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 Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 2 Problem description Multiple aircraft crossing ATM sector Minimum individual costs Maintain separation Maximum fairness Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 3 Problem formulation Determine the optimal control histories (aircraft index 𝑗 ) 𝐯 𝑗,𝑝𝑞𝑢 𝑢 ∈ ℝ 𝑛 𝑗 , 𝑗 = 1, … , 𝑂 the corresponding optimal state histories 𝐲 𝑗,𝑝𝑞𝑢 𝑢 ∈ ℝ 𝑜 𝑗 , 𝑗 = 1, … , 𝑂 and any additional parameters 𝐪 𝑗 ∈ ℝ 𝑙 𝑗 that minimize the Bolza cost functional 𝑢 𝑔 𝐾 = Φ 𝐲 𝑗 𝑢 𝑔 , 𝑢 𝑔 + ℒ 𝐲 𝑗 𝑢 , 𝐯 𝑗 𝑢 , 𝐪 𝑗 , 𝑢 𝑒𝑢 𝑢 0 subject to the state dynamics 𝐲 𝑗 𝑢 = 𝒈 𝑗 𝐲 𝑗 𝑢 , 𝐯 𝑗 𝑢 , 𝐪 𝑗 , 𝑢 𝑗 , 𝑗 = 1, … , 𝑂 the initial and final boundary conditions 𝛀 𝟏 ∈ ℝ 𝑞 𝛀 0 𝐲 𝑗 𝑢 0 , 𝑢 0 = 𝟏, 𝛀 𝒈 ∈ ℝ 𝑟 𝛀 𝑔 𝐲 𝑗 𝑢 𝑔 , 𝑢 𝑔 = 𝟏, and the equality and inequality path constraints 𝑫 𝑓𝑟 ∈ ℝ 𝑠 , 𝑫 𝑓𝑟 𝐲 𝒋 𝑢 , 𝐯 𝑗 𝑢 , 𝐪 𝑗 , 𝑢 = 𝟏, 𝑗 = 1, … , 𝑂 𝑫 𝑗𝑜 ∈ ℝ 𝑡 , 𝑫 𝑗𝑜 𝐲 𝑗 𝑢 , 𝐯 𝑗 𝑢 , 𝐪 𝑗 , 𝑢 ≤ 𝟏, 𝑗 = 1, … , 𝑂 Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 4 Optimization Process Optimization using direct methods: Discretize then Optimize! Finite parameter Infinite optimal Discretization optimization Solution Optimization control problem of problem problem Optimization Reduce infinite Optimization Optimization Resulting values parameters: problem by parameters: algorithm for for states and Continuous discretization Values at parameter controls at functions discrete points optimization discrete points 𝐯 i 𝑢 , 𝐲 i 𝑢 , 𝐪 𝑗 , 𝑢 𝑔 𝐯 𝑗,𝑙 , 𝐲 𝑗,𝑙 , 𝐪 𝑗 , 𝑢 𝑔 e.g. Trapezoidal problems can be Collocation e.g.: Sequential interpolated to quadratic the full state and Analytical Optimization programming control histories. optimality problem can be (WORHP, conditions solved SNOPT, IPOPT, cannot be numerically. fmincon) evaluated. Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 5 Simulation model Kinematic models are sufficient, dynamics far from envelope 2D models are used State dynamics 𝑦 𝑗 = 𝑊 𝐿,𝑗 ⋅ cos 𝜓 𝐿,𝑗 𝑧 𝑗 = 𝑊 𝐿,𝑗 ⋅ sin 𝜓 𝐿,𝑗 State and control vectors 𝐲 𝑗 = 𝑦 𝑗 , 𝑧 𝑗 T 𝑈 𝐯 𝑗 = 𝑊 𝐿,𝑗 , 𝜓 𝐿,𝑗 Separation path constraints (applied pair wise) = 5𝑂𝑁 2 ≤ (𝑦 𝑗 −𝑦 𝑘 ) 2 + (𝑧 𝑗 −𝑧 𝑘 ) 2 2 𝑒 min Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 6 Different flight times in one phase Aircraft need different times to cross sector Multi-Phase-Approach: Sequence needs to be known a priori Solution: Fading of aircraft dynamics = 𝑊 𝐿,𝑗 ⋅ cos 𝜓 𝐿,𝑗 𝑦 𝑗 𝐲 𝑗 = ∙ δ x ∙ δ y 𝑧 𝑗 𝑊 𝐿,𝑗 ⋅ sin 𝜓 𝐿,𝑗 With 1 δ x = ± 1 + 1 2 tanh 𝑏 ∙ 𝑦 − 𝑦 𝑔 0.9 a = 1e-1 2 a = 1e-2 0.8 a = 1e-3 0.7 δ 𝑧 = ± 1 + 1 2 tanh 𝑏 ∙ 𝑧 − 𝑧 𝑔 0.6 2 𝜀 𝑦 0.5 0.4 0.3 Steepness parameter 0.2 𝑏 = 1,0799 ∙ 10 −2 ⋅ 1/𝑛 0.1 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 𝑦 [NM] Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 7 Cost functions and fairness I Flight time is used as approximation for cost Fading of model: 𝑦 𝑗 𝑊 𝐿,𝑗 ⋅ cos 𝜓 𝐿,𝑗 𝑧 𝑗 𝐘 i = = ∙ δ x ∙ δ y 𝑊 𝐿,𝑗 ⋅ sin 𝜓 𝐿,𝑗 𝑢 𝑗 1 For comparison: Relative cost increase 𝑑 𝑗 = 𝑢 𝑗,𝑔𝑗𝑜𝑏𝑚 − 𝑢 𝑗,𝑔𝑗𝑜𝑏𝑚,𝑛𝑗𝑜 ∙ 100% 𝑢 𝑗,𝑔𝑗𝑜𝑏𝑚,𝑛𝑗𝑜 Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 8 Cost functions and fairness II Fairness means neat distribution of relative cost increases Leads to multi criteria optimization problem Different formulations: All cost increases should be minimized 𝑑 1 ⋮ 𝐊 = 𝑑 𝑂 The statistical values of the cost increases should be minimized 𝑑 𝑗 𝐊 = 𝑑 𝑡𝑣𝑛 = 𝑑 𝑤𝑏𝑠 1 𝑂 𝑑 𝑗 − 𝑑 𝑛 2 Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 9 Multi criteria optimization methods I Scalarization techniques Weighted sum (fairness cannot be considered) 𝑂 𝐾 𝑇𝑣𝑛 = 𝑥 𝑗 ⋅ 𝑑 𝑗 𝑗=1 p-Norm (all 𝑑 𝑗 ≥ 0 ) 1 𝑞 𝑂 𝑞 𝐾 𝑞 = 𝑑 𝑗 𝑗=1 Distance to target cost (find target by min of overall cost, 𝑙 𝑈 tuning parameter) 𝑂 𝑑 𝑗 − 𝑑 𝑈 2 𝐾 𝑈 = 𝑗=1 𝑑 𝑈 = min 𝑑 𝑗 ⋅ 𝑙 𝑈 Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 10 Multi criteria optimization methods II Min-Max optimization (two step method, 𝑙 𝑑 allows tuning) 1 𝑞 𝑂 𝑞 1. min 𝐾 𝑛𝑏𝑦 = min 𝑑 ∞ = min lim 𝑑 𝑗 = min max 𝑑 𝑗 𝑞→∞ 𝑗 𝑗=1 𝑂 𝑑 𝑗 ≤ 𝑙 𝑑 ⋅ 𝑑 max , 𝑙 𝑑 ≥ 1, 𝑗 = 1, … , 𝑂 2. min 𝐾 𝑇𝑣𝑛 = min 𝑑 𝑗 , 𝑡. 𝑢ℎ. 𝑗=1 Mean-Variance minimization (two step method, 𝑙 𝑑 allows tuning) 𝑂 1. min 𝐾 𝑡𝑣𝑛 = min 𝑑 𝑗 𝑗=1 𝑂 min 𝐾 𝑤𝑏𝑠 = min 1 𝑑 2 2. 𝑂 𝑑 𝑗 − 𝑡. 𝑢ℎ. 𝑑 𝑡𝑣𝑛 ≤ 𝑑 𝑡𝑣𝑛,𝑛𝑗𝑜 ⋅ 𝑙 𝑑 𝑙 𝑑 ≥ 1 𝑗=1 Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 11 Example Scenarios Example 1 Example 2 60 40 20 20 y [NM] 10 y [NM] 0 0 -10 -20 -20 -60 -40 -20 0 20 40 60 -40 x [NM] -60 -60 -40 -20 0 20 40 60 x [NM] Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 12 Results for Example 1 0.5 0.4 0.3 [%] 0.2 c 0.1 Mean Standard deviation c 0 c Min-Sum Min-Square Min-Max Min-Mean and Var Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 13 Results for Example 1 𝑂 min 𝐾 𝑤𝑏𝑠 = min 1 0.5 𝑑 2 𝑂 𝑑 𝑗 − 𝑗=1 0.4 𝑑 𝑡𝑣𝑛 ≤ 𝑑 𝑡𝑣𝑛,𝑛𝑗𝑜 ⋅ 𝑙 𝑑 0.6 0.3 Mean-limited variance minimization [%] 0.5 0.2 c 0.4 0.1 Mean [%] 0.3 Standard deviation c 0 c Min-Sum Min-Square Min-Max Min-Mean and Var 0.2 0.1 Mean Standard deviation 0 1 1.002 1.006 1.01 1.03 1.05 1.07 1.09 1.15 1.2 k c Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 14 Results for Example 1 Mean / Variance Pareto Front 0.34 Sum Square Target 0.32 Min-Max Limit Standard deviation Var 0.3 0.28 0.26 0.24 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 Mean [%] Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 15 Results for Example 2 4 Mean Standard deviation 3 [%] 2 1 0 Min-Sum Min-Square Min-Max Min-Mean and Var Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 16 Results for Example 2 Min-Max optimization 20 360 velocity [kts] 340 320 300 0 200 400 600 800 1000 time [s] course angle [deg] 50 0 -50 0 200 400 600 800 1000 time [s] Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
Optimization of ATM Scenarios Considering Overall and Single Costs 17 Results for Example 2 1.6 Sum 1.4 Square Target 1.2 Min-Max Standard deviation Limit 1 Var 0.8 0.6 0.4 0.2 0 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 Mean [%] Matthias Bittner Intitute of 28/05/2014 – ICRAT, Istanbul, Turkey Flight System Dynamics
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