## 6 Simulation and Verification of Operation Procedure in Apron U-Shaped Area  
### Time Constraints-Based Simulation Model for Apron Operations  
In order to evaluate the apron U-shaped area operation procedures and performance, this section constructs a simulation model of apron operation considering time constraints and performs simulation extrapolations of the three apron U-shaped area operation procedures designed in Section 3. The apron simulation environment is coded in Python 3.6 and run on a PC with Intel i7-9700 KF8C8T, 3.60 GHz and 16.0 GB RAM. The relevant parameters were set as follows:  
1. The runway operation mode is segregated parallel operation.
2. Aircraft taxiing speed by area: average taxiing speed in the maneuvering area is 30 km/h, the average taxiing speed in the apron is 10 km/h, and the average taxiing speed in the U-shaped area is 8 km/h.
3. The longitudinal safety separation of aircraft is 50 m.  
The Monte-Carlo method ([PERSON], et al., 2018; [PERSON], et al., 2020) [39; 40] was used to generate simulated flight flows in the apron U-shaped area. In order to analyze the operational performance of the U-shaped area at different traffic intensities, the Poisson distribution parameter was increased from 20 to 50 aircraft/h in increments of 5, and the ratio of arrival and departure sorties was designed to be 1:1, with 1000 simulations under each procedure.  
To make the apron operation simulation closer to the real situation, the following time constraints were established for the aircraft surface taxiing process.  
Assume that \(\mathcal{F}=\left\{f_{1},\ldots,f_{n}\right\}\) is the set of aircraft in the study period, \(R_{i}\) is the corresponding taxi path of the aircraft \(f_{i}\), the set of nodes of the path is \(R_{i}=\left\{u_{i1},u_{i2},\ldots,u_{i_{k}}\right\}\) where \(u_{ip}\in V\), \((u_{ip},u_{i,p+1})\in E\), and \(p=1,\ldots,k_{i-1}\), and \(t_{iu}\) is the time at which the aircraft \(f_{i}\) passes through the nodes \(u\) on the taxi path \(R_{i}\).  
We introduce the 0-1 variable \(x_{iju}\), and \(x_{iju}=1\) indicates that the aircraft \(f_{j}\) arrives at the node \(u\) immediately after the aircraft \(f_{i}\); otherwise, \(x_{iju}=0\). For \(\forall f_{i},f_{j}\in F,i\
eq j\), the time constraint is as follows:  
**Case 1:** If \(u\) is not the U-shaped area entrance node or the parking stand node, and aircraft \(f_{j}\) reached node \(u\) immediately after the aircraft \(f_{i}\) (i.e., \(x_{iju}=1\)), then we have  
\[t_{ju}>x_{iju}t_{iu},\quad\forall u\in R_{i}\cap R_{j} \tag{24}\]  
**Case 2:** If \(u\) is the U-shaped area entrance node, and other conditions are the same as in case 1, then  
\[t_{ju}>x_{iju}t_{iu}+w,\quad\forall u\in R_{i}\cap R_{j} \tag{25}\]  
where \(w\) is the waiting time of aircraft entering or leaving the U-shaped area due to different operation rules.  
In order to define the uniqueness of the sequential relationship between the aircraft \(f_{j}\) and the aircraft \(f_{i}\), the following constraint is established:  
\[x_{iju}+x_{juu}=1,\quad\forall u\in R_{i}\cap R_{j} \tag{26}\]  
In addition, for conflict detection and deconfliction during taxiing, constraints are established to avoid four types of conflict situations, namely, cross-conflict, tail-chase conflict, head-to-head conflict, and taxiway waiting conflict, respectively ([PERSON], 2011) [41], which are not described in detail in this section.