Multi-Objective Dynamic Optimization using Evolutionary Algorithms

Kanpur Genetic Algorithms LaboratoryIIT Kanpur

25, July 2006 (11:00 AM)

Multi-Objective Dynamic Multi-Objective Dynamic Optimization Optimization

usingusingEvolutionary AlgorithmsEvolutionary Algorithms

byUdaya Bhaskara Rao N.

under the guidance of

Dr. Kalyanmoy DebProfessor

Department of Mechanical Engineering


25, July 2006 (11:00 AM)

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.


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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 :

POFPOS

No change Change

No change Type IV Type IChange Type III Type II


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Introduction to DMO It is better to go for DMO, whenever the problem is time

dependent.

Advantages in using DMO:

1. By relating time with generation number, number of variables reduce i.e. the dimension of problem reduces.

2. Whenever problem changes, the new problem adopts the old solution, which helps in faster convergence.

3. Results for all the problems can be found in one run.


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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.


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Test problems in DMOFDA 1:

Type I


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Type III


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Type II


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Type I


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Type II


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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.


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Elitism removed

Modified NSGA-II algorithm-I


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Elitism introduced

interactively

Modified NSGA-II algorithm-II


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FDA 2 simulation


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FDA 3 simulation


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FDA 5 simulation


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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)


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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:


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Objective functionsEconomy:

Emission:


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ConstraintsPower balance constraints:

Water availability constraints:


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Constraint handlingStep 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,


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Constraint handlingStep 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.


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Constraint handlingStep 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


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NSGA-II application on hydrothermal scheduling problem

Population size = 240Number of generations = 1000Crossover probability = 0.9Mutation probability = 0.04Distribution index for crossover = 20Distribution index for mutation = 50

Input parameters :


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Analysis of obtained results

Ph1 vs. F1 Ph2 vs. F1


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Ps1 vs. F1 Ps2 vs. F1


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Ps3 vs. F1 Ps4 vs. F1


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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:

1. Term time should be removed as a variable.

2. All time variable parameters should be directly related with generation number.

The dimension of problem has reduced from 24 to 6.


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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:

1. Introducing new solutions at change by generating random solutions.

2. Introducing new solutions at change by mutating old solutions .


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Introducingnew

randomsolutions

atchange

First modification


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Introducingnew

mutatedsolutions

atchange

Second modification


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Interpolation


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Results and discussion

Comparison among three modified algorithms :

4 timeslots

8 timeslots


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16 timeslots

48 timeslots


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96 timeslots

192 timeslots


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4 timeslots

8 timeslots

Percentage of random new solutions verses performance index :


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16 timeslots

48 timeslots



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96 timeslots

192 timeslots



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4 timeslots

8 timeslots

Percentage of mutated new solutions verses performance index :


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16 timeslots

48 timeslots



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96 timeslots

192 timeslots



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Conclusions

1. Static to dynamic conversion of the problem, increases its convergence rate and simultaneously there also exists a possibility for dimensionality reduction.

2. Modified NSGA-II algorithms, has yielded better results for all test problems.

3. The reformulated hydrothermal scheduling problem has been solved efficiently.

4. The static analysis of hydrothermal scheduling problem with modified NSGA-II produced better results compared to previous works.


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Conclusions

5. Best results are produced when the hydrothermal scheduling problem is formulated into DMO problem, with considerable reduction in computational time over static problems.

6. 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.


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Future scope of work

1. Generalization of the proposed algorithm, would make it user friendly.

2. The hydrothermal scheduling problem defined for individual hydro units, can be extended for cascaded hydro units.

3. 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.


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Back up slides


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Parameter Analysis


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Nomenclature


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Input parameters


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Input parameters


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Pareto front Non-dominated front is Pareto-optimal front.

Trade-off of optimal solutions on F1 vs F2 plot.

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Multi-Objective Dynamic Optimization using Evolutionary Algorithms