Transcript Slide 1

Multi-Objective Dynamic Optimization
using
Evolutionary 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)
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)
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 multiobjective optimization in dynamic environment.
 Classification in DMOs :
POS
POF
No change
Change
No change
Type IV
Type I
Change
Type III
Type II
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
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)
Test problems in DMO
FDA 1:
Type I
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Test problems in DMO
FDA 2:
Type III
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Test problems in DMO
FDA 3:
Type II
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Test problems in DMO
FDA 4:
Type I
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Test problems in DMO
FDA 5:
Type II
Kanpur Genetic Algorithms Laboratory
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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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
Modified NSGA-II algorithm-I
Elitism removed
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25, July 2006 (11:00 AM)
Modified NSGA-II algorithm-II
Elitism introduced
interactively
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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)
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
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)
Objective functions
Economy:
Emission:
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Constraints
Power balance constraints:
Water availability constraints:
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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
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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
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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)
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
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Analysis of obtained results
Ph1 vs. F1
Ph2 vs. F1
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Analysis of obtained results
Ps1 vs. F1
Ps2 vs. F1
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Analysis of obtained results
Ps4 vs. F1
Ps3 vs. F1
Kanpur Genetic Algorithms Laboratory
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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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
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 .
Kanpur Genetic Algorithms Laboratory
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First modification
Introducing
new
random
solutions
at
change
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Second modification
Introducing
new
mutated
solutions
at
change
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Interpolation
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Results and discussion
Comparison among three modified algorithms :
4 timeslots
8 timeslots
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
Results and discussion
Comparison among three modified algorithms :
16 timeslots
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48 timeslots
Results and discussion
Comparison among three modified algorithms :
96 timeslots
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IIT Kanpur
25, July 2006 (11:00 AM)
192 timeslots
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)
Results and discussion
Percentage of random new solutions verses performance index :
16 timeslots
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
48 timeslots
Results and discussion
Percentage of random new solutions verses performance index :
96 timeslots
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
192 timeslots
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)
Results and discussion
Percentage of mutated new solutions verses performance index :
16 timeslots
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
48 timeslots
Results and discussion
Percentage of mutated new solutions verses performance index :
96 timeslots
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
192 timeslots
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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)
Back up slides
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IIT Kanpur
25, July 2006 (11:00 AM)
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.
Kanpur Genetic Algorithms Laboratory
IIT Kanpur
25, July 2006 (11:00 AM)