitly has the homogeneous assumption of the segment flight time distribution, which may differ from reality. Furthermore, the ensemble method needs fewer computation resources [22], so it is used in this study.

2.1. Time-Variant Wind Uncertainty

The EPS is a weather forecast ensemble with $\mathcal{K}$ members (such as PEARP (35 members, France), MOGREPS (12 members, UK), and ECMWF (51 members, Europe)) [19]. A wind grid is constructed to store the information (latitude $\lambda$, longitude $\varphi$, altitude $h$, latitudinal wind $W_E$, and longitudinal wind $W_N$) obtained from ECMWF. The dimensions of this grid are $\mathcal{N}\times \mathcal{M}$, with latitude ranging from ${\lambda }_{min}$ to ${\lambda }_{max}$ and longitude ranging from ${\varphi }_{min}$ to ${\varphi }_{max}$. The latitude interval is denoted as ${\mathsf{\Delta }}_{lat}$, and the longitude interval is denoted as ${\mathsf{\Delta }}_{lon}$. The wind angle ${\theta }_w$ (Figure 2a) and wind value $‖\overrightarrow{W}‖$ at each point can be calculated by Equation (1).

$${\theta }_w={\tan }^{-1}\left\{{\displaystyle \frac{W_E}{W_N}}\right\}, ‖\overrightarrow{W}‖=\sqrt{W_E^2+W_N^2}$$

(1)

The forecast time interval is $t_v$, and the moment when the wind condition changes is defined as a Time-Variant Moment (TVM). Therefore, as shown in Figure 3, $W_E\left(t\right)$ and $W_N\left(t\right)$ can be described as piecewise functions with the domain of $\left[0,T\right]$ and the discontinuity points at {$t_v,{2t}_v,{3t}_v,\cdots$}.

2.2. Segment Flight Time Prediction Model

Because the cruise phase occupies the majority of the flight time, it is common to optimize only the cruise trajectory. We constructed a rectangular Searching Area (SA) whose range and grid granularity are consistent with the wind grid. Each waypoint follows the rule of connection depicted in Figure 4. Furthermore, the grid nodes and connections inside the SA form a waypoint ensemble $W$ and a segment ensemble $A$, respectively. Then, a graph $G\left(W,A\right)$ can be modeled.

The trajectory of flight $f$, denoted as $r_f$, can be represented by a set of $n$ segments and their corresponding flight time ensembles, i.e.,

$$\begin{gathered} r_f=\left\{\left(a_{f,1},T_{f,1}\right),\left(a_{f,2},T_{f,2}\right),\cdots ,\left(a_{f,i},T_{f,i}\right),\cdots ,\left(a_{f,n},T_{f,n}\right)\right\}, \\ {a}_{f,i}\in A,a_{f,i}=\left(w_{f,i},w_{f,i+1}\right),w_{f,i}\in W \end{gathered}$$

(2)

where $a_{f,i}$ and $T_{f,i}$ are the segment and its flight time ensemble, respectively.

Now, we will discuss how to calculate $T_{f,i}$ using ensemble trajectory prediction considering time-variant wind uncertainty. The coordinates of the tail node and head node of $a_i$ are defined as $\left({\lambda }_i,{\varphi }_i\right)$ and $\left({\lambda }_{i+1},{\varphi }_{i+1}\right)$, respectively. The heading angle ${\theta }_i$ (see Figure 2b) and the length $l_i$ can be calculated as follows:

$$\tan {\theta }_i={\displaystyle \frac{{\varphi }_i-{\varphi }_{i+1}}{\ln \left[\frac{\tan \left(\frac{\pi }{4}-\frac{{\lambda }_i}{2}\right)}{\tan \left(\frac{\pi }{4}-\frac{{\lambda }_{i+1}}{2}\right)}\right]}}$$

(3)

$$l_i=\left\{\begin{array}{cc}{\displaystyle \frac{\left(R_E+h\right)\left({\lambda }_{i+1}-{\lambda }_i\right)}{\cos {\theta }_i}},& {\lambda }_{i+1}\ne {\lambda }_i\\ \left(R_E+h\right)\cos {\lambda }_i\left|{\varphi }_{i+1}-{\varphi }_i\right|,& {\lambda }_{i+1}={\lambda }_i\end{array}\right.$$

(4)

where $\lambda$ and $\varphi$ should be expressed in radians.

As shown in Figure 5, the aircraft is influenced by the along-trajectory wind ($W_{p,i}$) and the crosswind ($W_{c,i}$) on segment $a_i$. Then, converting $W_{c,i}$ into an equivalent headwind, the ground speed ($V_{g,i}$) and flight time $t_{f,i}$ of the aircraft are as follows:

$$V_{g,i}=\sqrt{V^2-W_{c,i}^2}+W_{p,i}$$

(5)

$${\displaystyle \frac{dl}{dt}}=V_{g,i}$$

(6)

where $V$ is the true air speed. Within the current validity period (i.e., $t\in \left[0,t_v\right]$ or $t\in \left(t_v,2t_v\right)$), under the impact of member $k$, $t_{f,i}^k$ can be calculated according to Equation (6). Then, we can obtain an ensemble $T_{f,i}\left(t\right)=\left(t_{f,i}^1,t_{f,i}^2,\cdots ,t_{f,i}^{\mathcal{K}}\right)$ by predicting independently based on each member in the EPS. The expectation $E_{f,i}\left(t\right)$ is

$$E_{f,i}\left(t\right)={\displaystyle \frac{1}{\mathcal{K}}}\sum _{k=1}^{\mathcal{K}}{t}_{f,i}^k$$

(7)

Let $t_{f,i}^o$ and $t_{f,i}^d$ be the expectation time that aircraft $f$ arrives at the origin and destination of $a_i$, respectively. Then,

$$t_{f,i}^o={\sigma }_f+∆T+\sum _{\overline{a}\in \overline{A}}{E}_{f,i}^{\overline{a}}\left(t\right)$$

(8)

$$t_{f,i}^d=t_{f,i}^o+E_{f,i}\left(t\right)$$

(9)

where ${\sigma }_f$ is the departure time, $∆T$ is the time interval from departure to entering FRA, and $\overline{A}$ represents the ensemble of passing segments before $a_i$.

Without loss of generality, let us consider two continuous segments ($a_i$, $a_{i+1}$) in FRA. Suppose that there are two wind conditions W1 and W2, which are valid within the period $\left[0,t_v\right]$ and $(t_v,2t_v)$, respectively.

Condition 1. 

$t_{f,i}^o<t_v$ and $t_{f,i}^d<t_v$.

Condition 2. 

$t_{f,i}^o<t_v$ and $t_{f,i}^d\ge t_v$.

Condition 1 indicates that there is no TVM on segment $a_i$. In this case, W1 is the only valid wind for both segments, as shown in Figure 6a. Subsequently,

$$\begin{gathered} T_{f,i}\left(t\right)=T_{f,i}^1 \\ T_{f,i+1}\left(t\right)=T_{f,i+1}^1 \end{gathered}$$

(10)

Condition 2 indicates that there is a TVM on segment $a_i$. In this case, the valid wind on $a_i$ is W1, whereas the valid wind on $a_{i+1}$ is W2, as shown in Figure 6b. To simplify this calculation, we might ignore the deviation caused by wind variation on a single segment. Subsequently,

$$\begin{gathered} T_{f,i}\left(t\right)=T_{f,i}^1 \\ T_{f,i+1}\left(t\right)=T_{f,i+1}^2 \end{gathered}$$

(11)

3. Trajectory Planning Model

3.1. Definitions and Mathematical Model

Before proposing the trajectory planning model, the definitions for the variables are introduced in advance.

Definitions 1. 

-

$\alpha$: reliability level, which represents the probability that a flight will arrive at its destination punctually.

-

${\mathit{T}}_f$: the stochastic vector of trajectory flight time, ${\mathit{T}}_f=(T_{f,1},\cdots ,T_{f,i},\cdots )$.

-

MEFT: the conditional expectation of the trajectory flight time exceeds the Minimum Travel Time (MTT) while satisfying the reliability level $\alpha$.

According to Definition 1 the MEFT can be described as

$${\phi }_{\alpha }(r_f,T_f)=E\left[\xi (r_f,T_f)\mid \xi (r_f,T_f)⩾{\zeta }_{\alpha }(r_f,T_f)\right]$$

(12)

where ${\phi }_{\alpha }(r_f,T_f)$ is the MEFT, ${\zeta }_{\alpha }(r_f,T_f)$ is the MTT, and $\xi (r_f,T_f)$ is the stochastic flight time.

${\zeta }_{\alpha }(r_f,T_f)$ can be calculated by the following:

$${\zeta }_{\alpha }(r_f,T_f)=min\{\zeta \mid \mathrm{P}\mathrm{r}[\xi (r_f,T_f)⩽\zeta ]⩾\alpha \}$$

(13)

where $\zeta$ is the flight time threshold of $r_f$.

The optimal trajectory, denoted as $r_f^{\mathrm{*}}$, can be defined as the trajectory that minimizes the value of the MEFT among the ensemble of feasible trajectories that satisfy the reliability level $\alpha$.

$${\phi }_{\alpha }\left(r_f^{\mathrm{*}},T_f^{\mathrm{*}}\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left[{\phi }_{\alpha }\right(r_f,T_f\left)\right]$$

(14)

According to Acerbi et al. [23], Equation (12) can be expressed as follows:

$${\phi }_{\alpha }(r_f,T_f)={\displaystyle \frac{1}{1-\alpha }}{\int }_{\alpha }^1 {\zeta }_{\tau }(r_f,T_f)\mathrm{d}\tau$$

(15)

Finally, we construct the following problem:

$$\underset{r}{min} {\phi }_{\alpha }(r_f,T_f)={\displaystyle \frac{1}{1-\alpha }}{\int }_{\alpha }^1 {\zeta }_{\tau }(r_f,T_f)\mathrm{d}\tau$$

(16)

$$\mathrm{s}. \mathrm{t}. \sum _j x_{ij}-\sum _j x_{ji}=\left\{\begin{array}{cc}1& i \mathrm{is} \mathrm{the} \mathrm{origin}\\ 0& i \mathrm{is} \mathrm{not} \mathrm{the} \mathrm{origin}\\ -1& i \mathrm{is} \mathrm{the} \mathrm{destination}\end{array}\right.$$

(17)

$$x_{ij}=\left\{\begin{array}{cc}1,& (w_i,w_j)\in r_f\\ 0,& (w_i,w_j)\notin r_f\end{array}\right.,\forall w_i,w_j\in W$$

(18)

3.2. The Calculation Method of the MEFT

Addressing the aforementioned mixed-integer programming model presents two challenges. Firstly, due to the heterogeneous nature of the segment flight time distributions, it is difficult to calculate them directly. Secondly, the objective function ${\phi }_{\alpha }(r_f,{\mathit{T}}_f)$ is excessively complex. To address these problems, we use the first four moments of the segment flight time instead of the PDF to portray its stochasticity and estimate the MEFT by the Cornish–Fisher expansion [24]. The expansion can be used to estimate the minimum quantile of a variable analytically based on its statistics. It was demonstrated that the approximation result can meet the accuracy requirement [25]. Then, ${\zeta }_{\alpha }(r_f,\mathit{T})$ is

$${\zeta }_{\alpha }\left(r_f,{\mathit{T}}_f\right)\approx E\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]+{\rho }_r\left(\alpha \right)\sqrt{D\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}$$

(19)

$$\begin{array}{c}{\rho }_r\left(\alpha \right)={\mathsf{\Phi }}^{-1}\left(\alpha \right)+{\displaystyle \frac{S\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{6}}+\\ \left\{{\left[{\mathsf{\Phi }}^{-1}\left(\alpha \right)\right]}^2-1\right\}\cdot {\displaystyle \frac{1}{24}}M\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]\left\{{\left[{\mathsf{\Phi }}^{-1}\left(\alpha \right)\right]}^3-3{\mathsf{\Phi }}^{-1}\left(\alpha \right)\right\}\\ -{\displaystyle \frac{1}{36}}{S}^2\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]\left\{2{\left[{\mathsf{\Phi }}^{-1}\left(\alpha \right)\right]}^3-5{\mathsf{\Phi }}^{-1}\left(\alpha \right)\right\}\end{array}$$

(20)

where ${\mathsf{\Phi }}^{-1}\left(\alpha \right)$ is the inverse function of the standard Gaussian distribution. $S\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ and $M\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ are the skewness and kurtosis coefficients, which can measure the asymmetry and sharpness of the PDF relative to the normal distribution, respectively. Equation (19) avoids assuming the form of the PDF in advance, which is universal for the study of air traffic networks. Specifically, this equation effectively solves the problem that the flight time distributions are heterogeneous under different segments, trajectories, and time periods.

The formulations of $S\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ and $M\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ are

$$S\left[T\right(r_f,\mathit{\xi }\left)\right]={\displaystyle \frac{E\left\{[\xi \left(r_f,{\mathit{T}}_f\right)-E(\xi \left(r_f,{\mathit{T}}_f\right)){]}^3\right\}}{\left\{D\right[\xi \left(r_f,{\mathit{T}}_f\right)]{\}}^{3/2}}}$$

(21)

$$M\left[T\right(r_f,\mathit{\xi }\left)\right]={\displaystyle \frac{E\left\{\xi \left(r_f,{\mathit{T}}_f\right)-E\left(\xi \left(r_f,{\mathit{T}}_f\right)\right){]}^4\right\}-3\left\{D\right[\xi \left(r_f,{\mathit{T}}_f\right)]{\}}^2}{\left\{D\right[\xi \left(r_f,{\mathit{T}}_f\right)]{\}}^2}}$$

(22)

according to Equations (15), (19), and (20),

$$\begin{gathered} {\phi }_{\alpha }\left(r_f,{\mathit{T}}_f\right)\approx {\displaystyle \frac{1}{1-\alpha }}{\int }_{\alpha }^1 \left\{E\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]+{\rho }_r\left(\tau \right)\sqrt{D\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}\right\} \\ ={\displaystyle \frac{1}{1-\alpha }}{\int }_{\alpha }^1 \left\{E\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]\right\}\mathrm{d}\tau +{\displaystyle \frac{1}{1-\alpha }}{\int }_{\alpha }^1 \left\{{\rho }_r\left(\tau \right)\sqrt{D\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}\right\}\mathrm{d}\tau \\ =E\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]+{\displaystyle \frac{\sqrt{D\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}}{1-\alpha }}{\int }_{\alpha }^1 {\rho }_r\left(\tau \right)\mathrm{d}\tau \end{gathered}$$

(23)

$$\begin{array}{c}{\int }_{\alpha }^1 {\rho }_r\left(\tau \right)\mathrm{d}\tau =-{\displaystyle \frac{S\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{6}}\left(1-\alpha \right)+\\ \left\{1-{\displaystyle \frac{M\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{8}}+\left.{\displaystyle \frac{5S^2\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{36}}\right\}{\int }_{\alpha }^1 {\mathsf{\Phi }}^{-1}\left(\tau \right)\mathrm{d}\tau +\right.\\ {\displaystyle \frac{S\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{6}}{\int }_{\alpha }^1 {\left[{\mathsf{\Phi }}^{-1}\left(\tau \right)\right]}^2\mathrm{d}\tau +\left\{{\displaystyle \frac{M\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{24}}-{\displaystyle \frac{S^2\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]}{18}}\right\}{\int }_{\alpha }^1 {\left[{\mathsf{\Phi }}^{-1}\left(\tau \right)\right]}^3\mathrm{d}\tau \end{array}$$

(24)

Different from $E\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ and $D\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$, both $S\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ and $M\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]$ do not satisfy the additivity under the assumption that the flight time distributions for each segment are independent. Therefore, we introduce the third and fourth cumulants which satisfy the additivity to simplify calculation (see Appendix A).

In summary, the MEFT trajectory planning model (from Equation (16) to (18)) can be rewritten as follows:

$$\underset{r}{\min }{\phi }_{\alpha }(r_f,{\mathit{T}}_f)\approx \sum _{a_{f,i}\in r_f} x_{ij}E\left(a_{f,i}\right)+{\displaystyle \frac{{\int }_{\alpha }^1 {\rho }_r\left(\tau \right)\mathrm{d}\tau }{1-\alpha }}\sqrt{\sum _{a_{f,i}\in r_f} x_{ij}D\left(a_{f,i}\right)}$$

(25)

$$\mathrm{s}.\mathrm{t}.\sum _j x_{ij}-\sum _j x_{ji}=\left\{\begin{array}{cc}1& i \mathrm{is} \mathrm{the} \mathrm{origin}\\ 0& i \mathrm{is} \mathrm{not} \mathrm{the} \mathrm{origin}\\ -1& i \mathrm{is} \mathrm{the} \mathrm{destination}\end{array}\right.$$

(26)

$$x_{ij}=\left\{\begin{array}{cc}1,& (w_i,w_j)\in r_f\\ 0,& (w_i,w_j)\notin r_f\end{array}\right.,\forall w_i,w_j\in W$$

$$\begin{array}{c}{\int }_{\alpha }^1 {\rho }_r\left(\tau \right)\mathrm{d}\tau =-{\displaystyle \frac{J}{6}}\left(1-\alpha \right)+\left(1-{\displaystyle \frac{G}{8}}+{\displaystyle \frac{5J^2}{36}}\right)\cdot {\int }_{\alpha }^1 {\mathsf{\Phi }}^{-1}\left(\tau \right)\mathrm{d}\tau +{\displaystyle \frac{J}{6}}{\int }_{\alpha }^1 {\left[{\mathsf{\Phi }}^{-1}\left(\tau \right)\right]}^2\mathrm{d}\tau \\ +\left({\displaystyle \frac{G}{24}}-{\displaystyle \frac{J^2}{18}}\right){\int }_{\alpha }^1 {\left[{\mathsf{\Phi }}^{-1}\left(\tau \right)\right]}^3\mathrm{d}\tau \end{array}$$

(27)

$$J=\sum _{a_{f,i}\in A} x_{ij}S\left(a_{f,i}\right){\left[D\left(a_{f,i}\right)\right]}^{3/2}/{\left[\sum _{a_{f,i}\in r_f} x_{ij}D\left(a_{f,i}\right)\right]}^{3/2}$$

(28)

$$G=\sum _{a_{f,i}\in A} x_{ij}M\left(a_{f,i}\right){\left[D\left(a_{f,i}\right)\right]}^2/{\left[\sum _{a_{f,i}\in r_f} x_{ij}D\left(a_{f,i}\right)\right]}^2$$

(29)

The analysis above highlights the calculation method of the MEFT, which can be estimated by the Cornish–Fisher expansion. Finally, the model is described as a shortest path problem, which will be addressed by the algorithm proposed in Section 4.

4. A Two-Stage Algorithm for the Proposed Model

A two-stage algorithm for a time-variant network is proposed in this study. First, we calculate the bounds of the optimal trajectory by analyzing the characteristics of the MEFT. Then, the feasible trajectory ensemble is obtained by the TVSP algorithm and the K-TVSP algorithm. In the second stage, we calculate and compare the MEFTs of feasible trajectories to identify the most optimal trajectory.

4.1. The Upper and Lower Bounds

Proposition 1. 

The MEFT satisfies subadditivity.

Proof. 

Assuming that ${\mathrm{r}}_{\mathrm{f}}$ consists of segments ${\mathrm{a}}_{\mathrm{f},1}$ and ${\mathrm{a}}_{\mathrm{f},2}$, if the MEFTs of ${\phi }_{\alpha }\left(a_{f,1}\right)$ and ${\phi }_{\alpha }\left(a_{f,2}\right)$ do not change with time (i.e., time-invariant network), according to reference [18],

$${\phi }_{\alpha }\left(a_{f,1}\right)+{\phi }_{\alpha }\left(a_{f,2}\right)\ge {\phi }_{\alpha }\left(r_f,{\mathit{T}}_f\right)$$

(30)

In a time-variant network, the MEFTs of segments $a_{f,1}$ and $a_{f,2}$ at time $t$ can be defined as ${\phi }_{\alpha }\left(a_{f,1},t\right)$ and ${\phi }_{\alpha }\left(a_{f,2},t\right)$. There is no doubt that they have a maximum value during the studied horizon $[0, T]$. According to Equation (30),

$$\mathrm{m}\mathrm{a}\mathrm{x}\left({\phi }_{\alpha }\left(a_{f,1},t\right)\right)+\mathrm{m}\mathrm{a}\mathrm{x}\left({\phi }_{\alpha }\left(a_{f,2},t\right)\right)\ge {\phi }_{\alpha }\left(r_f,{\mathit{T}}_f\right)$$

(31)

Let ${\mathsf{\Phi }}_{\alpha }(r_f,{\mathit{T}}_f)$ be the sum of the maximum value of the MEFTs of the segments consisting of $r_f$; then,

$${\phi }_{\alpha }\left(r_f,{\mathit{T}}_f\right)\le {\mathsf{\Phi }}_{\alpha }\left(r_f,{\mathit{T}}_f\right)=\sum _{a_{f,i}\in r_f} \max \left({\phi }_{\alpha }\left(a_{f,i}\right),t\right)$$

(32)

Therefore, the MEFT satisfies the subadditivity in a time-variant network. □

Proposition 2. 

${\phi }_{\alpha }\left(r_f,{\mathit{T}}_f\right)>E\left[\xi \left(r_f,{\mathit{T}}_f\right)\right]=\sum _{a_{f,i}\in r_f}E\left(a_{f,i}\right)$.

Proof. 

See the definition of the MEFT. □

Evidently, $r_f^{*}$ satisfies Propositions 1 and 2:

$$\sum _{a_{f,i}\in r_f^{*}}E\left(a_{f,i}\right)=E\left[\xi \left(r_f^{*},{\mathit{T}}_f^{\mathit{*}}\right)\right]<{\phi }_{\alpha }\left(r_f^{*},{\mathit{T}}_f^{\mathit{*}}\right)\le {\mathsf{\Phi }}_{\alpha }\left(r_f^{*},{\mathit{T}}_f^{\mathit{*}}\right)=\sum _{a_{f,i}\in r_f^{*}} \max \left({\phi }_{\alpha }\left(a_{f,i}\right),t\right)$$

(33)

Suppose that $r_{f,1}$ is the shortest trajectory in network $G(W,A,E(T\left)\right)$, and $r_{f,2}$ is the shortest trajectory in network $G(W,A,\mathrm{m}\mathrm{a}\mathrm{x}(\phi \left(T\right))$. Then,

$$\sum _{a_{f,i}\in r_{f,1}}E\left(a_{f,i}\right)<{\phi }_{\alpha }\left(r_f^{*},{\mathit{T}}_f^{\mathit{*}}\right)\le \sum _{a_{f,i}\in r_{f,2}} \mathrm{m}\mathrm{a}\mathrm{x}\left({\phi }_{\alpha }\left(a_{f,i}\right)\right)$$

(34)

$r_{f,1}$ and $r_{f,2}$ can be calculated by the TVSP algorithm. Additionally, the values of ${\phi }_{\alpha }(r_{f,1},{\mathit{T}}_{f,1})$ and ${\phi }_{\alpha }(r_{f,2},{\mathit{T}}_{f,2})$ can be obtained simultaneously. Therefore, the bounds of the optimal objective are

$$\sum _{a_{f,i}\in r_{f,1}}E\left(a_{f,i}\right)<\sum _{a_{f,i}\in r_f^{*}}E\left(a_{f,i}\right)<{\phi }_{\alpha }\left(r_f^{*},{\mathit{T}}_f^{\mathit{*}}\right)\le \mathrm{m}\mathrm{i}\mathrm{n}\left[{\phi }_{\alpha }\right(r_{f,1},{\mathit{T}}_{f,1}),{\phi }_{\alpha }(r_{f,2},{\mathit{T}}_{f,2}\left)\right]$$

(35)

The interpretation of Equation (35) is that in the network $G(W,A,E(T\left)\right)$, the trajectories satisfying Equation (36) constitute the selectable set $R_f$, $r_f^{*}\in R_f$. So, the issue is converted into a K-shortest path problem with a known upper bound, which may be solved by the K-TVSP algorithm. Currently, the optimal trajectory can be computed by the enumeration method.

$$\sum _{a_{f,i}\in r_{f,1}}E\left(a_{f,i}\right)<E\left(r_f\right)\le \mathrm{m}\mathrm{i}\mathrm{n}\left[{\phi }_{\alpha }\right(r_{f,1},{\mathit{T}}_{f,1}),{\phi }_{\alpha }(r_{f,2},{\mathit{T}}_{f,2}\left)\right]$$

(36)

4.2. Two-Stage Algorithm

This section presents the framework of the two-stage algorithm (see Algorithm 1) and proves its feasibility in a time-variant network.

Algorithm 1. The framework of the two-stage algorithm

Input: Airspace entry point $o$, Airspace exit point $d$, $\alpha$, $\zeta$

Output: Minimum MEFT 4D trajectory

//Stage 1

1: Initialize $R_f=\varnothing$;

2: Solve for the upper bound with the short path algorithm in network $G(W,A,\phi (T\left)\right)$;

3: Solve for the lower bound with the short path algorithm in network $G(W,A,E(T\left)\right)$;

4: Obtain $R_f$ with the K-short path algorithm in network $G(W,A,E(T\left)\right)$;

//Stage 2

5: Calculate the MEFT for each trajectory in $R_f$;

6: Select the minimum MEFT 4D trajectory $r_f^{*}$;

7: end

In trajectory planning, commonly used algorithms include the Dijkstra algorithm, the Floyd algorithm, heuristic algorithms, etc. However, these algorithms are not efficient enough and have few applications in time-variant air traffic networks. Fortunately, extensive research has been conducted on the TVSP problem. In this study, the TVSP algorithm proposed by Tan et al. [26] is used to solve the bounds. Additionally, the K-TVSP algorithm, based on Yen’s logic [27], is employed to calculate $R_f$.

The origin and destination of $r_f$ are defined as $w_o$ and $w_d$, respectively. The full steps that need to be taken to apply the K-TVSP algorithm are summarized as follows:

Step 1: Set $k=1$, and the shortest path is $r_1=(w_o,\cdots ,w_i,w_{j,1},\cdots ,w_d)$.

Step 2: Let $r_k=(w_o,\cdots ,w_i,w_{j,k},\cdots ,w_d)$ denote the $kth$ shortest path. Let $f_{id}\left(t\right)$ and $f_{ij}\left(t\right)$ represent the cost from $w_i$ to $w_d$ and $w_j$ at $t$ in the K-TVSP, respectively.

Step 3: Select a waypoint $w_i$ from $r_k$. Select a potential waypoint $w_{j,k+1}$ from the connect waypoint ensemble of $w_i$. Set the cost of segment $\left(w_i,w_{j,k}\right)$ as $+\infty$. Note that the problem is discrete, and $t\in \left((M-1)t_{vp},M{t}_{vp}\right),M\in N^{+}$. If $t+f_{ij,k}\left(t\right)<M{t}_{vp}$, then go to Step 4; else, if $t+f_{ij,k}\left(t\right)\ge M{t}_{vp}$ and $f_{ij,k}\left(t\right)\le f_{ij,k+1}\left(t\right)$, go to Step 5; if $t+f_{ij,k}\left(t\right)\ge M{t}_{vp}$ and $f_{ij,k}\left(t\right)>f_{ij,k+1}\left(t\right)$, go to Step 6.

Step 4: Apply the TVSP to calculate the optimal solution with $w_i$ as the origin and set it as $r_{k+1}$.

Step 5: In this case, $t+f_{ij,k+1}\left(t\right)\ge M{t}_{vp}$, and the time-variant wind condition on $(w_i,w_{j,k})$ and $(w_i,w_{j,k+1})$ is consistent, as is the successor airspace network. Apply the TVSP to calculate the optimal solution with $w_i$ as the origin and set it as $r_{k+1}$.

Step 6: In this case, if $t+f_{ij,k+1}\left(t\right)\ge M{t}_{vp}$, go to Step 4; if $t+f_{ij,k+1}\left(t\right)<M{t}_{vp}$, apply the TVSP to calculate the optimal solution with $w_o$ as the origin and set it as $r_{k+1}$.

Step 7: If the cost of $r_{k+1}$ meets Constraint (36), set $r_{k+1}$ into $R_f$ and go to Step 2; otherwise, stop the iteration.

According to Equation (11), we ignore the deviation on a single segment. Therefore, in Step 3, a classification discussion is not needed when $t+f_{ij,k}\left(t\right)<M{t}_{vp}$.

This section describes the framework of the proposed two-stage algorithm, which can obtain the shortest path in a time-variant network. In the next section, a case study will be conducted to demonstrate its feasibility.

5. Numerical Simulation

5.1. Simulation Data

The China Western Airspace (CWA) is designated as the target airspace, with its location indicated in Figure 7. The flight from Guangzhou Baiyun Airport (ICAO: ZGGG) in China to Amsterdam Schiphol Airport (ICAO: EHAM) in the Netherlands on 8 June 2019, is applied as an example. We focus on the trajectory within the CWA, and the longitudes and latitudes of the entry and exit points are $(105°18\prime \mathrm{E},28°29\prime \mathrm{N})$ and $(82°53\prime \mathrm{E},46°52\prime \mathrm{N})$, respectively. The cruising altitude is 10,100 m, and the expected entry time into the CWA is 8:00 a.m.

The time span of the wind data is from 8:00 a.m. to 12:00 a.m., with a granularity of 1 h and a forecast ahead of 24 h. The wind grid spans from $72° \mathrm{E}$ to $108° \mathrm{E}$ and from $22° \mathrm{N}$ to $50° \mathrm{N}$, with a granularity of $0.2°$. Figure 8 gives the mean value of 51 members in the wind forecast ensemble at 10,100 m altitude from 8:00 to 12:00. The arrow symbolizes the force and direction of the wind, while the color indicates the range of the wind speed (with a darker color indicating a stronger uncertainty). The CWA is located inside the westerlies, the maximum force of the forecast wind is 45 m/s, and the minimum is 2.4 m/s. Particularly, there is a propensity for uncertainty to be enhanced and force to weaken with time.

5.2. Heterogeneity of Segment Flight Time Distribution

The KS test is applied to verify the heterogeneity of the segment flight time distribution in terms of both time and space. In this study, this test is conducted with a confidence level $\beta =0.95$ and a corresponding parameter $c\left(\beta \right)=1.36$ [28]. Two equal-length segments A and B are selected from the ensemble. In the time dimension, a comparison of the cumulative distributions of segment A in the periods of 8:00–9:00 and 9:00–10:00 is given in Figure 9a. In the space dimension, a comparison of the cumulative distributions of segments A and B from 8:00 to 9:00 is given in Figure 9b. The size of the ensemble is 51 ($U_{A,1}=U_{A,2}=U_B$), and the KS statistic $\overline{P}$ is calculated by Equation (37),

$${\overline{P}}_1={\overline{P}}_2=c\left(\beta \right)·\sqrt{{\displaystyle \frac{U_{A,1}+U_{A,2}}{U_{A,1}·U_{A,1}}}}=1.36\times \sqrt{{\displaystyle \frac{2}{51}}}=0.27$$

(37)

According to Figure 9, $P_1=0.41>{\overline{P}}_1$, and $P_2=0.49>{\overline{P}}_2$. Thus, the flight time distributions of segment A are heterogeneous under different time periods, and the distributions of segments A and B during the same time period are heterogeneous, too.

5.3. The Necessity of Considering S and M

Figure 10 shows a simple airspace network containing three feasible trajectories. Given that $\alpha =0.95$, the corresponding $E, D, S, M$, and MEFT are shown in

Table 1. The $E, D, S, M$ and MEFT of three feasible trajectories.

Time	Characteristics	Trajectory 1	Trajectory 2	Trajectory 3
8:00–9:00	$E$	8.2	6.4	12.7
	$D$	14.8	22.4	18.6
	$S$	−0.4	0.8	1.2
	$M$	0.7	−0.5	1.4
	MEFT (min)	13.55	13.78	14.03
9:00–10:00	$E$	6.8	7.2	11.8
	$D$	16.9	28.4	21.7
	$S$	0.5	−0.7	0.8
	$M$	0.2	0.6	1.1
	MEFT (min)	14.92	15.62	14.33
10:00–11:00	$E$	7.2	5.6	11.2
	$D$	15.2	26.5	22.8
	$S$	1.1	−0.2	−0.4
	$M$	0.1	1.0	−0.8
	MEFT (min)	14.12	14.01	14.56
11:00–12:00	$E$	7.8	6.1	12.1
	$D$	13.9	24.3	20.6
	$S$	0.6	0.9	−0.8
	$M$	−0.5	−0.9	−1.6
	MEFT (min)	13.88	14.66	15.21

. $S<0, M<0$ indicates that the distribution has left-skewed characteristics with a low peak.

Trajectory 2 is the most optimal at 8:00–9:00 and 10:00–12:00, whereas trajectory 1 is the most optimal at 9:00–10:00 when $E$ is selected as the objective function. While considering $S$ and $M$, it is important to note that the MEFTs of the three trajectories have distinct advantages during different periods. Therefore, the optimal solution will be affected by $S$ and $M$ in the reliability trajectory planning problem.

5.4. Effectiveness Analysis

Figure 11 and Figure 12 show the great circle trajectory, the structured airspace trajectory, the upper bound trajectory, the lower bound trajectory, and the optimal trajectory for the flights ZGGG-EHAM and EHAM-ZGGG, respectively. The forecast data and observed data in the figures are the mean values during 4 h.

Table 2. The indicators of the ZGGG-EHAM trajectories.

$\mathbf{Indicators} (\mathit{\alpha }=0.95$)	$\mathit{E}$	$\mathit{D}$	$\mathit{S}$	$\mathit{M}$	MEFT	True Flight Time
Great circle trajectory	222.6	29.7	6.26	3.21	247.5	236.5
Structured airspace trajectory	252.8	22.7	8.24	2.25	263.4	260.2
Lower bound trajectory	217.6	21.8	−5.72	7.84	256.8	239.6
Upper bound trajectory	263.7	24.8	6.16	4.22	250.3	268.5
Optimal trajectory	234.2	15.4	4.41	2.91	238.1	236.1

and

Table 3. The indicators of the EHAM-ZGGG trajectories.

$\mathbf{Indicators} (\mathit{\alpha }=0.95$)	$\mathit{E}$	$\mathit{D}$	$\mathit{S}$	$\mathit{M}$	MEFT	True Flight Time
Great circle trajectory	187.5	20.3	4.2	1.2	222.3	198.5
Structured airspace trajectory	212.4	18.7	5.4	2.4	228.7	227.3
Lower bound trajectory	184.2	16.5	3.8	4.6	245.9	193.6
Upper bound trajectory	227.1	21.5	−2.4	3.6	238.1	221.4
Optimal trajectory	201.2	15.2	2.1	1.3	207.5	199.4

give the calculated values of $E,D,S,M$, and the MEFT using forecast data. Similarly, the true flight time is calculated using observed data. By checking the flight plan, the ${\zeta }_{ZE}$ and ${\zeta }_{EZ}$ of ZGGG-EHAM and EHAM-ZGGG are set to 232 min and 196 min, respectively.

According to Figure 11 and Table 2, the lower bound trajectory of the ZGGG-EHAM flight closely follows the great circle trajectory. As a result, it has a short voyage, with an $E$ value of 217.6 min. However, it passes through a region with strong uncertainty, leading to a large deviation between the true flight time and $E$, which is 22 min. On the contrary, the upper bound trajectory has a larger $E$ and smaller deviation because it shifts to the north to avoid the region with strong uncertainty. In contrast, the optimal trajectory achieves a balance between reliability and efficiency, ensuring a reliability rate of 99.2% between the true flight time and $E$. Compared with the structured airspace trajectory, the total flight time is reduced by 24.1 min, which is about 9.2%. Therefore, the proposed model is effective. The results in Figure 12 and Table 2 can further prove this conclusion.

Furthermore, we can also find an interesting phenomenon. As seen from Figure 11a and Figure 12a, the trajectory of the ZGGG-EHAM flight seems to be closer to the region with high uncertainty than the EHAM-ZGGG flight. Specifically, this is evident in the region spanning from $94° \mathrm{E}$ to $102° \mathrm{E}$. There are two primary factors contributing to this: first, the former is a downwind flight; second, wind varies with time. Note that the wind uncertainty shows a strengthening trend (see Figure 8). Consequently, when the former enters this region, the uncertainty is weak, and thus, it passes through selectively; whereas when the latter enters, the uncertainty becomes strong, and thus, it avoids it selectively. This phenomenon proves the proposed model’s sensitivity to time-variant wind uncertainty.

To validate the efficiency of the two-stage algorithm, the solution solved by the GA is shown in Figure 13. The maximum generations, the crossover rate, the mutation rate, and the population size are 250, 0.9, 0.2, and 100, respectively. As can be seen in Figure 13, the GA covers after about 120 iterations; the minimum fitness computed by the GA is consistent with the two-stage algorithm, which validates the efficiency of the proposed algorithm. However, the computational time required for the GA is 5.2 h, which is too much longer than the 2.3 min needed for the two-stage algorithm to find the optimal trajectory.

5.5. Sensitivity Analysis on $\alpha$

Figure 14 shows the optimal trajectories with different $\alpha$, and the corresponding MEFTs are given in

Table 4. The MEFTs of the optimal trajectory with different $\alpha$.

$\mathbf{Reliability} \mathbf{Parameter} \mathit{\alpha }$	MEFT (min)
0.95	238.1
0.9	234.5
0.85	228.6
0.8	224.5
0.5	210.2

.

It is evident that the MEFT grows gradually as the value of $\alpha$ increases. Meanwhile, the optimal trajectory continuously changes, indicating that the model is sensitive to variations in $\alpha$. Furthermore, let $\alpha =0.95, 0.9, 0.85$; the trajectories overlap in the region of $(28° \mathrm{N},37° \mathrm{N})$. This suggests that if we set a high-reliability requirement, the optimal trajectory in the region with high uncertainty is relatively stable. While in the region with low uncertainty, the aircraft can choose shorter routes or more favorable wind conditions. Comparing the three cases of $\alpha =0.95, 0.8, 0.5$, the optimal trajectory is constantly close to the great circle trajectory, and the voyage in the region with high uncertainty increases, which is consistent with the real situation and can prove the sensitivity to variations in $\alpha$.

6. Conclusions

This study proposes a trajectory planning model considering the influence of time-variant wind uncertainty and the heterogeneity of segment flight time distributions. The objective function of this model is the MEFT, which can represent the reliability that an aircraft reaches its destination punctually. We propose a segment flight time prediction model in which wind is defined as a discrete random variable. Additionally, we apply the first four moments to portray the stochasticity of the flight time distributions, avoiding the assumption of homogeneity. Finally, a two-stage algorithm is suggested to address this problem, and its convergence in a time-variant network is proven.

The simulation results show that the optimal trajectory has an extremely high level of reliability, reaching up to 99.2%. When comparing it with the structured airspace trajectory, the total flight time decreases by 24.1 min, about 9.2%. The sensitivity analysis on $\alpha$ indicates that if we impose a high-reliability requirement, the trajectory remains generally stable in the region with high uncertainty. Conversely, in the region with low uncertainty, the aircraft can choose a shorter route and take full advantage of the favorable wind. The solution time of the two-stage algorithm is only 2.3 min, which can satisfy the requirement of trajectory planning. Note that a high-reliable trajectory can enhance flight time predictability and reduce delays in a flight loop. Therefore, the proposed model and algorithm can effectively serve practical applications.

Future studies can extend in the following directions: Firstly, wind is recommended to be modeled as a continuous variable instead of a discrete one. Secondly, although the proposed two-stage algorithm can solve the model effectively, it is essential to develop algorithms with less complexity to reduce solution time. Thirdly, this study focuses on the horizontal trajectory optimization, ignoring the change in the flight profile during operation, which should be further considered in the future. Fourth, innovative models should be constructed to analyze the effects of trajectory coupling to be applied in the large-scale problem.