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
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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
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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%
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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
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