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Optimization of ATM Scenarios Considering Overall and Single Costs 1 Optimization of ATM Scenarios Considering Overall and Single Costs 28/05/2014 Matthias Bittner, Benjamin Fleischmann, Maximilian Richter, Florian Holzapfel Matthias Bittner

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SLIDE 1

Optimization of ATM Scenarios Considering Overall and Single Costs 1

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Optimization of ATM Scenarios Considering Overall and Single Costs

28/05/2014 Matthias Bittner, Benjamin Fleischmann, Maximilian Richter, Florian Holzapfel

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SLIDE 2

Optimization of ATM Scenarios Considering Overall and Single Costs 2

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Multiple aircraft crossing ATM sector
  • Minimum individual costs
  • Maintain separation
  • Maximum fairness

Problem description

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Optimization of ATM Scenarios Considering Overall and Single Costs 3

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Determine the optimal control histories (aircraft index 𝑗) 𝐯𝑗,𝑝𝑞𝑢 𝑢 ∈ ℝ𝑛𝑗, 𝑗 = 1, … , 𝑂 the corresponding optimal state histories 𝐲𝑗,𝑝𝑞𝑢 𝑢 ∈ ℝ𝑜𝑗, 𝑗 = 1, … , 𝑂 and any additional parameters 𝐪𝑗 ∈ ℝ𝑙𝑗 that minimize the Bolza cost functional 𝐾 = Φ 𝐲𝑗 𝑢𝑔 , 𝑢𝑔 +

𝑢0 𝑢𝑔

ℒ 𝐲𝑗 𝑢 , 𝐯𝑗 𝑢 , 𝐪𝑗, 𝑢 𝑒𝑢 subject to the state dynamics 𝐲𝑗 𝑢 = 𝒈𝑗 𝐲𝑗 𝑢 , 𝐯𝑗 𝑢 , 𝐪𝑗, 𝑢𝑗 , 𝑗 = 1, … , 𝑂 the initial and final boundary conditions 𝛀0 𝐲𝑗 𝑢0 , 𝑢0 = 𝟏, 𝛀𝟏 ∈ ℝ𝑞 𝛀

𝑔 𝐲𝑗 𝑢𝑔 , 𝑢𝑔 = 𝟏,

𝛀𝒈 ∈ ℝ𝑟 and the equality and inequality path constraints 𝑫𝑓𝑟 𝐲𝒋 𝑢 , 𝐯𝑗 𝑢 , 𝐪𝑗, 𝑢 = 𝟏, 𝑫𝑓𝑟 ∈ ℝ𝑠, 𝑗 = 1, … , 𝑂 𝑫𝑗𝑜 𝐲𝑗 𝑢 , 𝐯𝑗 𝑢 , 𝐪𝑗, 𝑢 ≤ 𝟏, 𝑫𝑗𝑜 ∈ ℝ𝑡, 𝑗 = 1, … , 𝑂

Problem formulation

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SLIDE 4

Optimization of ATM Scenarios Considering Overall and Single Costs 4

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Optimization using direct methods: Discretize then Optimize!

Optimization Process

Infinite optimal control problem Optimization parameters: Continuous functions 𝐯i 𝑢 , 𝐲i 𝑢 , 𝐪𝑗, 𝑢𝑔 Analytical

  • ptimality

conditions cannot be evaluated. Discretization

  • f problem

Reduce infinite problem by discretization e.g. Trapezoidal Collocation Finite parameter

  • ptimization

problem Optimization parameters: Values at discrete points 𝐯𝑗,𝑙, 𝐲𝑗,𝑙, 𝐪𝑗, 𝑢𝑔 Optimization problem can be solved numerically. Optimization Optimization algorithm for parameter

  • ptimization

problems e.g.: Sequential quadratic programming (WORHP, SNOPT, IPOPT, fmincon) Solution Resulting values for states and controls at discrete points can be interpolated to the full state and control histories.

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SLIDE 5

Optimization of ATM Scenarios Considering Overall and Single Costs 5

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Kinematic models are sufficient, dynamics far from envelope
  • 2D models are used

State dynamics 𝑦𝑗 = 𝑊

𝐿,𝑗 ⋅ cos 𝜓𝐿,𝑗

𝑧𝑗 = 𝑊

𝐿,𝑗 ⋅ sin 𝜓𝐿,𝑗

State and control vectors 𝐲𝑗 = 𝑦𝑗, 𝑧𝑗 T 𝐯𝑗 = 𝑊

𝐿,𝑗, 𝜓𝐿,𝑗 𝑈

Separation path constraints (applied pair wise) 𝑒min

2

= 5𝑂𝑁 2 ≤ (𝑦𝑗−𝑦𝑘) 2 + (𝑧𝑗−𝑧𝑘) 2

Simulation model

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SLIDE 6

Optimization of ATM Scenarios Considering Overall and Single Costs 6

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Aircraft need different times to cross sector
  • Multi-Phase-Approach: Sequence needs to be known a priori
  • Solution: Fading of aircraft dynamics

𝐲𝑗 = 𝑦𝑗 𝑧𝑗 = 𝑊

𝐿,𝑗 ⋅ cos 𝜓𝐿,𝑗

𝑊

𝐿,𝑗 ⋅ sin 𝜓𝐿,𝑗

∙ δx ∙ δy With δx = ± 1 2 tanh 𝑏 ∙ 𝑦 − 𝑦𝑔 + 1 2 δ𝑧 = ± 1 2 tanh 𝑏 ∙ 𝑧 − 𝑧𝑔 + 1 2 Steepness parameter 𝑏 = 1,0799 ∙ 10−2 ⋅ 1/𝑛

Different flight times in one phase

  • 2
  • 1.5
  • 1
  • 0.5

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a = 1e-1 a = 1e-2 a = 1e-3

𝑦 [NM] 𝜀𝑦

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SLIDE 7

Optimization of ATM Scenarios Considering Overall and Single Costs 7

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Flight time is used as approximation for cost
  • Fading of model:

𝐘i = 𝑦𝑗 𝑧𝑗 𝑢𝑗 = 𝑊

𝐿,𝑗 ⋅ cos 𝜓𝐿,𝑗

𝑊

𝐿,𝑗 ⋅ sin 𝜓𝐿,𝑗

1 ∙ δx ∙ δy

  • For comparison: Relative cost increase

𝑑𝑗 = 𝑢𝑗,𝑔𝑗𝑜𝑏𝑚 − 𝑢𝑗,𝑔𝑗𝑜𝑏𝑚,𝑛𝑗𝑜 𝑢𝑗,𝑔𝑗𝑜𝑏𝑚,𝑛𝑗𝑜 ∙ 100%

Cost functions and fairness I

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SLIDE 8

Optimization of ATM Scenarios Considering Overall and Single Costs 8

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Fairness means neat distribution of relative cost increases
  • Leads to multi criteria optimization problem
  • Different formulations:
  • All cost increases should be minimized

𝐊 = 𝑑1 ⋮ 𝑑𝑂

  • The statistical values of the cost increases should be minimized

𝐊 = 𝑑𝑡𝑣𝑛 𝑑𝑤𝑏𝑠 = 𝑑𝑗 1 𝑂 𝑑𝑗 − 𝑑𝑛 2

Cost functions and fairness II

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SLIDE 9

Optimization of ATM Scenarios Considering Overall and Single Costs 9

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Scalarization techniques

  • Weighted sum (fairness cannot be considered)

𝐾𝑇𝑣𝑛 =

𝑗=1 𝑂

𝑥𝑗 ⋅ 𝑑𝑗

  • p-Norm (all 𝑑𝑗 ≥ 0)

𝐾𝑞 =

𝑗=1 𝑂

𝑑𝑗

𝑞 1 𝑞

  • Distance to target cost (find target by min of overall cost, 𝑙𝑈 tuning parameter)

𝐾𝑈 =

𝑗=1 𝑂

𝑑𝑗 − 𝑑𝑈 2 𝑑𝑈 = min 𝑑𝑗 ⋅ 𝑙𝑈

Multi criteria optimization methods I

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SLIDE 10

Optimization of ATM Scenarios Considering Overall and Single Costs 10

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Min-Max optimization (two step method, 𝑙𝑑 allows tuning)

1. min 𝐾𝑛𝑏𝑦 = min 𝑑 ∞ = min lim

𝑞→∞ 𝑗=1 𝑂

𝑑𝑗

𝑞 1 𝑞

= min max

𝑗

𝑑𝑗 2. min 𝐾𝑇𝑣𝑛 = min

𝑗=1 𝑂

𝑑𝑗 , 𝑡. 𝑢ℎ. 𝑑𝑗 ≤ 𝑙𝑑 ⋅ 𝑑max , 𝑙𝑑 ≥ 1, 𝑗 = 1, … , 𝑂

  • Mean-Variance minimization (two step method, 𝑙𝑑 allows tuning)

1. min 𝐾𝑡𝑣𝑛 = min

𝑗=1 𝑂

𝑑𝑗 2. min 𝐾𝑤𝑏𝑠 = min 1 𝑂

𝑗=1 𝑂

𝑑𝑗 − 𝑑 2 𝑡. 𝑢ℎ. 𝑑𝑡𝑣𝑛 ≤ 𝑑𝑡𝑣𝑛,𝑛𝑗𝑜 ⋅ 𝑙𝑑 𝑙𝑑 ≥ 1

Multi criteria optimization methods II

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SLIDE 11

Optimization of ATM Scenarios Considering Overall and Single Costs 11

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Example 1 Example 2

Example Scenarios

  • 60
  • 40
  • 20

20 40 60

  • 60
  • 40
  • 20

20 40 60 x [NM] y [NM]

  • 60
  • 40
  • 20

20 40 60

  • 20
  • 10

10 20 x [NM] y [NM]

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SLIDE 12

Optimization of ATM Scenarios Considering Overall and Single Costs 12

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Results for Example 1

Min-Sum Min-Square Min-Max Min-Mean and Var 0.1 0.2 0.3 0.4 0.5 [%] Mean Standard deviation

c c c

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SLIDE 13

Optimization of ATM Scenarios Considering Overall and Single Costs 13

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Results for Example 1

Min-Sum Min-Square Min-Max Min-Mean and Var 0.1 0.2 0.3 0.4 0.5 [%] Mean Standard deviation

1 1.002 1.006 1.01 1.03 1.05 1.07 1.09 1.15 1.2 0.1 0.2 0.3 0.4 0.5 0.6

[%] kc

Mean Standard deviation

Mean-limited variance minimization min 𝐾𝑤𝑏𝑠 = min 1 𝑂

𝑗=1 𝑂

𝑑𝑗 − 𝑑 2 𝑑𝑡𝑣𝑛 ≤ 𝑑𝑡𝑣𝑛,𝑛𝑗𝑜 ⋅ 𝑙𝑑 c c c

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SLIDE 14

Optimization of ATM Scenarios Considering Overall and Single Costs 14

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

  • Mean / Variance Pareto Front

Results for Example 1

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.24 0.26 0.28 0.3 0.32 0.34

Mean [%] Standard deviation

Sum Square Target Min-Max Limit Var

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SLIDE 15

Optimization of ATM Scenarios Considering Overall and Single Costs 15

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Results for Example 2

Min-Sum Min-Square Min-Max Min-Mean and Var 1 2 3 4 [%] Mean Standard deviation

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SLIDE 16

Optimization of ATM Scenarios Considering Overall and Single Costs 16

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Results for Example 2

200 400 600 800 1000 300 320 340 360 time [s] velocity [kts] 200 400 600 800 1000

  • 50

50 time [s] course angle [deg]

20

Min-Max optimization

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SLIDE 17

Optimization of ATM Scenarios Considering Overall and Single Costs 17

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Results for Example 2

2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Mean [%] Standard deviation

Sum Square Target Min-Max Limit Var

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SLIDE 18

Optimization of ATM Scenarios Considering Overall and Single Costs 18

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Conclusions

  • Modelling of ATM scenarios as multi criteria optimal control problems
  • Optimal control problem solved using direct collocation
  • Use of fading dynamics to model different sector crossing times
  • Implementation of different methods to solve multi criteria optimization problems
  • Results show:
  • Methods are comparable, but depend on scenario
  • “Overall cost” for the increase of fairness strongly depends on scenario
  • Approximation of Pareto front combining the methods

Outlook

  • Multi criteria optimization methods can be improved (e.g. Tchebychev scalarization)
  • More elaborate model / wind may be added
  • Controller workload, number of maneuvers etc. may be considered

Conclusions and outlook

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SLIDE 19

Optimization of ATM Scenarios Considering Overall and Single Costs 19

Intitute of Flight System Dynamics

Matthias Bittner 28/05/2014 – ICRAT, Istanbul, Turkey

Matthias Bittner Institute of Flight System Dynamics Technische Universität München Boltzmannstraße 15 D-85748 Garching bei München Deutschland / Germany Phone: +49 89 289-16080 Email: m.bittner@tum.de

Thank you very much!