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Multi-Objective Dynamic Optimization using Evolutionary Algorithms

Multi-Objective Dynamic Optimization using Evolutionary Algorithms. by Udaya Bhaskara Rao N. under the guidance of Dr. Kalyanmoy Deb Professor Department of Mechanical Engineering. Birds view. Introduction to DMO. Test problems in DMO. NSGA-II application in DMO.

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Multi-Objective Dynamic Optimization using Evolutionary Algorithms

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  1. Multi-Objective Dynamic Optimization usingEvolutionary Algorithms by Udaya Bhaskara Rao N. under the guidance of Dr. Kalyanmoy Deb Professor Department of Mechanical Engineering Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  2. Birds view • Introduction to DMO. • Test problems in DMO. • NSGA-II application in DMO. • Introduction to hydrothermal scheduling problem. • NSGA-II application on hydrothermal scheduling problem. • Hydrothermal scheduling problem formulation as DMO. • Modifications in the proposed algorithm. • Results and discussion. • Conclusions. • Future scope of work. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  3. Introduction to DMO • Dynamic optimization is optimization in dynamic environment. • i.e. either objective function or constraints are time dependent. • The dynamic multi-objective optimization (DMO) is multi-objective optimization in dynamic environment. • Classification in DMOs : Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  4. Introduction to DMO • It is better to go for DMO, whenever the problem is time dependent. Advantages in using DMO: • By relating time with generation number, number of variables reduce i.e. the dimension of problem reduces. • Whenever problem changes, the new problem adopts the old solution, which helps in faster convergence. • Results for all the problems can be found in one run. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  5. Test problems in DMO • The following test problems are formulated by Farina et. al. (2004). FDA 1 : Constant convex Pareto-optimal front in objective space and linear change in solution space. FDA 2 : Pareto-optimal front changes from convex to non convex and no change in solution space. FDA 3 : Change in Pareto-optimal front but all convex and linear change in solution space. FDA 4 : Constant non convex Pareto-optimal front and linear change in solution space which is three Dimensional space. FDA 5 : Change in Pareto-optimal front but all non convex and linear change in solution space which is three dimensional space. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  6. Test problems in DMO FDA 1: Type I Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  7. Test problems in DMO FDA 2: Type III Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  8. Test problems in DMO FDA 3: Type II Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  9. Test problems in DMO FDA 4: Type I Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  10. Test problems in DMO FDA 5: Type II Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  11. NSGA-II application in DMO • Present algorithm is developed based on NSGA-II. • NSGA-ll algorithm can not be applied straightaway on DMO problems. • Elitism, restricts the upward movement of Pareto-optimal front in NSGA-II and hence removed. • The term time is correlated with generation number. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  12. Modified NSGA-II algorithm-I Elitism removed Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  13. Modified NSGA-II algorithm-II Elitism introduced interactively Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  14. FDA 2 simulation Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  15. FDA 3 simulation Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  16. FDA 5 simulation Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  17. Introduction to hydrothermal scheduling problem • In hydrothermal systems both hydroelectric and thermal generating units are to be utilized together to meet the total power demand. • The hydrothermal problem here consists of Ns number of thermal and Nh number of hydroelectric generating units sharing the total power demand. • Minimizing both fuel cost and emission of nitrogen oxides from the thermal generating units. • The static problem formulation is taken from the work done by M. Basu (2005). (Weighted sum approach using simulated annealing) Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  18. Introduction to hydrothermal scheduling problem • In this present work the problem is formulated for two hydraulic units and four thermal units. • Problem is defined for four timeslots of each 12 hours. • So the total number of variables are 24. • The demand values for these four time slots are as follows: Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  19. Objective functions Economy: Emission: Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  20. Constraints Power balance constraints: Water availability constraints: Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  21. Constraint handling Step 1 : The procedure is to be started with the two water available constraints, as they are independent of variables related to thermal units. Step 2 : Constraint equation can be written as, Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  22. Constraint handling Step 3 :Start with h = 1, m = 1 • For finding Phm value from constraint equation, first rewrite the equation in terms of Phm by taking all four Ph values of the present hydro unit from GA solution and finding out the ratios with respect to Phm. • The obtained quadratic equation in terms of Phm is solved algebraically to get Phm value. Subsequently the positive value is chosen, so that the lower limit is satisfied automatically. If it is also satisfied the upper limit go to Step 5, else go to Step 4. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  23. Constraint handling Step 4 : m = m + 1, and if m ≤ 4 repeat Step 3 else go to Step 6. Step 5 : Change all four Ph value by using previously calculated ratios and h = h + 1, if h ≤ 2 repeat Step 3, else Exit. Step 6 : The constraint is not satisfied, so for the present variable values, the fitness function is to be penalized with the constraint violation. • If both water availability constraints are satisfied through the above process, similar analysis is to be done on power balance constraints Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  24. NSGA-II application on hydrothermal scheduling problem Input parameters : Population size = 240 Number of generations = 1000 Crossover probability = 0.9 Mutation probability = 0.04 Distribution index for crossover = 20 Distribution index for mutation = 50 Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  25. Analysis of obtained results Ph1 vs. F1 Ph2 vs. F1 Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  26. Analysis of obtained results Ps1 vs. F1 Ps2 vs. F1 Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  27. Analysis of obtained results Ps4 vs. F1 Ps3 vs. F1 Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  28. Hydrothermal scheduling problem reformulation as DMO • As the problem parameters change with time, this problem comes under DMO. • Few modifications required in problem formulation, they are as follows: • Term time should be removed as a variable. • All time variable parameters should be directly related with generation number. • The dimension of problem has reduced from 24 to 6. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  29. Modifications in the proposed algorithm • The proposed algorithm can handle the changes occurring after every 12 hours. • The algorithm is further modified to handle frequent changes. • The main modifications are as follows: • Introducing new solutions at change by generating random solutions. • Introducing new solutions at change by mutating old solutions . Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  30. First modification Introducing new random solutions at change Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  31. Second modification Introducing new mutated solutions at change Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  32. Interpolation Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  33. Results and discussion Comparison among three modified algorithms : 4 timeslots 8 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  34. Results and discussion Comparison among three modified algorithms : 16 timeslots 48 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  35. Results and discussion Comparison among three modified algorithms : 192 timeslots 96 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  36. Results and discussion Percentage of random new solutions verses performance index : 4 timeslots 8 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  37. Results and discussion Percentage of random new solutions verses performance index : 16 timeslots 48 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  38. Results and discussion Percentage of random new solutions verses performance index : 192 timeslots 96 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  39. Results and discussion Percentage of mutated new solutions verses performance index : 4 timeslots 8 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  40. Results and discussion Percentage of mutated new solutions verses performance index : 16 timeslots 48 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  41. Results and discussion Percentage of mutated new solutions verses performance index : 192 timeslots 96 timeslots Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  42. Conclusions • Static to dynamic conversion of the problem, increases its convergence rate and simultaneously there also exists a possibility for dimensionality reduction. • Modified NSGA-II algorithms, has yielded better results for all test problems. • The reformulated hydrothermal scheduling problem has been solved efficiently. • The static analysis of hydrothermal scheduling problem with modified NSGA-II produced better results compared to previous works. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  43. Conclusions • Best results are produced when the hydrothermal scheduling problem is formulated into DMO problem, with considerable reduction in computational time over static problems. • The final proposed algorithm has increased the possibility in achieving Pareto-optimal front within short time period and performs best up to one hour time slot. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  44. Future scope of work • Generalization of the proposed algorithm, would make it user friendly. • The hydrothermal scheduling problem defined for individual hydro units, can be extended for cascaded hydro units. • The present algorithm is used to search for Pareto-optimal front, this algorithm can be slightly modified to get reliable and robust Pareto-optimal solutions also. Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  45. Thank you

  46. Back up slides Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  47. Parameter Analysis Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  48. Nomenclature Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  49. Input parameters Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

  50. Input parameters Kanpur Genetic Algorithms Laboratory IIT Kanpur 25, July 2006 (11:00 AM)

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