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AIAA GNC Conf.

AIAA GNC Conf. Airspace Throughput Analysis Considering Stochastic Weather. Jimmy Krozel, Ph.D., Joseph S.B. Mitchell, Ph.D., and Valentin Polishchuk Aug. 24, 2006. Motivation. Estimating the capacity of an airspace: Sector or Center Flow Evaluation Area (FEA)

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AIAA GNC Conf.

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  1. AIAA GNC Conf. Airspace Throughput Analysis Considering Stochastic Weather Jimmy Krozel, Ph.D., Joseph S.B. Mitchell, Ph.D., and Valentin Polishchuk Aug. 24, 2006

  2. Motivation • Estimating the capacity of an airspace: • Sector or Center • Flow Evaluation Area (FEA) • Flow Constrained Area (FCA) • is a fundamental research question that we can study either theoretically or empirically • This problem is at the root of Traffic Flow Management (TFM) • How do you know that you have a TFM problem: • Demand > Airspace Capacity • unless you have a good way of estimating the airspace capacity? AIAA GNC Conference, Keystone, CO

  3. Weather Avoidance Algorithms for En Route Aircraft Sector 1 Flow AIAA GNC Conference, Keystone, CO

  4. Weather Avoidance Algorithms for En Route Aircraft Sector 2 Flows AIAA GNC Conference, Keystone, CO

  5. Weather Avoidance Algorithms for En Route Aircraft Sector 3 Flows AIAA GNC Conference, Keystone, CO

  6. Weather Avoidance Algorithms for En Route Aircraft Sector Too Close! Violation of Sector Boundary Constraint! 4 Flows AIAA GNC Conference, Keystone, CO

  7. FCA Capacity Estimate for Airspace Flow Programs (AFPs) FCA Flights may be routed through the FCA using weather-avoidance routing. FCA How many air lanes can I plan across the FCA? AIAA GNC Conference, Keystone, CO

  8. Weather Forecasting • We consider: • Deterministic Forecasting: Good for short look-ahead times, e.g., 30 min – 1 hr look ahead (e.g., based on extrapolation of weather observations data) • Probabilistic Forecasting: Good for mid to long term look-ahead times, e.g.,1-2 hours, based on extrapolation of weather sensor observations, explicit forecast of convection in a Numerical Weather Prediction (NWP) model, and heuristics • Scenario-based Forecasting: Good for 2+ hours look-ahead times; a set of N weather forecast Scenarios are given to span the forecast possibilities based on an ensemble of NWP models and heuristics AIAA GNC Conference, Keystone, CO

  9. Network Capacity Problem • What can we learn from a simplified network flow problem? AIAA GNC Conference, Keystone, CO

  10. Network Capacity Problem • What can we learn from a simplified network flow problem? • The solution to our problem resides in Maxflow / Mincut Theory AIAA GNC Conference, Keystone, CO

  11. Extend the Problem to a Continuous Version • Change problem from 1D  2D • A Mincut / Maxflow Theory for this 2D Continuous Problem exists too; (Mitchell, 1990) gives a proof and an algorithm AIAA GNC Conference, Keystone, CO

  12. ModelingIssues • In TFM, we are not dealing with a continuous fluid flow • We identify a maximum integer number of air lanes that can pass through the airspace • Solution is “Directional” – e.g., East-West vs North-South • Theory applies to any polygon shape (sectors, centers, FEAs, or FCAs) at a given Flight Level • Model static weather (no motion) first • Later, introduce moving weather that is growing or decaying, and consider how to identify (define) routes that allow traffic to flow through the airspace over a given period of time AIAA GNC Conference, Keystone, CO

  13. Map the Theory to Aviation • How can we use this theoretical result? AIAA GNC Conference, Keystone, CO

  14. Algorithm: Mincut (Deterministic) 1. Airspace with Hazardous Weather Constraints AIAA GNC Conference, Keystone, CO

  15. Algorithm: Mincut (Deterministic) 2. Define Critical Graph – connect closest points (B, T, a  g) AIAA GNC Conference, Keystone, CO

  16. Algorithm: Mincut (Deterministic) 2. Define Critical Graph – connect closest points Assign cost using floor function | x | AIAA GNC Conference, Keystone, CO

  17. Algorithm: Mincut (Deterministic) 3. Search for Shortest Path Tree within Critical Graph AIAA GNC Conference, Keystone, CO

  18. Algorithm: Mincut (Deterministic) 3. Search for Shortest Path Tree AIAA GNC Conference, Keystone, CO

  19. Algorithm: Mincut (Deterministic) 4. Shortest B-T Path in Shortest Path Treedefines the mincut AIAA GNC Conference, Keystone, CO

  20. Algorithm: Mincut (Deterministic) 5. Find maximum number of air lanes through the mincut AIAA GNC Conference, Keystone, CO

  21. Algorithm: Mincut (Deterministic) 6. Quantify Capacity (Maximum Throughput) as: a. Continuous flow (length of mincut) b. Maximum integer Number of Air Lanes c. (Maximum integer Number of Air Lanes) x (Number of aircraft “head to tail” across the sector (length of each air lane)) f (flow speed; separation or Miles-in-Trail (MIT) requirements) Air Lane 1 Air Lane 2 AIAA GNC Conference, Keystone, CO

  22. Algorithm: Mincut (Probability Distribution) • Airspaces with Hazardous Weather Constraints are given for N forecasts (scenarios) and probability pk that the kthforecast is going to be the final outcome • For each Scenario k (k=1,2,…,N): • Create critical graph • Search for mincut (use Floor Function for air lanes) • Record mincut length xk • Output (Capacity = X): • - E(X) = Skxkpk Expected Value • - Var(X) = Skxk2pk - [E(X)]2 Variance • - FX(x) = Sk:xk<xpk Cumulative Distribution Function AIAA GNC Conference, Keystone, CO

  23. Example: Mincut (Probability Distribution) NWP Model 1 NWP Model 2 NWP Model 3 4 air lanes 4 air lanes 7 air lanes Output (Capacity = X): - E(X) = Skxkpk = 70%(4) + 10%(4) + 20%(7) = 4.6 air lanes - Var(X) = Skxk2pk - [E(X)]2 = 1.44 - FX(x) = Sk:xk<xpk  P(X=4) = 80% P(X=7) = 20% AIAA GNC Conference, Keystone, CO

  24. Algorithm Examples • We developed a Matlab algorithm for Continuous 2D Flow Problem and for the integer Number of Air Lanes • Identifies the real-time flow “Bottleneck” • For growing or moving weather cases, the mincut moves around over time AIAA GNC Conference, Keystone, CO

  25. Popcorn Convection vs Squall Line Organization flow flow 44800 Synthetic Weather Data Experiments AIAA GNC Conference, Keystone, CO

  26. Popcorn Convection vs Squall Line Organization flow flow L In the limit… as d0 The Squall Line Model Approaches the 1-D Case Capacity = L – Wx_Coverage d AIAA GNC Conference, Keystone, CO

  27. Popcorn Convection vs Squall Line Organization RNP - 3 RNP - 5 RNP-10 AIAA GNC Conference, Keystone, CO

  28. Conclusions: Synthetic Weather Data Experiments • Trends: Capacity  0 as Weather Severity  100% • Capacity  0 prior to 100% Weather Severity • Capacity  0 faster with larger (less precise) RNP • Capacity  0 faster for Squall Line Convection Model • Results (quadratic vs linear decline) depend on organization of convection: • 1D: Capacity = 1 – Wx_Coverage (%) • 2D: Capacity ~ 1 - Wx_Coverage (%) Popcorn Convection • 2D: Capacity ~ 1 - Wx_Coverage (%) Squall Line Convection • Solution is “Directional” – e.g., East-West vs North-South AIAA GNC Conference, Keystone, CO

  29. Real World Weather Scenario RNP-3 RNP-5 RNP-10 Linear  looks like Squall Line Convection AIAA GNC Conference, Keystone, CO

  30. Probability x Air Lanes will get through? • How many air lanes can we get through the FCA when we have a probabilistic weather forecast model? • Probability that x air lanes get through decreases with increasing RNP requirements 1 1 2 2 3 4 3 5 6 4 5 AIAA GNC Conference, Keystone, CO

  31. Design of Air Lanes Valid over a 30 min Time Period • A FCA needs fixed (spatial) routes for a given period of time (e.g., 30 minutes) FCA AIAA GNC Conference, Keystone, CO

  32. Design of Air Lanes Valid over a 30 min Time Period • A FCA needs fixed (spatial) routes for a given period of time (e.g., 30 minutes) FCA AIAA GNC Conference, Keystone, CO

  33. Design of Air Lanes Valid over a 30 min Time Period • A FCA needs fixed (spatial) routes for a given period of time (e.g., 30 minutes) FCA AIAA GNC Conference, Keystone, CO

  34. Design of Air Lanes Valid over a 30 min Time Period • A FCA needs fixed (spatial) routes for a given period of time (e.g., 30 minutes) FCA AIAA GNC Conference, Keystone, CO

  35. Design of Air Lanes Valid over a 30 min Time Period • A FCA needs fixed (spatial) routes for a given period of time (e.g., 30 minutes) FCA AIAA GNC Conference, Keystone, CO

  36. Moving Weather • For each weather forecast (e.g., every 5 minutes); analyze the estimated capacity • We want to define one set of air lanes that remain fixed over some planning horizon (e.g., 30 minutes) • Plan routes based on the local minimum over the planning horizon • Switch “direction” of flow if desirable to meet needs of users 7 6 5 Capacity Number of Air Lanes 5 Air Lanes Open 4 3 2 2 Air Lanes Open 1 Time (Min) 0 30 60 Hold Air Lanes Fixed For 30 Minutes Hold Air Lanes Fixed For 30 Minutes AIAA GNC Conference, Keystone, CO

  37. Conclusions • We developed a theoretical approach and algorithm for estimating airspace capacity (deterministic and stochastic) • The solution is based on the Maxflow/Mincut Theory • We determined the maximum integer number of air lanes that can pass through a 2D airspace with severe weather hazards • The solution is “directional” (e.g., East-West vs North-South) • The solution identifies the theoretical “bottleneck” of the weather constraints defined by a Critical Graph/Shortest Path Tree • Capacity is dependent on the weather organization (Popcorn convection vs Squall Line models were compared) • Capacity for Popcorn convection quadratically decreases to 0 with increasing Severe Weather Coverage and Squall Lines linearly decrease to 0 with increasing Severe Weather Coverage AIAA GNC Conference, Keystone, CO

  38. Future Research • Further Research is planned for: • Estimated Capacity over a period of time with Moving Weather (Deterministic & Stochastic Cases) • Practical MIT Spacing between aircraft and RNP Requirements between air lanes for use in today’s AFPs • Well stated theoretical problem statements for defining the air lanes based on Computational Geometry • Mixed RNP requirements matching User Demand Capabilities (e.g., 20% RNP-10; 30% RNP-5; 50% RNP-3) • Proof of Concept for real-world data Scenario-based weather forecasting with multiple NWP models • Transfer technology to Metron Aviation Applications (e.g., AFP benefits from an automated estimate for the capacity of an FCA) AIAA GNC Conference, Keystone, CO

  39. Future Research • Connect with Actual 2-hr Probabilistic Forecast Shade of red indicates the probability that the convection will be NWS Level 3 or above AIAA GNC Conference, Keystone, CO

  40. Point of Contact krozel@metronaviation.com AIAA GNC Conference, Keystone, CO

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