Dynamic Routing of Aircraft under weather uncertainty Arnab Nilim Laurent El Ghaoui Mark Hansen University of California, Berkeley Vu Duong Euro Control Experimental Centre 0 Introduction Delays in commercial air travel Weather induced enroute delays Shortcomings of existing programs 1 Delay Distribution Delay Distribution, June 2001 Delay Distribution , June 2000 3% 5% 0% 1% 9% 7% 10% 6% 80% 1 2 3 4 79% 5 1:Weather 3: Equipment 2: Volume 4: Runway 1 2 3 4 5 5: Other 2 Previous work Deterministic traffic flow management (Bertsimas (98), Goodhart (99), Burlingame(94) etc) Automation tool: explicitly dealing with the dynamics and the stochasticity of the weather Optimization under uncertainty 3 New architecture (???) Airspace vs Trajectory based architecture 4 Research Agenda Dynamic Routing strategy of a single aircraft. Robust solution w.r.t. estimation of storm probabilities error. System level solution 5 Dynamic Routing of Aircraft under Uncertainty 6 Simple optimization minφ {l1 + p (m1 + m2 ), l1 + (1 − p )l2 } 7 Uncertainty 8 Stochastic dynamic Programming/Markov Decision Processes xk +1 = f k ( xk , µ k , wk ), ∀k = 0,1,...., n − 1 State: Control: → µ = [ µ 0 , µ1 ,....., µ n −1 ] Expected cost function: n −1 J µ ( x0 ) = E( wk ∀k =0,1,...n −1) ( g n ( xn ) + ∑ g k ( xk , µ k ( xk ), wk )) k =0 Markovian uncertainty: N v(i, n) = min (1≤ k ≤ Ai ) [q + ∑ pijk v( j , n − 1)] k i j =1 9 Weather Model 10 Gridding 11 Algorithm Step 1:Calculate the total number of stages nmax T T − mod[ ] 15 + 1 = 15 Step 2: Discretize (airspace) T: Worst case time Step 3: Pruning 12 Algorithm Step 4: Next points 13 Algorithm Step 5: Assigning appropriate cost c(i, x, y, z j , w j ) : Cost to go from ( x, y) to ( z j , w j ) in state i Step 6:Defining value function v(i, x, y, n) : Expected minimum distance to go if the aircraft is at the point (x,y), with the state i and it has n stages to go to reach the destination point Step 7: Assigning boundary conditions 14 Algorithm Step 8: v (i , x , y , n ) = min ( z1 , w1 ),....( z a , wa ) V Where, 2 c(i, x, y, z1 , w1 ) + ∑ pij v( j , z1 , w1 , n − 1) j =1 M 2 pij v( j , z 2 , w2 , n − 1) V = c(i, x, y, z 2 , w2 ) + ∑ j =1 .......... .... M 2 c(i, x, y, z a , wa ) + ∑ pij v( j , z a , wa , n − 1) j =1 M 15 Simulation 16 Improvements I.M. of our model over TS1 I.M. of our model over TS2 Scenario 1 66.42% 42.76% Scenario 2 54.78% 49.81% 17 Conclusion Less circuitous route Less overloading in the neighboring sectors Complexity Robust solution w.r.t. errors in estimation of the storm probabilities Routing of multiple aircraft under uncertainty 18 Acknowledgements Stuart Dreyfus, Christos Pappadimitrou, Pravin Varaiya, Jim Evans, Jim Wetherly 19