IE-OR Seminar April 18, 2006

1. IE-OR SeminarApril 18, 2006 Evolutionary Algorithms in Addressing Contamination Threat Management in Civil Infrastructures Ranji S. Ranjithan Department of Civil Engineering, NCSU

2. Many security threat problems in civil infrastructure systems

3. Contamination threat problemin water distribution networks

4. Water distribution networks… • Solve for network hydraulics (i.e., pressure, flow) • Depends on • Water demand/usage • Properties of network components • Uncertainty/variability • Dynamic system • Solve for contamination transport • Depends on existing hydraulic conditions • Spatial/temporal variation • time series of contamination concentration

5. Water distribution networks…

6. Water distribution networks…

7. Water distribution networks… • Explain the contamination issues • Show animation

8. Water distribution networks…

9. Why is this an important problem? • Potentially lethal and public health hazard • Cause short term chaos and long term issues • Diversionary action to cause service outage • Reduction in fire fighting capacity • Distract public & system managers

10. What needs to be done? • Determine • Location of the contaminant source(s) • Contamination release history • Identify threat management options • Sections of the network to be shut down • Flow controls to • Limit spread of contamination • Flush contamination

11. What needs to be done? • Determine • Location of the contaminant source(s) • Contamination release history • Identify threat management options • Sections of the network to be shut down • Flow controls to • Limit spread of contamination • Flush contamination

12. Example math formulation • Find: L(x,y), {Mt}, T0 • Minimize Prediction Error • ∑i,t || Cit(obs) – Cit(L(x,y), {Mt}, T0) || • where • L(x,y) – contamination source location (x,y) • Mt – contaminant mass loading at time t • T0 – contamination start time • Cit(obs) – observed concentration • Cit(L(x,y), {Mt}, T0) – concentration from system simulation model • i – observation (sensor) location • t – time of observation

13. Example math formulation • Find:L(x,y), {Mt}, T0 • Minimize Prediction Error • ∑i,t || Cit(obs) – Cit(L(x,y), {Mt}, T0) || • where • L(x,y) – contamination source location (x,y) • Mt – contaminant mass loading at time t • T0 – contamination start time • Cit(obs) – observed concentration • Cit(L(x,y), {Mt}, T0) – concentration from system simulation model • i – observation (sensor) location • t – time of observation • unsteady • nonlinear • uncertainty/error

14. Example math formulation • estimate solution state with currently available data • identify possible solutions that fit the data • assess confidence in current estimate of solution(s) • Find:L(x,y), {Mt}, T0 • Minimize Prediction Error • ∑i,t || Cit(obs) – Cit(L(x,y), {Mt}, T0) || • where • L(x,y) – contamination source location (x,y) • Mt – contaminant mass loading at time t • T0 – contamination start time • Cit(obs) – observed concentration • Cit(L(x,y), {Mt}, T0) – concentration from system simulation model • i – observation (sensor) location • t – time of observation

15. Interesting challenges • Non-unique solutions • Due to limited observations (in space & time) Resolve non-uniqueness • Incrementally adaptive search • Due to dynamically updated information stream Optimization under dynamic environments • Search under noisy conditions • Due to data errors & model uncertainty Optimization under uncertain environments

16. Interesting challenges • Non-unique solutions • Due to limited observations (in space & time) Resolve non-uniqueness • Incrementally adaptive search • Due to dynamically updated information stream Optimization under dynamic environments • Search under noisy conditions • Due to data errors & model uncertainty Optimization under uncertain environments

17. Evolutionary algorithm-based solution approach • Evolutionary algorithms (EAs) for numeric search • Genetic algorithms, evolution strategies • Key characteristics • Population-based probabilistic search • Directed “random” search • Conditional sampling of decision space • Updated statistics/likelihood values • Based on quality of prior solutions (samples)

18. Resolving non-uniqueness • Underlying premise • In addition to the “optimal” solution, identify other “good” solutions that fit the observations • Are there different solutions with similar performance in objective space? Search for alternative solutions [work conducted by Dr. Emily Zechman]

19. f(x) x Resolving non-uniqueness… • Search for alternative solutions

20. f(x) x Resolving non-uniqueness… • Search for different solutions that are far apart in decision space

21. Resolving non-uniqueness… • Effects of uncertainty f(x) x

22. Resolving non-uniqueness… • Search for solutions that are far apart in decision space and are within an objective threshold of best solution f(x) x

23. Resolving non-uniqueness…EAs for Generating Alternatives (EAGA) Create n sub populations Sub Pop 1 Sub Pop 2 . . . Evaluate obj function values Evaluate obj function values Best solution (X*, Z*) Feasible/Infeasible? Evaluate pop centroid (C1) in decision space Evaluate distance in decision space to other populations Selection (obj fn values) & EA operators Selection (feasibility, dist) & EA operators . . . Y N Y N STOP? STOP? Best Solutions

24. EAGA…Illustration using a test function • y = [(1 - 10x)*sin(11*x)]2 / [2.83*(10x)1.46]

25. EAGA…Illustration using a test function • y = [(1 - 10x)*sin(11*x)]2 / [2.83*(10x)1.46] • Generate 3 different solutions • Optimal and two alternatives • Within a 75% threshold of the optimal solution • Search using Evolution Strategies

26. EAGA…Illustration using a test function • y = [(1 - 10x)*sin(11*x)]2 / [2.83*(10x)1.46]

27. EAGA…Illustration using a test function

28. c t Contaminant source identification Groundwater contamination problem 1

29. Resolving non-uniqueness

30. Resolving non-uniqueness…

31. Resolving non-uniqueness…Using EAGA 1 • Objective function: • minimize prediction error • EAGA settings: • four different solutions • evolution strategies •  = 200, µ = 100 • 40 generations • subpopulation size 100 • 30 random trials • Decision Variables: • center of source (x, y) • size in x direction • size in y direction • concentration

32. Resolving non-uniqueness, using EAGA… Observations from Well 1 only

33. Resolving non-uniqueness, using EAGA… Observations from Well 1 only… Predictions At Well 1

34. Resolving non-uniqueness, using EAGA… Observations from Well 1 only… Predictions At Well 2

35. Resolving non-uniqueness, using EAGA… Observations from Wells 1 & 2

36. Resolving non-uniqueness, using EAGA… Observations from Wells 1 & 2… Predictions At Well 1

37. Resolving non-uniqueness, using EAGA… Observations from Wells 1 & 2… Predictions At Well 2

38. Interesting challenges • Non-unique solutions • Due to limited observations (in space & time) Resolve non-uniqueness • Incrementally adaptive search • Due to dynamically updated information stream Optimization under dynamic environments • Search under noisy conditions • Due to data errors & model uncertainty Optimization under uncertain environments

39. Dynamic optimization • Minimize Prediction Error • ∑i,t || Cit(obs) – Cit(L(x,y), {Mt}, T0) || • Cit(obs) – streaming data • Objective function is dynamically updated • Dynamically update estimate of source characteristics

40. Dynamic optimization… • Underlying premise • Predict solutions using available information at any time step • Search for a diverse set of solutions (EAGA) • Current solutions are good starting points for search in the next time step [work conducted by Ms. Li Liu]

41. Dynamic optimization… t = 1 f(x) x

42. Dynamic optimization… t = 2 f(x) x

43. Dynamic optimization… t = 3 f(x) x

44. Dynamic optimization…Adaptive Dynamic OPt Technique (ADOPT) • Set time step t=0 • Initialize sub-populations with random solutions • Construct obj function for time step t+1 • Apply EAGA to all sub-populations • Merge solutions to identify unique set of solutions • If t < Tmax, go to Step 3 • Record solution and stop

45. ADOPT…Illustration using a test function • Test function • where • B(x) is a time-invariant “basis” landscape • P is the function defining the shape of peak i • each of peak has its own time-varying parameters • h (height) • w (width) • p (shift) • 35 time steps

46. ADOPT…Illustration using a test function 2-D case

47. ADOPT…Results for the test function • 5-D case; avg error & std over all time steps, & 30 random trials

48. ADOPT…Illustration using a test function • 5-D case; avg error & std dev over all time steps

49. Contaminant source identification 1 t c t

50. ADOPT…Contaminant source identification • Minimize Prediction Error • ∑i,t || Cit(obs) – Cit(L(x,y), {Mt}) || • Cit(obs) – streaming data • Objective function is dynamically updated • Is available information sufficient to be confident about current solution?