Efficient multi-objective optimizers by meta-heuristics for power system

control

GHOURAF DJAMEL EDDINE1 , NACERI ABDELLATIF2

Department of Electrical Engineering,

National Polytechnic School

1SCAMRE Laboratory, BP1523 EL M’naour, Oran 31000, ALGERIA

2IRECOM Laboratory, BP 98 22000 ALGERIA

Abstract: - This paper proposes the Meta-heuristics approaches using genetic algorithms (GA) and particle swarm optimization (PSO) for tuning

power system stabilizer PSS parameters. In this work we have proposed a multi-objective function based on two objectives: first maximize the

stability margin by increasing the damping factors and second minimize the eigenvalues real parts. For the effectiveness function proposed

check, we compared it with mono-objective function. The simulation results, by comparative study between genetic algorithms and particle

swarm optimizations techniques via multi objective and mono objective functions proved the efficiency of the PSS adapted by multi-objective

function based genetic algorithms in comparison with particle swarm optimization, it’s enhanced stability of power system works under

different operating modes and different network configurations. The simulation results obtained under developed graphical user interface (GUI)

Keywords- Turbo-Alternator, Genetic Algorithms GA, Particle Swarm Optimization PSO, multi-objective function, mono-objective function,

robustness, graphical interface GUI.

Received: November 23, 2021. Revised: October 11, 2022. Accepted: November 13, 2022. Published: December 5, 2022.

1. Introduction

The electrical energy has become the major form of

energy for end use consumption in today’s world. There is

always a need to make electric energy generation and

transmission, both more economic and reliable. The

voltages throughout the system are also controlled to be

within ±5% of their rated values by automatic voltage

regulators acting on the generator field exciters, and by the

sources of reactive power in the network, [1].

Stability and robustness are considered essential

requirements for friability and continuity of electrical

energy production this latter produced by a series of systems

with very complex mathematical models called power

systems. Since these systems are installed in complex

environmental conditions they are exposed to a variation of

uncertainty which is affected directly in the operation of

these systems and therefore the stability of the energy

production, the power system stabilizer PSS plays an

important role to improve the power systems stability, [2].

The parameters of CPSS are determined based on the

linearized model of the power system. Providing good

damping over a wide operating range, the CPSS parameters

should be fine-tuned in response to both types of

oscillations. Since power systems are highly non-linear

systems, with configurations and parameters which alter

through time, the CPSS design based on the linearized

model of the power system cannot guarantee its

performance in a practical operating environment, [3].

Therefore, an adaptive PSS which considers the nonlinear

nature of the plant and adapts to the changes in the

environment is required for the power system, [3]. In order

to improve the performance of CPSSs, numerous techniques

have been proposed for designing them, such as intelligent

optimization methods and fuzzy logic method [7, 8].

Meta-heuristic techniques are a new family of

stochastic algorithms which aim to solve difficult

optimization problems. Used to solve various

applicative problems, these methods have the advantage to

be generally efficient on a large number of

problems.GA and PSO belong to population approaches.

Meta-heuristics are generally used to solve a simplified

OPF (Optimal Power Flow) problem such as the classic

economic dispatch, security - constrained economic

power dispatch, and reactive optimization problem, as

well as optimal reconfiguration of an electric

distribution network. [4],[6].

Genetic algorithms (GAs) were invented by John

Holland in the 1960s and were developed by Holland

and his students and colleagues at the University of

Michigan in the 1960s and the 1970s. In contrast with

evolution strategies and evolutionary programming,

Holland's original goal was not to design algorithms to

solve specific problems, but rather to formally study the

phenomenon of adaptation as it occurs in nature and to

develop ways in which the mechanisms of natural

adaptation might be imported into computer systems, [5].

The Particle Swarm Optimization (PSO) strategy is

a new class of algorithms proposed to solve continuous

optimization problems . The Particle Swarm Optimizer was

introduced by James Kennedy and Russell Eberhart in 1995.

Inspired by social behavior and movement dynamics of

insects, birds and fish, it is also related, however, to

evolutionary computation, and has links to both genetic

algorithms and evolution strategies, [4], [5].

In this paper, the robust PSS design is realized using

multi-objective function optimization GA and PSO applied

in the automatic excitation regulator of powerful

synchronous generators

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2. Power Systems Model

The dynamic performance study and stability analysis of

power systems requires faithful mathematical models, we

used in our work permeances networks modeling based on

the PARK-GARIVE model of powerful synchronous

generators for simplifying hypotheses and testing the

control algorithm. The PSG model defined by the following

equations [2, 15]:

Figure 1 Standard system IEEE type SMIB with excitation control

of powerful synchronous generators

 Currents equations

qsr

aqq

qsr

aqq

srd

add

adf





























 Voltage equations

qqddq

ddqqd

riEiXU





sfqad

fqsfq

sfdsfad

fdsfd

fsf

XXX

E11

111











 Flow equations:

fff

qsqdaqdsdqad

iRU

iXXEiXXE

""""

)( )(























 Mechanical equations

 

eTjTdaqqadjMMs

TMIIs

T -ou .. 

 Automatic Voltage Regulator model (AVR)

, FrefE

REA

RVVV

VVK

V





 Power system stabilizer model (PSS)

1input

KV PSSPSS 







3. Meta-Heuristics

The new paradigms were called meta-heuristics and

were first introduced in the mid-80s as a family of searching

algorithms able to approach and solve complex optimization

problems, using a set of several general heuristics. The term

meta-heuristic was proposed in [16], to define a high level

heuristic used to guide other heuristics for a better evolution

in the search space. Although traditional stochastic search

methods are mainly guided by chance (solutions change

randomly from one step to another), they can be used in

combination with meta-heuristic algorithms to guide the

search process and to accelerate the convergence.

Most meta-heuristics algorithms are only

approximation algorithms, because they cannot always find

the global optimal solution, [9]. But the most attractive

feature of a meta-heuristic is that its application requires no

special knowledge on the optimization problem to be

solved, hence it can be used to define the concept of a

general problem solving model for optimization problems or

other related problems, [17], [18]. Since their introduction in

the mid-80s till now, meta-heuristic methods for solving

optimization problems have been continuously developed,

allowing addressing and solving a growing number of such

problems, previously considered difficult or even

impossible to solve. These methods include simulated

annealing, tabu search, evolutionary computation

techniques, artificial immune systems, genetic algorithms,

particle swarm optimization, ant colony algorithm,

differential evolution, harmony search, honey-bee colony

optimization etc. The next section presents a brief review of

basic issues for the most commonly used meta-heuristics

cited above. Several applications of these methods in the

field of power systems, [10].

In this work we are based on genetic algorithms, particle

swarm optimization techniques.

III.1.Genetic algorithms

Genetic Algorithm (GA) is a search technique that mimics

the mechanisms of natural selection, discovered by John

Holland in 1970, [11], [19].Cell is the building unit of all

living organisms. In each cell there is a set of chromosomes

which are strings of DNA. Every chromosome consists of

genes which encode a particular protein. During

reproduction, crossover first occurs. Genes from parents

form in some way the whole new chromosome. However,

the new created offspring can be mutated. Mutation occurs

when the elements of DNA are a bit changed.

These changes are mainly caused by errors in copying

genes from parents. The fitness of an organism is measured

AVR

PSS

Uter

Exciter

Δω

Uref

Infinity bus

(1)

(2)

(3)

(4)

(5)

(6)

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by success of the organism in its life. With generations, the

good characteristics remain and the bad ones died which

represents “The survival of the fittest”.

Much work have been done on optimization by genetic

algorithms to tune power system stabilizer parameters for

adaptation and reliability of these techniques to power

systems.

III.2. Particle swarm optimization

Particle swarm optimization is a population based

stochastic optimization method, [12].Explores for the

optimal solution from a population swarm of moving

particle vectors, based on a fitness function. Each ith

particle vector represents a potential answer and has a

position (Xik) and a velocity (Vik) at the kth iteration in the

problem space. Each ith vector keeps a record of its

individual best position (Pik), which is associated with its

own best fitness it has achieved so far, at any kth step in the

iteration process. This value is known as pbesti. Moreover,

the optimum position among all the particles obtained so far

in the swarm is stored as the global best position (Pgk). This

location is called gbest. The new velocity of particle will be

updated according to the following equation, [13]:

   

lXPrcXPrcwvv



2211

where w is an inertia weight in the first part that

represents the memory of a particle during a search, c1 and

c2 are positive numbers illustrating the weights of the

acceleration terms that guide each particle toward the

individual best and the swarm best positions respectively, r1

and r2 are uniformly distributed random numbers in (0, 1),

and N is the number of particles in the swarm. The second

and the third parts of (8) represent cognitive and social parts

respectively. The inertia weighting function in (7) is usually

calculated using the following equation:

max

minmaxmax )(

iter

iterWWW

W



Where wmax and wmin are the maximum and minimum

values of w respectively, itermax is the maximum number of

iterations and iter is the current iteration number. The first

term in (7) enables each particle to perform a global search

by exploring a new search space. The last two terms in (7)

enable each particle to perform a local search around its

individual best position and the swarm best position. Each

particle changes its position based on the updated velocity

according to the following equation:

11   k

iVXX

III.3.The difference between GA and PSO

The PSO algorithm shares many common points with

the genetic algorithm (GA). Both algorithms start with a

population of individuals randomly generated; all both have

objective function values for evaluating the population.

Both algorithms start with the population and seek optimum

random techniques. The two systems do not guarantee

success. They also have the memory, which is important for

the algorithm. Such as genetic algorithms, PSO is based on

populations that slowly converge to one or more solutions.

However, with PSO, the particles are preserved throughout

the entire process; they do not die. Contrary to the genetic

algorithm, this is based on competition for the best chance

of survival and reproduction. PSO uses a type of

cooperation between the molecules; this is realized by

exchanging the coordinates of the best solutions which have

been produced up to this point. PSO traditionally has no

crossover between individuals, and has no mutation and the

particles are never replaced by other individuals during

execution. Instead of that PSO refines its research by

attracting the particles [14, 20].

Table 1 gives us the difference between GA and PSO, [21].

Table 1 a comparative between GA and PSO

I. TUNING POWER SYSTEM STABILIZER PARAMETERS

BASED GA AND PSO .

IV.1 Objectives functions

The objective functions choice based on the needs of our

controlled system, [21].

To study of the influence choice of objective functions

in the controlled system performances we have realized a

comparative study between two objective functions:

 Mono objective function.

 Multi objective function

IV.1.1 Mono-objective function.

The aim of using PSS is to ensure a satisfactory damping

of the oscillations and to guarantee the overall stability of

the system for different operating points.

To meet this goal, we have used for the first time a mono

objective function to minimize the real parts of the system

eigenvalues. Therefore, all eigenvalues will be in D area of

stability (figure 2)

(9)

(8)

(7)

PSO

Base

Nature

Principle

Algorithm

Invidious

Chromosome

Bird, insect ...

selection

Utilizable

No utilizable

crossing

Utilizable

No utilizable

mutation

Utilizable

No utilizable

Number of

individuals

generated each

iteration (example

30 individuals in

a population)

60 individuals

(30 individuals

of crossing and

30 individuals

of mutation)

individuals

Excursion Time

Court

Average

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Figure 2 Stability areas.

To understand this notion, we consider two systems with

same imaginary parts ωs1 = ωs2 and the deferent real part σ:

 System 1 : P1, 2=-6±6j

 System 2 : P1, 2=-1±6j

Figure 3 The σ influence in the controlled system stability

In this result we can see that the decrease of the real part

improved the dynamic performance and system stability.

Depending on this notion we proposed the flowing mono

objective function which must minimize the real parts of the

eigenvalues system.

Fobj= min(σ)

IV.1.2.Optimization results

To optimize and study of power system we created a

graphical user interface GUI (figure 4) under MATLAB

allows to:

 Optimize controller parameters using genetic

algorithms and particle swarm optimization by

mono and multi objective.

 View system regulation results and simulation.

 Calculate the system dynamics parameters.

 Test system stability and robustness.

A. GA optimization method

To run optimization by genetic algorithms under GUI

we use: optimization /GA /PSS/ mono objective

Figure 4 PSS parameters syntheses using GA mono objective

under GUI MATLAB

The below optimization result for: 10 generations and 10

individuals obtained using realized graphical interface.

_________________________________________________________________________________________________

********* Creating the initial population ********

_________________________________________________________________________________________________

********* 1st step coding and initialization ********

_________________________________________________________________________________________________

N ind K1 K2 T1 T2 Segma

_________________________________________________________________________________________________

Individu:01 +05.3255 +03.4588 0.0118 0.0726 -0.9124

Individu:02 +05.3255 +05.3529 0.0785 0.0216 -0.9127

Individu:03 +02.3059 +00.5216 0.0313 0.0397 -0.9197

Individu:04 +06.6431 +02.7176 0.0633 0.0138 -0.9099

Individu:05 +06.4784 +00.6039 0.0064 0.0698 -0.9101

Individu:06 +00.2471 +02.9373 0.0169 0.0028 -0.5860

Individu:07 +00.4392 +05.4902 0.0559 0.0616 -0.5478

Individu:08 +03.6235 +02.0588 0.0528 0.0236 -0.9167

Individu:09 +06.8078 +01.0431 0.0906 0.0040 -0.9089

Individu:10 +05.4902 +04.5843 0.0988 0.0514 -0.9112

********* 2nd step selection ********

_________________________________________________________________________________________________

N ind K1 K2 T1 T2 Segma

_________________________________________________________________________________________________

Individu:01 +05.3255 +05.3529 0.0785 0.0216 -00.9127

Individu:02 +02.3059 +00.5216 0.0313 0.0397 -00.9197

Individu:03 +02.3059 +00.5216 0.0313 0.0397 -00.9197

Individu:04 +06.4784 +00.6039 0.0064 0.0698 -00.9101

Individu:05 +06.4784 +00.6039 0.0064 0.0698 -00.9101

Individu:06 +00.2471 +02.9373 0.0169 0.0028 -00.5860

Individu:07 +03.6235 +02.0588 0.0528 0.0236 -00.9167

Individu:08 +03.6235 +02.0588 0.0528 0.0236 -00.9167

Individu:09 +05.4902 +04.5843 0.0988 0.0514 -00.9112

Individu:10 +02.3059 +00.5216 0.0313 0.0397 -00.9197

********* 3rd step Crossing ********

_________________________________________________________________________________________________

N ind K1 K2 T1 T2 Segma

_________________________________________________________________________________________________

Individu:01 +05.8745 +05.7922 0.0801 0.0154 -00.9116

Individu:02 +01.7569 +00.0824 0.0298 0.0459 -00.9209

Individu:03 +02.9647 +00.6314 0.0313 0.0444 -00.9182

Individu:04 +05.8196 +00.4941 0.0064 0.0651 -00.9118

Individu:05 +05.4902 +01.1529 0.0173 0.0526 -00.9126

Individu:06 +01.2353 +02.3882 0.0060 0.0201 -00.9966

Individu:07 +03.6235 +02.0588 0.0528 0.0236 -00.9167

Individu:08 +03.6235 +02.0588 0.0528 0.0236 -00.9167

Individu:09 +05.8196 +04.0353 0.0801 0.0655 -00.9102

Individu:10 +01.9765 +01.0706 0.0501 0.0256 -00.9203

********* 4st Step Mutation ********

_________________________________________________________________________________________________

N ind K1 K2 T1 T2 Segma

_________________________________________________________________________________________________

Individu:01 +04.0078 +01.8392 0.0902 0.0044 -00.9157

Individu:02 +01.7569 +03.6784 0.0793 0.0271 -00.9282

Individu:03 +00.3294 +00.4941 0.0095 0.0444 -00.5863

Individu:04 +02.3059 +00.7137 0.0048 0.0710 -00.9196

Individu:05 +03.8980 +01.5373 0.0485 0.0655 -00.9155

Individu:06 +05.2431 +02.3608 0.0294 0.0099 -00.9138

Individu:07 +05.7647 +02.6078 0.0520 0.0691 -00.9107

Individu:08 +03.2941 +03.6784 0.0863 0.0150 -1.41140

Individu:09 +00.7412 +04.0902 0.0563 0.0628 -00.6215

Individu:10 +05.9294 +02.8275 0.0926 0.0256 -00.9107

********* Optimization Results ********

_________________________________________________________________________________________________

N Pop K1 K2 T1 T2 Segma

_________________________________________________________________________________________________

Population:01 +03.2941 +03.6784 +00.0863 0.0150 -1.4114

Population:02 +06.5333 +06.6431 +00.0856 0.0330 -2.0971

Population:03 +06.5333 +06.6431 +00.0856 0.0330 -2.0971

Population:04 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Population:05 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Population:06 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Population:07 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Population:08 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Population:09 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Population:10 +06.5882 +06.7529 +00.0294 0.0177 -3.2822

Optimization is completed .......

The optimized parameters K1= +06.5882 K2= +06.7529 T1=+00.0294 T2= 0.0177 Segma= -3.2822

D area

σcr

Imaginary

Real

(10)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Step Res ponse

Time (sec )

Amplitude

System 2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Step Res ponse

Time (sec )

Amplitude

System 1

-8 -7 -6 -5 -4 -3 -2 -1 0

-8

-6

-4

-2

0.080.170.280.380.50.64

0.8

0.94

0.080.170.280.380.50.64

0.8

0.94

Pole-Zero Map

Real Axis

Imaginary Axis

Systeme 1

Systeme 2

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0 1 2 3 4 5 6 7 8

-1.5

-1

-0.5

0.5

1.5

2x 10-3 GLISSEMENT

temps en sec

glissement

PSS-GA-MONOOBJ

PSS-PSO-MONOOBJ

0 1 2 3 4 5 6 7 8

1.32

1.34

1.36

1.38

1.4

1.42

1.44

DELTA

temps en sec

delta

PSS-GA-MONOOBJ

PSS-PSO-MONOOBJ

B. PSO optimization method

To run particle swarm optimization under GUI we use:

optimization /PSO /PSS/ mono objective

Figure 5 PSS parameters syntheses using PSO mono objective under GUI

MATLAB

__________________________________________________________________________________________

********* PSO initialization ********

__________________________________________________________________________________________

N ind K1 K2 T1 T2 Segma

__________________________________________________________________________________________

Individu:01 +02.2669 +03.7478 0.0196 0.0735 -1.2297

Individu:02 +01.8199 +03.4169 0.0887 0.0380 -0.8399

Individu:03 +03.5118 +01.0559 0.0077 0.0258 -0.9175

Individu:04 +00.9394 +02.0283 0.0995 0.0431 -0.6079

Individu:05 +00.9999 +04.9862 0.0136 0.0070 -0.9263

Individu:06 +01.3419 +01.7628 0.0087 0.0457 -1.0159

Individu:07 +05.4810 +01.0900 0.0381 0.0693 -0.9117

Individu:08 +01.9461 +06.7947 0.0322 0.0798 -1.0203

Individu:09 +03.0305 +01.3165 0.0950 0.0563 -0.9171

Individu:10 +05.9236 +04.6694 0.0731 0.0946 -0.9093

__________________________________________________________________________________________

***** PSO algorithm *****

__________________________________________________________________________________________

N Pop K1 K2 T1 T2 Segma

__________________________________________________________________________________________

Iteration:01 +02.2669 +03.7478 0.0196 0.0735 -1.2297

Iteration:02 +01.9678 +03.6840 0.0197 0.0479 -1.2695

Iteration:03 +02.8359 +05.3230 0.0241 0.0581 -1.5411

Iteration:04 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Iteration:05 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Iteration:06 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Iteration:07 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Iteration:08 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Iteration:09 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Iteration:10 +04.2027 +04.9842 0.0396 0.0528 -1.9376

Optimization is completed .......

The optimized parameters K1= +04.2027 K2= +04.9842 T1=+00.0396 T2= 0.0528 Segma= -1.9376

Figure 6 Optimization result of GA and PSO using mono

objective function

The optimization results obtained show that the GA (σ =

-3.2822) more reliable compared to PSO (σ = -1.9376)

IV.1.3.Simulation results

For SMIB system stability study we have performed

perturbation in turbine torque (ΔTm =15% at 0.5 second)

We simulated SMIB system under

 Different operations regimes: under-excited, the

nominal and the over-excited.

 Different electrical network: long, court and

average

 Different synchronous generators: TBB 200,

500, 1000 and BBC720.

We optimized the controller parameters by GA and PSO

under different conditions cited above.

The following results were obtained by SMIB studied

with following cases: closed loop System with PSS_GA_

mono objective and PSS_PSO_ mono objective

Figures 7 and 8 show simulation results of power system

under critical regime (under excited and long transmission

line network)

Figure 7 Variable speed

Figure 8 Internal angle

From the obtained results it can be seen that:

 The parameters optimization of power system stabilizer

PSS using mono objective GA and PSO gives the SMIB

system a considerable improvements in the stability and

dynamics performances

 Concerning the optimization method, the GA is well

adapted with the system SMIB compared to the PSO.

1 2 3 4 5 6 7 8 9 10

-3.5

-3

-2.5

-2

-1.5

-1

itération

segma

Convergence vers la solution optimale méthode GA et PSO

PSO

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IV.2.1 Multi objective function.

The system SMIB is stable based on mono objective

function, but it contains a disadvantage especially if the two

factors σ and damping coefficient ζ are minimal

simultaneously. The dynamic behavior of such a system

depends on two values: σ and especially the damping

coefficient ζ. To study the influence of damping coefficient

ζ on the controlled system we consider two systems with

real part σs1= σs2 and ωs1 ≠ ωs2 (ω imaginary part):

System 21: P21, 2=-2±j with ζ = 0.8944

System 22: P22, 2=-2±8j with ζ = 0.2425

The systems poles and step responses match each system

shown in figure 9.

Figure 9 the ζ influence to controlled system

From the obtained results, it can be seen that:

The increase of the damping coefficient ζ improves

system stability. Based on these results we propose a new

objective function composed by two functions. This

function must maximize stability margin by increasing the

damping factors while minimizing the real parts of the

system eigenvalues, and second function must maximize the

set of two objective functions.

max(ζ)-min(σ)

FMult_obj=max (max(ζ)-min(σ))

IV.2.2.Optimization results

A. GA optimization method

To run GA multi objective optimization under GUI we

use: optimization /GA /PSS/ multiobjective

Figure 10 PSS parameters syntheses using GA multi objective

under GUI MATLAB

Optimization example using GA technique with Number

of individuals=10 , Number of population =10

__________________________________________________________________________________________

********* Creating the initial population ********

__________________________________________________________________________________________

********* 1st step coding and initialization ********

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N ind K1 K2 T1 T2 Sigma ksi multi-obj

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Individu:01 +02.2588 +10.5882 0.0106 0.0843 -1.3181 +0.1054 +1.4234

Individu:02 +10.9647 +09.2706 0.0329 0.0459 -0.8998 +0.9959 +1.8957

Individu:03 +00.2824 +09.4118 0.0622 0.0890 -0.4643 +0.0381 +0.5023

Individu:04 +02.0706 +08.0941 0.0473 0.0432 -1.1622 +0.0932 +1.2554

Individu:05 +09.1765 +07.8118 0.0711 0.0659 -0.9022 +0.9960 +1.8982

Individu:06 +05.9765 +11.4353 0.0602 0.0792 -1.4711 +0.1082 +1.5794

Individu:07 +00.5647 +03.6706 0.0294 0.0702 -0.6460 +0.0535 +0.6995

Individu:08 +02.6353 +02.4471 0.0906 0.0154 -0.9187 +0.9989 +1.9176

Individu:09 +03.8118 +02.5412 0.0501 0.0095 -0.9165 +0.9964 +1.9129

Individu:10 +05.9765 +11.1529 0.0828 0.0706 -1.2001 +0.0891 +1.2892

********* 2nd step selection ********

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N ind K1 K2 T1 T2 Sigma ksi multi-obj

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Individu:01 +10.9647 +09.2706 0.0329 0.0459 -00.8998 +0.9959 +1.8957

Individu:02 +10.9647 +09.2706 0.0329 0.0459 -00.8998 +0.9959 +1.8957

Individu:03 +02.0706 +08.0941 0.0473 0.0432 -01.1622 +0.0932 +1.2554

Individu:04 +09.1765 +07.8118 0.0711 0.0659 -00.9022 +0.9960 +1.8982

Individu:05 +09.1765 +07.8118 0.0711 0.0659 -00.9022 +0.9960 +1.8982

Individu:06 +05.9765 +11.4353 0.0602 0.0792 -01.4711 +0.1082 +1.5794

Individu:07 +02.6353 +02.4471 0.0906 0.0154 -00.9187 +0.9989 +1.9176

Individu:08 +02.6353 +02.4471 0.0906 0.0154 -00.9187 +0.9989 +1.9176

Individu:09 +03.8118 +02.5412 0.0501 0.0095 -00.9165 +0.9964 +1.9129

Individu:10 +02.6353 +02.4471 0.0906 0.0154 -00.9187 +0.9989 +1.9176

********* 3rd step Crossing ********

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croissement state

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Pc = 0.267 0 1 1 1 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 1 0 0 -----> Pc < PC: There is a crossing

Pc = 0.521 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 1 1 -----> Pc < PC: There is a crossing

Pc = 0.521 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 -----> Pc < PC: There is a crossing

Pc = 0.766 0 0 1 1 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 -----> Pc > PC: no crossing ……..

Pc = 0.766 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 -----> Pc > PC: no crossing….......

Pc = 0.571 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 0 -----> Pc < PC: There is a crossing

Pc = 0.765 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 -----> Pc > PC: no crossing ……..

Pc = 0.765 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 1 -----> Pc > PC: no crossing ……____..