GAtoolbox: a Matlab-based Genetic Algorithm …GAtoolbox: a Matlab-based Genetic Algorithm Toolbox for Function Optimization Code: 27.001 Justo José Roberts1,3, Agnelo Marotta Cassula2,

GAtoolbox: a Matlab-based Genetic

Algorithm Toolbox for Function

Optimization

Code: 27.001

Justo José Roberts1,3, Agnelo Marotta Cassula2,

José Luz Silveira, Pedro Osvaldo Prado3,

José Celso Freire Junior2

1IPBEN-UNESP Guaratinguetá, Brazil 2São Paulo State University (UNESP), Brazil 3National University of Mar del Plata (UNMdP), Argentina

Structure

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I. Introduction and Objectives

II. Theoretical Background

III. Description of GAtoolbox

IV. Implementation

V. Conclusion and Future Work

I. Introduction

• It has become very popular in engineering activities, primarily

because of the availability and affordability of high speed

computers.

12/11/2017 3

• Optimization is the “determination of values for design variables

which minimize (maximize) the objective, while satisfying all

constraints”.

• Determine how desirable or undesirable the performance of a

system or a design is every day engineers decision.

• Genetic Algorithms (GA) constitutes an important topic in

Engineering Optimization undergraduate/postgraduate course.

I. Objectives

• Describe the main features of the developed

GAtoolbox.

• Present the results of its implementation in a

practical study case.

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Classification of optimization algorithms

Algorithms

Deterministic

Linear programming

Non-linear programming

Gradient-based

Gradient-free

...

Stochastic Heuristic

Metaheuristic

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Genetic Algorithm


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Genetic Algorithms (GA)

• Probabilistic algorithms

Evolutionary algorithms

population based naturally

inspired Metaheuristic.

• Conceived in 1960 by John Holland

species adaptation and natural

selection computers.

• Later, in the 1980s David E.

Goldberg first success in

industrial application with Gas.

Chromosome

DNA

Gene

Start

Generate first

populationEvaluate fitness

Are optimization

criteria met?Best individuals

Translate

End

Selection

Recombination

Mutation

Generate first

population


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Simple Genetic Algorithm

No

Yes


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Four main

modules Subroutines

1

2

3

4

· Linear ranking

· Non-linear ranking

· Roulette wheel selection

· Stochastic universal sampling

· Tournament selection

· Normalized geometric selection

· Discrete recombination

· Single-point crossover

· Double-point crossover

· Multi-point crossover

· Shuffle crossover

· Blend recombination (BLX-α)

· Line recombination

· Arithmetic recombination

· Flat recombination (BLX-0.0)

· Intermediate recombination

· Binary mutation

· Breeder mutation

· Boundary mutation

· Uniform mutation

· Multi-uniform mutation

· Non-uniform mutation

· Multi-non-uniform mutation

· Maximum time and generations

· Hitting a bound

· Running mean

· Standard deviation

· Best-worst

· Phi

· Uniform reinsertion

· Worst and fitness-based reinsertion

Parameter

definition

END

START

· Real decimal

· Integer decimal

· Binary

Variables setting

· Number of variables

· Population size

Geration of initial population

Fitness assignment

Selection

Recombination

Mutation

Satisfy optimization

criteria?

Evolution

Reinsertion

mf x

0, 1,...,jg x j J + K

, 1,...,l u

i i ix x x i n

1 2, ,...,T

nx x x x S

m 1,...,MMinimize

subjected to

Optimization problem definition

Problem Definition Module


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1

( )mf x 1,2,...,m M

( ) 0jg x 1,2,...,j J

( ) 0kh x 1,2,...,k K

i i ixLB x xUB 1,2,...,i n

x S

Subjected to

Minimize Mono-objective

Multi-objective

Unconstrained

Constrained



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1

min ( )f xMono-objective

Multi-objective 1 2min ( ) ( ), ( ),..., ( )m Mf x f x f x f x

A

F

E

D

Non-dominated solution

Dominated solutions

Pareto FrontRegion with no

feasible

solutions

1

4

2

3

B

C

f1

f2

*

* *

Dominace :

Pareto Optimal : |

Pareto Front :

x x

x S x x

PF x S

x S

NSGA-II

Nondominated Sorting Genetic Algorithm II



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1

( ) 0jg x 1,2,...,j J

( ) 0kh x 1,2,...,k K

i i ixLB x xUB 1,2,...,i n

Constrained problem

Approach adopted: dominance-based penalty-free

0

j

m

worst j

j

f x if g x

x

f g x otherwise

Method of Deb

Niching strategy based on Euclidean

distance

Method of Coello Coello and Montes

Variable number of competitors and not

deterministic

Variable Setting Module


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2

Genotype space = {0,1} Phenotype space

Encoding

(representation)

Decoding

(inverse representation) 01110101

01000100

10010010

10010001

Representation critical decision in any application, namely that of

deciding how best to represent a candidate solution of the algorithm.

Generally accepted that it is better to encode numerical variables directly as:

• Integers

• Floating point variables

Variable Setting Module


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2

Variable representations supported by GAtoolbox

GA works on Variable representation Need conversion?

binary real decimal Yes

binary integer decimal Yes

real decimal real decimal No

integer decimal integer decimal No

Integer representation truncation method 0,5

0,5

i i

i i

i

x se x

x x se r

x se r

Generation of Initial Population Module


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3

First step in the GA is to create the initial population of individuals

Generate a required number of individuals uniformly distributed in the

desire range

11 12

1000 1100

0100 0011

1010 1011

x x

pop

Binary coding

11 12

2.33 23.6

1.35 0.55

10.1 1.23

x x

pop

Real representation

Evolution Module


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4

Fitness assignment

Selection

Recombination

Mutation

Satisfy optimization

criteria?

Reinsertion

Evolution Module


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4

Linear ranking

Non-linear ranking higher selective pressure

Fitness

assignment

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Evolution Module


18

4

Roulette wheel selection Stochastic universal sampling

Normalized geometric selection

Spin

Spin

Tournament method Ranking Fitness Chromosome

1 2,1 Q

2 5 A

3 7,5 Z

4 10 W

5 11,2 S

6 20 R

7 2,3 F

8 24 I

11

1 1

irank x

popSize

q qP rank

q

Selection

Fitness Value Chromosome

1 Q

2 A A

3 Z

4 W A

5 S

6 R R

7 F

8 I I

Pick the best

parent Pick K at

random

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Evolution Module


19

4

Recombination

Operator Variable representation

Discrete recombination All representations

Single-point crossover Binary and decimal integer

Double-point crossover Binary and decimal integer

Multi-point crossover Binary and decimal integer

Shuffle crossover Binary and decimal integer

Blend recombination (BLX-α) Decimal real

Line recombination Decimal real

Arithmetic recombination Decimal real

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Evolution Module


20

4

Single-point crossover Binary and decimal integer

Double-point crossover Binary and decimal integer

Multi-point crossover Binary and decimal integer

Shuffle crossover Binary and decimal integer

Multi-point crossover Single-point crossover

Recombination

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Evolution Module


21

4

Discrete recombination All representations

Line recombination Decimal real

Arithmetic recombination Decimal real

Blend recombination (BLX-α) Decimal real

Recombination

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Evolution Module


22

4

Mutation

Operator Variable representation

Binary mutation Binary

Breeder GA Decimal real and integer

Boundary mutation Decimal real and integer

Uniform mutation Decimal real and integer

Multi-uniform mutation Decimal real and integer

Non-uniform mutation Decimal real and integer

Multi-non-uniform mutation Decimal real and integer

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Evolution Module


23

4

Breeder GA

Uniform mutation

Non-uniform mutation

Mutation

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Evolution Module


24

4

Termination

criteria

Maximum time or

Nº of generations

Hitting bound

Running mean

Standard deviation

Best-worst

Phi

max 0

0

t t

maxGen nGen

* lim

obj objF t F

1

*1

last

last

t*

obj objti t t

F t F i

2

1 1

1 1

pop pop

pop pop

N N

TC obj objN Ni i

F i F i

* *max 1 obj obj popF F i i N

*

1

1

1 1

pop

pop

obj

N

objN

i

F

F i

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Evolution Module


25

4

Reinsertion

Pure reinsertion offspring (λ) = parents (μ)

Uniform reinsertion offspring (λ) < parents (μ)

Elitist reinsertion offspring (λ) < parents (μ)

Worst and fitness-based offspring (λ) = parents (μ)

λbest ... ... ... ... λworst μbest ... ... ... ... μworst

λ∙genGap λ∙(1-genGap) μ∙genGap μ∙(1-genGap)

λbest ... ... μbest ... ...

Parents Offspring

New population

Old population

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Parallel Computing


26

• To reduce the computational time in the

execution of the optimization algorithm,

GAtoolbox allows the possibility of

parallel computation.

• Parallel Computing Toolbox (PCT)

• Run as many as eight Matlab workers on

your local machine in addition to your

Matlab client session.

parfor (parallel for-loops)

parfor

local workers

IV. Implementation

Step 1: setting up the optimization model

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% OPTIMIZATION MODEL

GaOptions = ga_options();

GaOptions.popSize = 50;

GaOptions.maxGen = 100;

GaOptions.numObj = 1;

GaOptions.numVar = 2;

GaOptions.numCons = 2;

GaOptions.xLB = [0,0];

GaOptions.xUB = [6,6];

GaOptions.varType = [2];

GaOptions.objFun = @test_fun305c;

% COMPONENTS' NAME

GaOptions.nameObj = {'Obj_1'};

GaOptions.nameVar = {'x_1','x_2'};

% SELECTION

GaOptions.selFun = 'coelloSelec';

GaOptions.coelloSr = 0.7;

% GENETIC ALGORITHM OPERATORS

GaOptions.recombiFun = 'blxRecombin';

GaOptions.BLXalpha = 0.5;

GaOptions.pXover = 0.9;

GaOptions.mutateFun = 'nonUniformMutate';

GaOptions.shapeB = 5;

GaOptions.pMut = 0.2;

% PARALLEL COMPUTING

GaOptions.useParallel = 'no';

% PLOT RESULTS

GaOptions.bestFitMeanPlot = 1;

% CALL GA FUNCTION

[xBest,bestObjValue,timeRun,GaOptions,Result]

= ga_constraint_optimization(GaOptions);

Step 2: create the objective function

function [y, cons] = test_fun305c(x)

y = [0];

cons = [0,0];

% OBJECTIVE FUNCTION

% MIN problem objective function

y(1) = (x(1).^2 + x(2) - 11).^2 + (x(1) + x(2).^2 - 7).^2;

% CONSTRAINTS

% Compute the constraint violation

c = 4.84 - (x(1) - 0.05).^2 - (x(2) - 2.5).^2;

cons(1) = (c<0)*abs(c);

c = x(1).^2 + (x(2) - 2.5).^2 - 4.84;

cons(2) = (c<0)*abs(c);

Step 3: Results

#Generation 50 / 50

currentGen 50

evaluateCount 2500

totalTime 588

Nwt Twt Npv Tpv Ndg NPVc

0 2 1023 4 3 7,99E+05

0 3 1020 4 3 8,00E+05

0 2 1017 4 3 8,00E+05

0 2 1016 4 3 8,01E+05

0 2 1040 4 3 8,01E+05

0 2 1030 4 3 8,03E+05

0 2 1028 4 3 8,03E+05

IV. Implementation – Study Case

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≈

=

CONV

AC DC

PV

WT

BT

DG

LAC

Ewt(kWh)

t(h)

Epv(kWh)

t(h)

SOC(%); Qbt(kWh)

t(h)

ηconv(%)

Pconv/Pconv,r (%)

Fdg(L/h)

Pdg(kW)

ηdg(%)

Pdg(kW)

Eload(kWh)

t(h)

Optimal design of Hybrid Power Systems (HPS)

Stochastic behavior of renewable resources and demand

Non-linear characteristic of some components Complexity

12/11/2017 29

START

Input parameters: component characteristics, renewable resources, load

END

Set of efficient solutions (configurations of HPS)

Compute the energy balance of the

system for one year of operation:

8760

1

i

i

E t

NPVc

LPSP

EMCO2

fren

fexess

SOCmin

Restrictions

Objetive/s

Compute:

Simulation Module Optimization Module (GAtoolbox)

Assess population

Select best individuals

Stopping

criteria met? Yes

No

Create first generation

Generate new population

(recombination, mutation)

Nwt

Npv

Ndg

Nbt

Nconv


12/11/2017 30

Simulation Module

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0

5

10

15

20

25

30

35

40

45

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61

SO

C (

%)

En

erg

y (

kW

h)

Hours (h)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0,0

5,0

10,0

15,0

20,0

25,0

30,0

1 2 3 4 5 6 7 8 9 10 11 12 13

SO

C (

%)

En

erg

ia (

kW

h)

Horas (h)

WT PV DG Load Unmet Load Energy Excess SOC


12/11/2017 31

Input data

Meteorological measurements

Demand load curve

0 50 100 150 200 250 3000

5

10

15

20

25

Pow

er

(kW

)Hour (h)

Weekday Weekend WeekendWeekday

HPS components

PV panel

Rated power 0.11 kWp

Area 1.02 m2

Initial capital (IC) 2900 $/kW

O&M 1% of IC $/year

Lifetime 25 year

Life cycle emissions 0.059 kgCO2e/kWh

Wind turbine

Rated power 7 kW

Tower height 17.5 m

Initial capital (IC) 3669 $/kW

O&M 3% of IC $/year

Lifetime 20 year

Life cycle emissions 0.02 kgCO2e/kWh


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Minimization of Cost and Maximization of Reliability

( , )m CMIN f NPV LPS

Restrictions

LPSPadm [%] 5 %

fren.min [%] 0 %

fexess,adm [%] 100 %

Fdg,adm [L/year] Inf

EMCO2,adm [t/year] Inf

[Nwt,min ; Nwt,max] [0,15]

[Npv,min ; Npv,max] [0,300]

[Nbt,min ; Nbt,max] [0,50]

[Ndg,min ; Ndg,max] [0,3]

[Nconv,min ; Nconv,max] [0,6]

6,677,248 combinations

Net Present Value of costs

Loss of Power Supply Probability


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Optimization Graphical Results ( , )m CMIN f NPV LPS

0 410000 820000 1230000 1640000 20500000

10000

20000

30000

40000

50000

Generation 1

Generation 2

Generation 3

Generation 4

Generation 5

LP

S (

kW

h/y

ear)

NPVc ($)

0,0

1,0

2,0

3,0

4,0

5,0

6,0

LP

SP

(%

)

288000 306000 324000 342000 3600000

500

1000

1500

2000

2500

3000

3500

4000

Generation 10

Generation 30

Generation 50

LP

S (

kW

h/y

ear)

NPVc ($)

0,0

1,0

2,0

3,0

4,0

5,0

6,0

LP

SP

(%

)

Sol 1

Sol 2

Sol 3


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Optimization Numerical Results ( , )m CMIN f NPV LPS

WT

(kW)

PV

(kW)

DG

(kW)

BT

(kWh)

CONV

(kW)

Eexess

(%)

RF

(%)

Hdg

(h/yr)

Fdg

(L/yr)

LCOE

($/kWh)

LPSP

(%)

NPVc

($)

Sol 1 7x7 0 0 16x4 4x5 52.35 100 -- -- 0.2693 4.99 290,773.55

Sol 2 9x7 0 0 23x4 4x5 61.79 100 0 0 0.3080 1.83 343,610.00

Sol 3 7x7 0 2x10 21x4 4x5 51.53 98.22 653 1654.4 0.3160 0 359,081.14

WT

(kW)

PV

(kW)

DG

(kW)

BT

(kWh)

CONV

(kW)

Eexess

(%)

RF

(%)

Hdg

(h/yr)

Fdg

(L/yr)

LCOE

($/kWh)

LPSP

(%)

NPVc

($)

Sol 1 7x7 0 0 16x4 4x5 52.35 100 -- -- 0.2693 4.99 290,773.55

Sol 2 9x7 0 0 23x4 4x5 61.79 100 0 0 0.3080 1.83 343,610.00

Sol 3 7x7 0 2x10 21x4 4x5 51.53 98.22 653 1654.4 0.3160 0 359,081.14

Improve reliability

+ 23% investment

V. Conclusion

• The GAtoolbox is a useful tool to teach the

basics of GA in an undergraduate/postgraduate

optimization course.

• Advantages of implementing the code in Matlab

students become familiar with it aid to

develop their final projects.

12/11/2017 35

• The implementation of the toolbox in open

architecture software such as Matlab meets

the expectations of a flexible tool for

researching purposes as well.

V. Future Work

Other functionalities will be integrated in future

versions of the GAtoolbox:

• Friendly graphical user interphase.

• Other metaheuristics such as Particle Swarm

Optimization, Tabu Search, Differential

Evolution, among others.

• Help documentation.

12/11/2017 36

12/11/2017 37

Thank you!

Justo José Roberts [email protected]

Documents

GAtoolbox: a Matlab-based Genetic Algorithm …GAtoolbox: a Matlab-based Genetic Algorithm Toolbox for Function Optimization Code: 27.001 Justo José Roberts1,3, Agnelo Marotta Cassula2,