# Traffic Flow Funnels Based on Aircraft Performance for Optimized Departure Procedures

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Towards 3D Trajectory Optimization

#### 1.2. Sources of Trajectory Uncertainty during Climb

#### 1.3. Trajectory Clustering and Traffic Flow Funnels

#### 1.4. Optimization Inside Traffic Flow Funnels

## 2. Materials and Methods

#### 2.1. Concept of Performance-Optimized Departure Funnels

- Historical departure flights from ADS-B data are clustered to various funnels incorporating the actual procedures as a reference scenario to identify benefits of the funnel concept (ADS-B funnel);
- These ADS-B flights are optimized without any route restriction considering wind, fuel, and direct operating costs with the multi-criteria optimization of TOMATO in a newly developed 3D search grid for departures. This ensures that most aircraft reach their optimal trajectory in the presence of uncertainty sources, such as the individual flight performance of various aircraft types, masses, and weather forecasts;
- These optimized trajectories are clustered to the optimized funnels with the same clustering algorithm;
- The ADS-B and optimized funnels are simulated with the same historical flight schedule, where the flights are restricted to fly inside the given funnels and optimized according to the weather data valid for this period. For this, the so-called 3D funnel grid is developed, which determines the most suitable funnel per flight and then limits pathfinding to the portion of airspace inside the funnel;
- The flight efficiency of the two funnel sets is compared to quantify the efficiency gain of the optimized funnels.

#### 2.2. Trajectory Optimization

#### 2.2.1. Tabulated Aircraft Performance for Pathfinding Algorithms

^{−1}], the true airspeed $TAS$ in [m s

^{−1}], and the fuel flow $FF$ in [kg s

^{−1}] are necessary to compute the time and fuel costs, as well as the possible altitude change on a given edge length. These altitude-dependent performance values are determined in advance to estimate the aircraft-specific flight performance as well as possible, despite the limited availability of aircraft state information in the graph search.

^{−1}] or the Mach number M, which must be converted to $TAS$ for pathfinding. With Equation (1), $TAS$ is obtained from the $CAS$ with the air density $\rho $ in [kg m

^{−3}], and pressure p in [Pa] at the current altitude, the air density ${\rho}_{0}=1.225\phantom{\rule{4.pt}{0ex}}\mathrm{kg}\phantom{\rule{4.pt}{0ex}}{\mathrm{m}}^{-3}$, and pressure ${p}_{0}=101,325\phantom{\rule{4.pt}{0ex}}\mathrm{Pa}$ at mean sea level (MSL), as well as the adiabatic index $\kappa =1.4$.

#### 2.2.2. Multi-Objective 3D Pathfinding for Departures

- The next node ${n}_{u+1}$ is placed in the direction of ${\nabla}_{u}$ at the next altitude ${h}_{u+1}$ from $\mathbb{A}$ to account for a straight climb without track change. For this, Equation (10) is used to determine the climb duration ${t}_{u,u+1}$ from ${h}_{u}$ to ${h}_{u+1}$ to then calculate the longitudinal climb distance $SA{D}_{u,u+1}$ with Equation (13).
- Further nodes are added at ${h}_{u+1}$ using $SA{D}_{u,u+1}$, but permitting left and right turns in $a\in \mathbb{N}$ angular steps of $\theta $ up to the maximum angular difference ${\psi}_{max}={\dot{\psi}}_{s}\xb7{t}_{u,u+1}$ to avoid an ROT that exceeds ${\dot{\psi}}_{s}=3\xb0{s}^{-1}$.
- To permit level segments, additional nodes are inserted at ${h}_{u}$ with turns permitted at a steps of $\theta $ up to $\pm {\psi}_{max}$. Furthermore, $SA{D}_{u,u1}$ projected on the ground is used to ensure that these nodes are located perpendicularly below the previously inserted nodes.

^{−1}]. The effect of wind is added to the costs with the headwind component $HW$ in [m s

^{−1}] at the given location and altitude. Equation (15) describes the calculation of C for a single edge:

#### 2.2.3. Vertical Profile with Sophisticated Aircraft Performance Model (SOPHIA)

#### 2.3. Algorithm for the Traffic Flow Funnels

- With the inner clustering, the trajectories are assigned to a runway threshold;
- The outer clustering determines a common end area on the E-TMA radius;
- The preliminary clustering groups all trajectories with the same runway threshold and end area;
- The sub-clustering separates groups of trajectories, which have the same runway threshold and end area, but do not follow a similar route in between;
- For each of these clusters, a mean trajectory is computed; and
- Finally, gates are placed along the mean trajectory to define the lateral and vertical extent of the funnel.

#### 2.4. Allocation and 3D Pathfinding Inside the Traffic Flow Funnels

#### 2.5. Evaluation Metrics

## 3. Results

#### 3.1. Scenario Definition

#### 3.2. Weather Effect on Optimized Trajectories and Height of Funnel Gates

#### 3.3. Funnels of ADS-B and Optimized Clustered Flights

#### 3.4. Flight Efficiency

## 4. Discussion

#### 4.1. Key Findings

#### 4.2. Assumptions and Simplifications

#### 4.3. Operational Applicability

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Clustered ADS-B radar tracks of departing flights using runway 26R at Munich Airport. Colors represent clusters where gray flights are outliers.

**Figure 2.**Comparison of the vertical profile from the ADS-B track data (green) with the OpenAP performance tables (red) and the optimization of SOPHIA (blue) of the same aircraft type.

**Figure 3.**Top view (

**left**) and 3D view (

**right**) of a simplified 3D path search grid for the climb with $\theta =45\xb0$ and three different optimization altitudes $\mathbb{A}=\{{h}_{1},{h}_{2},{h}_{3}\}$. The grid expands at the extended runway centerline of node $III$, which is also the current node n, resulting in a projected neighbor node ${n}_{u}$.

**Figure 4.**3D departure search grid from Munich’s runway 26R to a northern destination with $\theta =30\xb0$, ${\dot{\psi}}_{s}=3\xb0{s}^{-1}$ and $\mathbb{A}=\{{h}_{1},\dots ,{h}_{6}\}$ optimization altitudes. The color indicates the altitude through the $SAD$-dependent estimated costs to destination (magenta highest). Map: Google, ©2021, GeoBasis-DE/BKG.

**Figure 5.**Computation steps for the traffic flow funnels, where the first four clustering steps group trajectories with a common runway threshold, end point at the E-TMA radius and similar routing in between, while the last two steps define the funnels based on a mean trajectory with gates for the lateral and vertical extent.

**Figure 6.**Simplified 3D funnel grid for two rectangular gates g and $g+1$ with $\alpha =6$ lateral and $\beta =6$ vertical sections, resulting in 49 grid points for each gate.

**Figure 7.**Aggregated plot from the vertical departure profiles in 23 wind scenarios (grey) and distance dependent heights of gates of the optimized funnels for A320-211, 58,000 kg only. Maximum altitude difference is 270 m at 20 km from threshold.

**Figure 8.**Four of seven 3D departure funnels with rectangular gates (red with unique border colors per funnel) from clustered flight data (09/2020), with the raw ADS-B data points (blue), and the mean trajectory per funnel (red lines). Blue points outside funnels are outliers or other runways.

**Figure 9.**Clustered flight tracks of departing flights using runway 26R at Munich Airport, with the ADS-B clusters excluding outliers on the left and the optimized clusters on the right side.

**Figure 10.**Funnel in the direction of north-west with a free optimized flight profile (yellow) and restricted to the funnel (green).

**Figure 11.**Boxplot of flight efficiency (fuel burn, ground distance route extension) as the result of a transition from ADS-B to optimized calculated departure funnels.

**Table 1.**Difference in gate width due to wind at 10, 20, and 60 km along-track distance from runway threshold for several aircraft configurations and all freely optimized flights.

Aircraft | 10 km | 20 km | 60 km | |||
---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | |

[m] | [m] | [m] | [m] | [m] | [m] | |

A320 | 2088 | 1822 | 6533 | 5593 | 13,156 | 10,993 |

B737 | 804 | 679 | 1900 | 1634 | 3784 | 3369 |

B777 | 821 | 709 | 2247 | 1922 | 5274 | 4286 |

**Table 2.**Difference in gate height due wind at 10, 20, and 60 km along track distance from runway threshold for several aircraft configurations and all freely optimized flights.

Aircraft | Gross Mass [kg] | 10 km | 20 km | 60 km | |||
---|---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | ||

[m] | [m] | [m] | [m] | [m] | [m] | ||

A320 | 39,000 | 355 | 69 | 490 | 110 | 197 | 52 |

58,000 | 204 | 40 | 246 | 52 | 336 | 86 | |

77,000 | 183 | 37 | 226 | 46 | 227 | 57 | |

B737 | 41,500 | 395 | 77 | 429 | 96 | 196 | 52 |

60,200 | 221 | 44 | 214 | 45 | 328 | 85 | |

79,000 | 185 | 38 | 221 | 45 | 344 | 92 | |

B777 | 145,000 | 320 | 61 | 511 | 111 | 215 | 57 |

246,500 | 157 | 32 | 236 | 49 | 727 | 207 | |

347,500 | 83 | 16 | 188 | 37 | 268 | 69 |

**Table 3.**Dimension of departure funnels and grid point spacing per gate from the ADS-B data and optimized flights.

Percentile | Gates of ADS-B Funnels | Gates of Optimized Funnels | |||
---|---|---|---|---|---|

Size | Spacing | Size | Spacing | ||

[m] | [m] | [m] | [m] | ||

Width | 0.05 | 2112 | 106 | 1874 | 94 |

0.5 | 4793 | 240 | 10,556 | 528 | |

0.95 | 7542 | 377 | 19,797 | 989 | |

Height | 0.05 | 1367 | 34 | 434 | 11 |

0.5 | 2946 | 73 | 1719 | 43 | |

0.95 | 4612 | 115 | 3558 | 89 |

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**MDPI and ACS Style**

Lindner, M.; Zeh, T.; Braßel, H.; Rosenow, J.; Fricke, H.
Traffic Flow Funnels Based on Aircraft Performance for Optimized Departure Procedures. *Future Transp.* **2022**, *2*, 711-733.
https://doi.org/10.3390/futuretransp2030040

**AMA Style**

Lindner M, Zeh T, Braßel H, Rosenow J, Fricke H.
Traffic Flow Funnels Based on Aircraft Performance for Optimized Departure Procedures. *Future Transportation*. 2022; 2(3):711-733.
https://doi.org/10.3390/futuretransp2030040

**Chicago/Turabian Style**

Lindner, Martin, Thomas Zeh, Hannes Braßel, Judith Rosenow, and Hartmut Fricke.
2022. "Traffic Flow Funnels Based on Aircraft Performance for Optimized Departure Procedures" *Future Transportation* 2, no. 3: 711-733.
https://doi.org/10.3390/futuretransp2030040